May 25, 2026
We prove the soliton resolution conjecture for the Benjamin-Ono (BO) equation with an explicit error bound in the \(L^\infty\)-norm. For the finite-order multisoliton case, the explicit \(L^\infty\)-norm errors are bounded by \(\mathcal{O}(|t|^{-\frac{1}{4}(1-\frac{1}{2s})})\) with initial data \(u_0 \in H^{s,\alpha}(\mathbb{R})\) for any \(s>1/2\) and \(\alpha \geqslant 1\). For the infinite-order multisoliton case, the explicit \(L^\infty\)-norm errors are bounded by \(\mathcal{O}(|t|^{-1/3})\) when \(u_0\) is expressed as an infinite sum of soliton profiles. Recently, Gassot, Gérard, and Miller (arXiv:2601.10488, 2026) proved an implicit error bound in \(H^1\)-norm of the soliton resolution in the finite-order multisoliton case with \(u_0 \in H^{1,1}\left( \mathbb{R} \right)\), requiring extra condition \(x^2u_0(x) = c_0 + v_0(x), c_0\in \mathbb{R}, v_0(x) \in L^2(\mathbb{R})\). In the infinite-order multisoliton case, Gassot and Gérard (arXiv:2603.15419, 2026) proved an implicit error bound in \(L^\infty\)-norm for the soliton resolution when \(u_0\) is expressed as an infinite sum of soliton profiles. Notably, they highlighted the inverse spectral problem for the Lax operators as an interesting open problem. In order to address the soliton resolution with the explicit error in \(L^\infty\)-norm for finite/infinite-order multisoliton, there exist many open problems concerning initial conditions, error accuracy, and other related issues. Solving these open problems is the central objective of our work. In order to enlarge the initial data space and remove the extra conditions, we employ Kato-Rellich theorem to transform the soliton resolution conjecture into an error estimation problem between the sequence and the solution. It is worth noting that we solve the open inverse spectral problem for the Lax operator by constructing a trace-class operator based on the discrete spectrum.
In this work, we consider the Cauchy problem for the Benjamin-Ono (BO) equation \[\label{problem} \left\{ \begin{array}{l} \partial_t u - \partial_x \lvert D_x \rvert u + \partial_x (u^2) = 0, \\ u(0,x) = u_0(x), \end{array} \right. \quad (t,x) \in \mathbb{R}^2,\tag{1}\] where the real-valued function \(u(t,x)\) denotes the potential, \[\left| D_x \right|u\left( x \right) :=\frac{1}{\pi}\mathrm{P}.\mathrm{V}.\int_{\mathbb{R}}^{}{\frac{u^{\prime}\left( y \right)}{x-y}dy},\]and \[\mathrm{P.V.} \int_{\mathbb{R}} \frac{f(y)}{x-y} dy := \lim_{\epsilon \to 0^+} \int_{|x-y| > \epsilon} \frac{f(y)}{x-y} dy.\] Benjamin [1] (see also Davis and Acrivos [2]) derived the BO equation, which describes the weakly nonlinear propagation of internal gravity waves along the interface of a stratified two-layer fluid of great total depth. The potential function \(u\) measures the interface’s elevation relative to equilibrium. For a rigorous and comprehensive derivation of the BO equation, we refer readers to the monograph by Saut and Klein [3].
Investigating the functional spaces in which the BO equation admits a unique solution has been an important problem ever since its derivation. Saut was the first to prove the global well-posedness for the BO equation in the space \(H^s(\mathbb{R})\) with \(s \geqslant 2\) [4]. Following this seminal work, numerous studies investigated the Cauchy problem for the BO equation in lower regularity spaces [3], [5]–[10]. Most notably, Gérard, Kappeler, and Topalov [11] achieved a milestone result for the BO equation on the torus \(\mathbb{T}\). Via the construction of Birkhoff coordinates, they proved sharp global well-posedness for \(s > -1/2\) and ill-posedness at \(s = -1/2\), which additionally revealed remarkable qualitative dynamics. Based on the groundbreaking work [11], as well as the explicit formula for the BO equation derived by Gérard [12], Killip, Laurens, and Vişan [13] established global well-posedness on both the real line and the torus for initial data in \(H^s\) with \(s > -1/2\).
Following the achievement of well-posedness results for the BO equation, the next natural and important topic is to explore the long-time asymptotics of its solutions. Employing the celebrated explicit formula proposed by Gérard [12], Gérard and his collaborators systematically analyzed the long-time dynamics of the BO equation. For example, Blackstone, Gassot, Gérard, and Miller reduced the operator resolvent within the formula into finite-dimensional determinants and addressed the long-time dynamics for the BO equation with rational initial data in detail [14], [15]. Moreover, for a class of initial data for which the number of solitons is infinite, Gassot and Gérard investigated the long-time asymptotic behavior of the BO equation on the real line [16]. Notably, in the investigation of long-time dynamics, the soliton resolution conjecture is a central and fundamental open problem.
The investigation of the soliton resolution conjecture originated from the long-time asymptotic phenomenon of numerical solutions observed by Zabusky and Kruskal in the Korteweg-de Vries (KdV) equation [17]. This phenomenon profoundly reveals the dynamical behavior of nonlinear wave equations, and since then has continuously attracted widespread attention from numerous prominent scholars, including Tao [18] and Kenig [19]. In 2008, Tao [20] pointed out the possibility of establishing the soliton resolution conjecture and explicitly stated:
“More generally, if one starts with arbitrary (but smooth and decaying) initial data, what usually happens (numerically, at least) with evolutions of equations such as \(u_t+u_{xxx}+6uu_x=0\) is that some non-linear (and chaotic-seeming) behaviour happens for a while, but eventually most of the solution radiates away to infinity and a finite number of solitons emerge, moving away from each other at different rates.”
It is widely believed that the soliton resolution conjecture asserts that for sufficiently large times, the solution of a nonlinear dispersive equation can be decomposed into a finite number of solitons and a radiation term. For the vast majority of dispersive equations, this conjecture has long been considered an open problem. In 1993, Deift and Zhou investigated the asymptotic properties of solutions to the modified Korteweg-de Vries equation with decaying initial data [21]. Subsequently, based on the seminal work of Deift and Zhou, the soliton resolution conjecture was rigorously established for a broad class of classical integrable systems, including the celebrated KdV equation and nonlinear Schrödinger equation equation [22]–[30]. The second author Tian and his collaborators have also established results concerning the soliton resolution conjecture and long-time asymptotics for the Wadati-Konno-Ichikawa equation [27], [28], the spin-1 Gross-Pitaevskii equation [29], and the modified Camassa-Holm equation [30].
However, the soliton resolution conjecture for some integrable equations with a nonlocal Hilbert transform remains an open problem. Recently, Gassot, Gérard, and Miller [31] proved the following result in the finite-order multisoliton case:
Proposition 1 (Gassot, Gérard, and Miller [31]). Let \({u}_{0} \in {H}^{1}\left( \mathbb{R}\right)\) be a real-valued function such that \(x{u}_{0} \in {H}^{1}\left( \mathbb{R}\right)\) and \[\label{tiaojian} x^2u_0\left( x \right) =c_0+v_0\left( x \right) ,\enspace c_0\in \mathbb{R} ,v_0\left(x \right) \in L^2(\mathbb{R} ).\qquad{(1)}\] Then there exist a nonnegative integer \(N\) , elements \({p}_{1},\ldots ,{p}_{N}\) of \({\mathbb{C}}_{ + }\) , with \(\operatorname{Im}\left( {p}_{1}\right) < \cdots < \operatorname{Im}{p}_{N}\) , and real-valued functions \({u}_{\infty }^{ \pm }\) in \({H}^{1}\left( \mathbb{R}\right)\) such that \[\left\| r\left( t,\cdot \right) \right\| _{H^1\left( \mathbb{R} \right)}:=\left\| u\left( t,\cdot \right) -\sum_{j=1}^N{R_{p_j}\left( \cdot -\frac{1}{\mathrm{Im}p_j}t \right)}-\mathrm{e}^{t\partial _x\left| D \right|}u_{\infty}^{\pm} \right\| _{H^1\left( \mathbb{R} \right)}\rightarrow 0,\enspace t\rightarrow \infty,\] where \[\label{rp} R_p\left( x-\frac{1}{\mathrm{Im}p}t \right) =\frac{2\mathrm{Im}p}{\left| x-\frac{1}{\mathrm{Im}p}t+p \right|^2}.\qquad{(2)}\]
The above results of Gassot, Gérard, and Miller demonstrate that, in the sense of the \(H^1\)-norm, the solution for the BO equation asymptotically decomposes into a sum of solitons for the form \(R_p(x-\frac{1}{\mathrm{Im}p}t)\) and a dispersive term \(\mathrm{e}^{t\partial_x|D|}u_{\infty}^{\pm}\), with the asymptotic error \(\left\| r\left( t,\cdot \right) \right\| _{H^1\left( \mathbb{R} \right)}\rightarrow 0,\enspace t\rightarrow \pm \infty\).
Furthermore, for initial data given by the infinite-order multisoliton form ?? , corresponding to the special case where the Lax operator \(L_{u_0}\) without a continuous spectrum, Gassot and Gérard [16] proved the following result:
Proposition 2 (Gassot and Gérard [16]). The initial data \({u}_{0}\) given by \[\label{form1} u_0\left( x \right) :=\sum_{j=1}^{\infty}{\frac{2\mathrm{Im}\left( p_j \right)}{\left| x+p_j \right|^2}},\enspace x\in \mathbb{R} ,\qquad{(3)}\] under the condition \[\label{form2} \sum_{j=1}^{\infty}{\sum_{k=1}^{\infty}{\frac{\mathrm{Im}\left( p_j \right)}{\left| p_j-\overline{p_k} \right|^2}}}<\infty\qquad{(4)}\] belong to \({H}^{s}\left( \mathbb{R}\right)\) for every \(s \in \mathbb{R}\). Moreover, there is a sequence \({( p_{j}^{\infty} ) }_{j \geq 1}\) of complex numbers in \({\mathbb{C}}_{ + }\) with increasing imaginary parts \(0<\mathrm{Im}\left( p_{1}^{\infty} \right) <\mathrm{Im}\left( p_{2}^{\infty} \right) <\cdots\), such that there holds \[\underset{N\rightarrow \infty}{\lim}\underset{t\rightarrow \infty}{\lim\mathrm{sup}}\left\| u\left( t,\cdot \right) -\sum_{j=1}^{\infty}{R_{p_{j}^{\infty}}\left( \cdot -c_{p_{k}^{\infty}}t \right)} \right\| _{L^2}=0.\] Furthermore, for every \(t \in \mathbb{R}\) , the formula \[\label{usol} u_{\mathrm{sol}}\left( t,\cdot \right) :=\sum_{j=1}^{\infty}{R_{p_{j}^{\infty}}\left( x-c_{p_{j}^{\infty}}t \right)},\enspace x\in \mathbb{R},\qquad{(5)}\] defines a continuous function \(u_{\mathrm{sol}}\left( t,\cdot \right)\) on \(\mathbb{R}\), tending to 0 as \(x\rightarrow \infty\), and belonging to all the homogeneous Sobolev spaces \(\dot{H}^s\left( \mathbb{R} \right)\) for \(s\geqslant 1/2\). Finally, we have, for every \(s\geqslant 1/2\), \[\left\| u\left( t,\cdot \right) -u_{\mathrm{sol}}\left( t,\cdot \right) \right\| _{L^{\infty}}+\left\| u\left( t,\cdot \right) -u_{\mathrm{sol}}\left( t,\cdot \right) \right\| _{H^s}\underset{t\rightarrow \infty}{\longrightarrow}0.\]
However, we remark that the applicability of Propositions 1 and 2 is significantly constrained by their underlying assumptions. Firstly, the requirements in Proposition 1, specifically \(u_0 \in H^{1,1}(\mathbb{R})\) and the decay condition \(x^2u_0(x) = c_0 + v_0(x)\), are restrictive, posing a substantial barrier to establishing the soliton resolution conjecture in a general setting. To address this, we develop a novel construction of an isospectral smooth approximating sequence. This approach effectively overcomes the aforementioned technical obstacles, enabling us to prove the conjecture for a much broader and more physically relevant class of initial data.
Secondly, the works [16], [31] provide the implicit error bounds. As seen in the Proposition 1 and 2, this metric presents a gap between mathematical theory and physical observability The critical observable parameter is the localized peak amplitude of the wave packet, which is a quantity governed by the \(L^\infty\)-norm. Theoretical physicists and oceanographers are concerned not only with the stable decomposition of complex perturbations into solitons and radiation, but also with the precise decay rate of the radiation’s amplitude [32], [33]. Consequently, relying solely on an implicit energy-based limit fails to provide a quantitative timescale for how fast this maximal wave height subsides, leaving observers unable to predict exactly when the residual background perturbations will physically flatten out. This implies that the absence of an explicit \(L^\infty\) decay rate for the error term would limit the practical applicability of mathematical proofs concerning internal wave evolution in physical and engineering contexts. Therefore, providing an explicit formula for the asymptotic error in \(L^\infty\) is a natural question.
Thirdly, Gassot and Gérard [16], by analogy with the previous work [15] on proving the distorted Plancherel formula, proposed the following inverse spectral problem concerning the Lax operator \(L_{u_0}\) for the infinite-order multisoliton form ?? :
“In view of the distorted Plancherel formula [BGGM25, Theorem 26] proven for initial data \({u}_{0}\) with \(x{u}_{0} \in {L}^{2}\left( \mathbb{R}\right)\), it is natural to ask whether all initial data \({u}_{0} \in {L}^{2}\left( \mathbb{R}\right)\) such that the eigenvalues the Lax operator \({L}_{{u}_{0}}\) satisfy \(2\pi \sum_{k=1}^{\infty}{\left| \lambda _k \right|}=\left\| \Pi u_0 \right\| _{L^2}^{2}\) are of the form ?? . This seems to be an open problem.”
This open problem is equivalent to identifying the exact range of the mapping \(( p_j )_{j \geqslant 1} \mapsto ( p_{j}^{\infty} )_{j \geqslant 1}\), which is of significant value in elucidating the analytical structure of the BO equation with initial data taking the infinite-order multisoliton form ?? .
These naturally lead to the following open problems summarized from the work of Gérard and his collaborators [16], [31] and other related works:
Problem 1. In [31], Gassot, Gérard, and Miller analyzed the oscillatory integrals associated with the explicit formula via the van der Corput estimate and stationary phase asymptotics in \(H^{1,1}\left(\mathbb{R} \right)\). However, this method necessitates sufficiently smooth and rapidly decaying initial data, resulting in a stringent initial space. Consequently, how to prove the soliton resolution conjecture within a sufficiently low regularity space?
Problem 2. In [31], Gassot, Gérard, and Miller estimated the soliton limit and the radiation limit with Lebesgue theorem, thereby obtaining the implicit error in \(H^1\)-norm. However, Lebesgue theorem falls short of providing an explicit formula for the asymptotic error. Thus, in order to rigorously describe the decay rate of the error, how can one establish an explicit formula for the asymptotic error in the \(L^\infty\)-norm?
Problem 3. An additional condition ?? was required for the initial data in [31]. However, this condition is restrictive and is not applicable to the study of long-time asymptotics for general initial data. To validate the soliton resolution conjecture for a broader class of initial data, how to establish the soliton resolution for the BO equation without the condition \(\forall x \in \mathbb{R}, x^2 u_0(x) = c_0 + v_0(x)\)?
Problem 4. An open problem about the inverse spectral problem for the Lax operators was formulated in [16]: Are all initial data \({u}_{0} \in {L}^{2}\left( \mathbb{R}\right)\) satisfying the spectral condition \(2\pi \sum_{k=1}^{\infty}{\left| \lambda _k \right|}=\left\| \Pi u_0 \right\| _{L^2}^{2}\) for \({L}_{{u}_{0}}\) of the form ?? ?
Problem 5. The weak convergence of the resolvent operator was established in [16] by employing the explicit formula and conservation laws of the BO equation, which consequently led to an implicit error estimate. However, in order to rigorously characterize the influence of the infinitely many discrete spectra of the Lax pair on the long-time behavior of the BO equation solutions, a natural question arises, how can one derive an explicit formula for the asymptotic error in the \(L^\infty\)-norm?
In order to establish the soliton resolution conjecture for the BO equation with explicit error bounds, it is necessary to overcome certain analytical limitations present in recent works [16], [31]. Unlike classical integrable systems, the BO equation features a nonlocal dispersive structure, and its inverse scattering transform is formulated on the Hardy space via the nonlocal Lax operator \(L_{u_0} = D - T_{u_0}\). Consequently, the long-time evolution of the continuous spectrum is governed by an explicit resolvent formula involving singular oscillatory integrals.
In the finite-order multisoliton case, the analysis of these oscillatory integrals in [31] relies on the stationary phase method and the Lebesgue dominated convergence theorem. This approach requires sufficient regularity and rapid spatial decay to ensure absolute convergence, restricting the analysis to initial data \(u_0 \in H^{1,1}(\mathbb{R})\) satisfying the additional spatial constraint \(x^2u_0(x) = c_0 + v_0(x)\). Furthermore, the reliance on the dominated convergence theorem prevents the derivation of explicit decay rates, yielding only an implicit asymptotic error bound \(\|r(t)\|_{H^1} \to 0\).
In the infinite-order multisoliton case, Gassot and Gérard [16] established an implicit soliton resolution by analyzing the weak convergence of the resolvent operator in the Hardy space. However, the spectral trace condition 4 constrains only the imaginary parts of the poles \(\operatorname{Im}( p_{j}^{\infty} )\), but lacks control over the real parts \(\operatorname{Re}( p_{j}^{\infty} )\) and the exact distribution of the discrete spectrum. This lack of precise information regarding the spatial localization of infinitely many interacting solitons results in a qualitative resolution limit in the \(L^\infty\)-norm, leaving the inverse spectral problem for the Lax operator open.
In summary, the fundamental difficulties and challenges can be explicitly formulated as follows:
(I) In order to relax the restrictive condition ?? and lower the initial regularity to the threshold space, a standard approach is to approximate rough initial data via spatial truncations. However, such truncations inevitably perturb the discrete spectrum of the associated Lax operator. Because the imaginary part of the spectral parameter \(p_j\) determines the group velocity of the corresponding soliton via \(c_j = 1/\operatorname{Im}(p_j)\), any non-zero perturbation \(\delta c_j\) is amplified as \(t \to \pm\infty\). This induces a secular spatial shift \(\Delta x_j \sim t \delta c_j\), which destroys the uniformity of long-time estimates.
(II) In order to solve Problem 4 and solve the inverse spectral problem concerning the Lax operator proposed in [16], we need to deal with the condition 4 . However, this condition only constrains \(\mathrm{Im}( p_{j}^{\infty} )\) and does not constrain \(\mathrm{Re}( p_{j}^{\infty} )\). Consequently, the distribution of \(\mathrm{Re}( p_{j}^{\infty} )\) remains unknown, which requires us to introduce other methods to control \(\mathrm{Re}( p_{j}^{\infty} )\).
(III) The condition 4 lacks control over \(\mathrm{Re}( p_{j}^{\infty} )\), which further leads to the absence of the necessary spectral gap condition when analyzing the error in the \(L^\infty\)-norm between the solution with initial data ?? and ?? .
To overcome the aforementioned difficulties, we develop the following framework. For the finite-order multisoliton case, based on the special spectral structure of the BO equation and Kato’s perturbation theory, we construct an isospectral approximating sequence. This transforms the soliton resolution conjecture into an error estimation problem between the exact solution and the approximating sequence, thereby circumventing the difficulty that the soliton resolution conjecture cannot be proved using the stationary phase method due to initial data with low regularity and slow decay. Based on the explicit formula, we decompose the error estimation problem into an error estimate for the discrete spectrum and an error estimate for the continuous spectrum. For the discrete spectrum, relying on the isospectral property of the approximating sequence, we prove that its error can be bounded by a polynomial function. For the continuous spectrum, by extending the Hölder continuity of the Lax operator to the weighted \(L^2\) space and combining it with the \(L^2\) conservation law of the BO equation, we establish the error estimate in the \(L^2\)-norm. Finally, we use the Gagliardo-Nirenberg interpolation inequality to lift this error to the \(L^\infty\)-norm, thereby overcoming the aforementioned difficulties in obtaining the pointwise \(L^\infty\)-norm error in low-regularity spaces. For the infinite-order multisoliton case, based on the expression of the initial data, we construct a class of trace-class operators, thereby transforming the inverse spectral problem into the computation of the traces of these trace-class operators. By employing the fundamental idea of evaluating the trace in two different ways, we solve the inverse spectral problem. Furthermore, relying on the finite-rank operators generated by the trace-class operators, we transform the long-time asymptotics into an error analysis between the exact solution and the finite-rank operators, as well as between the finite-rank operators and the infinite-order soliton solution. Finally, by using the spectral analysis properties of the BO equation, we obtain the explicit error in the \(L^\infty\)-norm.
It is worth noting that there are some highlights in our work as follows.
In Theorem 3, under the condition that the initial data belongs to weighted \(L^2\) space, we prove the soliton resolution conjecture for the BO equation
In Theorem 5, we extend the initial data space \(H^{1,1}(\mathbb{R})\) in Proposition 1 to \(u_0 \in H^{s,\alpha}, \enspace s>1/2, \enspace \alpha \geqslant 1\), and prove that the \(L^\infty\)-norm of the explicit error decays at the rate of \(|t|^{-\frac{1}{4} (1-\frac{1}{2s})}\).
In both of the above theorems, we removed the condition \(\forall x \in \mathbb{R}, x^2 u_0(x) = c_0 + v_0(x)\), thereby proving the soliton resolution conjecture for a broader class of initial data.
The equivalence between the Lax operator satisfying 4 and the initial data satisfying ?? is proved in Theorem 7, thereby solving the open problems regarding the inverse spectral problem for the Lax operator posed in [16].
An \(L^\infty\)-norm error that decays at the rate \(\left| t \right|^{-\frac{\alpha -1}{3\alpha +4}}\) is proved in Theorem 9 for the soliton resolution in the infinite-order multisoliton case, under the assumption that the discrete spectrum of \(L_{u_0}\) satisfies the gap condition 6 .
In this work, we prove the soliton resolution conjecture for the BO equation with an explicit error bound in the \(L^\infty\)-norm, and solve the open inverse spectral problem for the Lax operator. Our main results read as follows.
In order to prove the soliton resolution conjecture for a broader class of initial data in \(L^2\) and solve Problem 1 and Problem 3, we construct an approximating function sequence. Through this method, we convert the analysis of the long-time behavior for the BO equation into an error estimation problem between the initial data and the isospectral smooth approximating sequence, which provides the essential framework for proving the following Theorem 3.
Theorem 3 (Soliton resolution conjecture in weighted \(L^{2}\) for the finite-order multisoliton). Let the initial data \(u_0\in L^2\left( \mathbb{R} \right)\) be a real-valued function such that \[\label{thm1tiaojian} xu_0\left( x \right) \in L^2\left( \mathbb{R} \right).\tag{2}\] Given \(\{p_j\}_{j=1}^N \subset \mathbb{C}\) satisfying \(0 < \mathrm{Im}(p_1) < \cdots < \mathrm{Im}(p_N)\). Then, there exist real-valued functions \(u_{\infty}^{\pm}\) in \(L^2 \left(\mathbb{R} \right)\) such that the solution \(u\left(t,x \right)\) can be decomposed as follows: \[\lim_{t \to \pm \infty} \left\| u(t) - \sum_{j=1}^{N} R_{p_j} - e^{t\partial_x|D|}u_\infty^\pm \right\|_{L^2} = 0 ,\] where the soliton terms \(R_{p_j}\) are defined by ?? , the dispersive term \(\mathrm{e}^{t\partial_x|D|}u_{\infty}^{\pm}\) is given by\[\mathrm{e}^{t\partial_x|D|}u_{\infty}^{\pm} = \frac{1}{2\pi} \int_{\mathbb{R}} e^{ix\xi + it\xi|\xi|} \widehat{u_{\infty}^{\pm}}(\xi) \, d\xi.\]
Remark 4. Comparing Theorem 3 with Proposition 1, the initial data space of Theorem 3 is equivalent to \(H^{0,1}\left( \mathbb{R} \right)\). Proposition 1 requires the initial data to belong to the \(H^{1,1}\left(\mathbb{R} \right)\) space, and additionally requires condition ?? . Our result broadens the initial data space and removes the condition \(x^2u_0\left( x \right) =c_0+v_0\left( x \right) ,\enspace c_0\in \mathbb{R} ,\enspace v_0\left(x \right) \in L^2(\mathbb{R} )\).
In order to solve Problems 1-3, we establish an explicit formula for the asymptotic error in \(L^\infty\)-norm under the condition \(u_0\in H^{s,\alpha}\left( \mathbb{R} \right) ,\enspace s>1/2,\enspace \alpha \geqslant 1\). We improve the error estimation method for the isospectral smooth approximating sequence by incorporating the Gagliardo-Nirenberg interpolation inequality, providing the crucial analytic tool to prove the following Theorem 5.
Theorem 5 (Explicit error bound in \(L^\infty\)-norm of the Soliton resolution conjecture for the finite-order multisoliton). Let the initial data \(u_0\in H^{s,\alpha}\left( \mathbb{R} \right) ,\enspace s>1/2, \enspace \alpha \geqslant 1\) be a real-valued function. Given \(\{p_j\}_{j=1}^N \subset \mathbb{C}\) satisfying \(0 < \mathrm{Im}(p_1) < \cdots < \mathrm{Im}(p_N)\). Then, there exist real-valued functions \(u_{\infty}^{\pm}\) in \(L^2 \left(\mathbb{R} \right)\) such that the solution \(u\left(t,x \right)\) can be decomposed as follows: \[u\left( t,x \right) =\sum_{j=1}^N{R_{p_j}\left( x-\frac{1}{\mathrm{Im}p_j}t \right)}+\mathrm{e}^{t\partial _x\left| D \right|}u_{\infty}^{\pm}+\mathcal{E} \left( t,x \right) ,\enspace t\rightarrow \pm \infty,\] where \[\label{wucha2} \left\| \mathcal{E} \left( t,\cdot \right) \right\| _{L^{\infty}}=\mathcal{O}\Big(|t|^{-\frac{1}{4}(1-\frac{1}{2s})}\Big),\enspace t\rightarrow \pm \infty .\tag{3}\]
Remark 6. Theorem 5 achieves significant improvements for Problems 1-3. For Problem 1, we broaden the initial data space to \(H^{s,\alpha}\left( \mathbb{R} \right) ,\enspace s>1/2, \enspace \alpha \geqslant 1\) in Theorem 5, thereby proving the soliton resolution conjecture within a low regularity space. For Problem 2, we provide an explicit formula for the asymptotic error in \(L^\infty\) of \(\mathcal{O}\Big(|t|^{-\frac{1}{4}(1-\frac{1}{2s})}\Big)\). For Problem 3, we remove the strict decay condition ?? , and thus prove the soliton resolution conjecture for a larger class of initial data.
In contrast to the well-developed spectral analysis of the Lax operator in the finite-order case, the understanding of its discrete spectrum in the infinite-order case remains incomplete, leaving several open problems unresolved. An example is the inverse spectral problem formulated by Gassot and Gérard in [16]. To further investigate the soliton resolution conjecture for the BO equation in the infinite-order setting, it is imperative to first address this inverse spectral problem. Thus, we construct a class of trace-class operators and reduce the inverse spectral problem to the analysis of these operators. Consequently, we establish that the characterization of the discrete spectrum of the Lax operator satisfying 4 is equivalent to the representability of the initial data \(u_0(x)\) in the form of ?? , as detailed in Theorem 7.
Theorem 7 (The inverse spectral problem for the infinite-order multisoliton). Let the Lax operator \(L_{u_0}\) admits an infinite number of negative eigenvalues \(\left( \lambda _k \right) _{k\geqslant 1}\) with \(\lambda _1<\lambda _2<\cdots <0\). Then all initial data \(u_0 \in L^2\left(\mathbb{R} \right)\) such that \(L_{u_0}\) satisfy \[\label{laxtezheng} 2\pi \sum_{k=1}^{\infty}{\left| \lambda _k \right|}=\left\| \Pi u_0 \right\| _{L^2}^{2}\tag{4}\] are of the form \[u_0\left( x \right) =\sum_{j=1}^{\infty}{\frac{2\mathrm{Im}\left( p_j \right)}{\left| x+p_j \right|^2},\enspace x\in \mathbb{R}},\]where \[\sum_{j=1}^{\infty}{\sum_{k=1}^{\infty}{\frac{\mathrm{Im}\left( p_j \right)}{\left| p_j-\overline{p_k} \right|^2}}}<\infty ,\enspace \mathrm{Im}p_j\geqslant 0.\]
Remark 8. Theorem 7 solves the open problem proposed by Gassot and Gérard [16], namely Problem 4, proving that if and only if the discrete spectrum of the Lax operator \(L_{u_0}\) satisfies condition 4 , the initial data can be expressed in the form ?? . Combined with the work of [16], this implies that 4 is equivalent to the initial data taking the form ?? , which also corresponds to the result of the distorted Plancherel formula in [15].
The trace-class operators constructed in the proof of Theorem 7 provide the foundation for estimating the error in the \(L^\infty\)-norm. This approach constitutes the core of Theorem 9, thereby enabling us to solve Problem 5 by deriving an explicit \(L^\infty\) asymptotic error formula under conditions ?? and ?? .
Theorem 9 (Explicit error bound in \(L^\infty\)-norm of the Soliton resolution for the finite-order multisoliton). Suppose the initial data \(u_0(x)\) satisfies ?? and ?? . Then \[\label{minimax95bound} \left\| u(t, \cdot) - u_{\mathrm{sol}}(t, \cdot) \right\|_{L^\infty(\mathbb{R})}=\mathcal{O} \left(\inf_{N \ge 1} \left\{ \sum_{k=N+1}^\infty |\lambda_k| + \frac{1}{\left| t \right|} \sum_{\substack{j, k, l \le N \\ j \neq l, k \neq l}} \frac{|\lambda_k|}{|\lambda_j - \lambda_l|^2 \cdot |\lambda_k - \lambda_l|} \right\} \right) , \enspace t\to \pm \infty.\tag{5}\] where \[u_{\mathrm{sol}}(t,x) := \sum_{k=1}^\infty R_{p_k^\infty}(x - c_{p_k^\infty} t).\] Furthermore, assume that there exist constants \(C_1, C_2 > 0\) and \(\alpha > 1\) such that \[\label{gap95condition} |\lambda _k|\leqslant C_1k^{-\alpha},\quad |\lambda _j-\lambda _k|\geqslant C_2|j-k|\max\mathrm{(}j,k)^{-\alpha -1},\enspace j,k\geqslant 1,\enspace j\ne k.\tag{6}\] Then \[\label{linfty95bound} \left\| u(t,\cdot )-u_{\mathrm{sol}}(t,\cdot ) \right\| _{L^{\infty}(\mathbb{R} )}=\mathcal{O} \left( \left| t \right|^{-\frac{\alpha -1}{3\alpha +4}} \right) ,\enspace t\rightarrow \pm \infty .\tag{7}\]
Remark 10. Comparing Theorem 9 with Proposition 2, we provide the explicit error estimation in the \(L^\infty\)-norm for the case of the infinite-order multisoliton and prove that the decay rate of the error depends on the distribution of the discrete spectrum of the Lax operator \(L_{u_0}\), thereby achieving significant improvements for Problem 5.
For the finite-order multisoliton, we use the spectral analysis of the BO equation in low-regularity spaces and Kato-Rellich theorem, thus transforming the problem of soliton resolution conjecture and long-time asymptotics for the BO equation into the error estimation between the solutions evolved from the smooth approximating sequence and the low-regularity solutions. By applying truncation and interpolation methods to the distorted Fourier transform of the explicit formula, we obtain an explicit formula for the asymptotic error in the \(L^\infty\)-norm, relying on the Hölder continuity of the Lax operator for the BO equation in weighted \(L^2(\mathbb{R})\). This stands in stark contrast to the approach in [31], which relies on the classical stationary phase method and the Lebesgue theorem to derive implicit errors.
For the infinite-order multisoliton, based on the condition 4 , we construct a class of trace-class operators and transform the inverse spectral problem associated with the Lax operator into the analysis of these operators. By employing Lidskii’s theorem, we solve the open problem proposed in [16]. Furthermore, by analyzing the time evolution of these trace-class operators, we obtain the explicit error estimation in the \(L^\infty\)-norm for the case of the infinite-order multisoliton.
The structure of this work is as follows. Section 2 provides the complete proof of Theorem 3. The proof extends the translation formula from smooth data to the \(H^{0,1}\) space by means of the distorted Plancherel theorem of [15], and employs the Volterra integral equation to transform the oscillatory integral containing the phase \(\mathrm{e}^{\mathrm{i}tu^2}\) into a standard Dirichlet-type integral. Then, Theorem 5 follows by a similar argument as in the proof of Theorem 3, combined with the Gagliardo-Nirenberg inequality. In Section 3, we construct a class of trace-class operators and, combined with Lidskii’s theorem, prove that 4 and ?? are equivalent. In Section 4, we analyze the time evolution of these operators in conjunction with the explicit formula. Under the condition that the discrete spectrum of the Lax operator satisfies 6 , we provide the explicit error estimation in the \(L^\infty\)-norm for the infinite-order multisoliton.
Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No. 12371255, the Fundamental Research Funds for the Central Universities of CUMT under Grant No. 2024ZDPYJQ1003, and the Postgraduate Research & Practice Program of Education & Teaching Reform of CUMT under Grant No. 2025YJSJG031.
This section presents the proof of Theorem 3 under the assumption that \(u_0\in L^2(\mathbb{R})\) is real-valued and \(\langle x\rangle u_0\in L^2(\mathbb{R})\). To simplify notation, we only give the proof for \(t\to+\infty\); the case \(t\to-\infty\) is entirely analogous.
For \(u_0\in H^{0,1}(\mathbb{R})\), by the Distorted Plancherel Theorem, we have \[\zeta_{u_0}(\lambda):=\langle\Pi u_0,\,m_-(\cdot,\lambda;u_0)\rangle\] belonging to \(L^2(0,\infty)\), where \(m_-(\cdot,\lambda)\) is the Jost function satisfying \(L_{u_0}m_-=\lambda m_-\) and \(\mathrm{e}^{-\mathrm{i}\lambda x}m_-(x,\lambda)\to1\) as \(x\to-\infty\).
Define \(u_\infty^+\in L^2_+(\mathbb{R})\) by \[\label{eq:u95infty95def} \widehat{u_\infty^+}(\xi):=\zeta_{u_0}(\xi)\cdot\mathbf{1}_{(0,\infty)}(\xi).\tag{8}\] By Plancherel’s Theorem, \(u_\infty^+\) exists and is unique.
According to [31], we have \[\label{eq:explicit} \Pi u(t,z)=\Omega_t^{u_0}(\Pi u_0):=\frac{1}{2\pi\mathrm{i}}\,I_+\Big((G-2t\mathcal{L}_{u_0}-z\,\mathrm{Id})^{-1}\Pi u_0\Big),\quad \operatorname{Im}z>0,\tag{9}\] where \(G\) is the adjoint of the multiplication operator by \(x\) on \(L^2_+(\mathbb{R})\) and \(\mathcal{L}_{u_0}=D-T_{u_0}\), \(D=-\mathrm{i}\partial_x\), \(T_{u_0}f=\Pi(u_0 f)\), \(I_+(f)=\widehat{f}(0^+)\). By [34], we have
Lemma 1. For any \(u_0\in L^2(\mathbb{R})\) and \(t\in\mathbb{R}\), \(\Omega_t^{u_0}\) on \(L^2_+(\mathbb{R})\) satisfies \[\|\Omega_t^{u_0}(f_0)\|_{L^2}\le\|f_0\|_{L^2},\quad\forall f_0\in L^2_+(\mathbb{R}).\]
The expression 9 can be reduced to \(\Pi u_0=\sum_{j=1}^N c_j\varphi_j+f_c\), where \(\varphi_j\) are the eigenfunctions corresponding to the discrete spectrum, and \(f_c\in\mathcal{H}_c(L_{u_0})\) is the continuous spectrum component.
For \(\Pi u_c(t)=\Omega_t^{u_0}(f_c)\), by [13], we have \(\|u(t)\|_{L^2}=\|u_0\|_{L^2}\). Moreover, we have \(\|\Pi u(t)\|_{L^2}^2=\frac{1}{2}\|u(t)\|_{L^2}^2\). By the Lax pair structure, \(L_{u(t)}\) is unitarily equivalent to \(L_{u_0}\). Applying Theorem 26 of [15] to \(u(t)\) and \(u_0\) respectively, we obtain \[\|\Pi u(t)\|_{L^2}^2=\sum_{j=1}^N 2\pi|\lambda_j|+\frac{1}{2\pi}\|\zeta_{u(t)}\|_{L^2(0,\infty)}^2, \qquad \|\Pi u_0\|_{L^2}^2=\sum_{j=1}^N 2\pi|\lambda_j|+\frac{1}{2\pi}\|\zeta_{u_0}\|_{L^2(0,\infty)}^2.\] Subtracting the two equations yields \(\|\zeta_{u(t)}\|_{L^2}=\|\zeta_{u_0}\|_{L^2}\). By the unitarity of the Distorted Fourier Transform on the subspace, we obtain \[\label{eq:mass95cons} \|\Omega_t^{u_0}(f_c)\|_{L^2}=\|\Pi u_c(t)\|_{L^2}=\frac{1}{\sqrt{2\pi}}\|\zeta_{u(t)}\|_{L^2} =\frac{1}{\sqrt{2\pi}}\|\zeta_{u_0}\|_{L^2}=\|f_c\|_{L^2}=\|u_\infty^+\|_{L^2}.\tag{10}\]
Next, we prove that the following weak limit holds \[\label{eq:weak95goal} \lim_{t\to\infty}\langle\Omega_t^{u_0}(f_c),\,U_0(t)h\rangle=\langle u_\infty^+,\,h\rangle,\quad\forall h\in L^2_+(\mathbb{R}),\tag{11}\] where \(U_0(t)=\mathrm{e}^{t\partial_x|D|}\), whose Fourier symbol on \(L^2_+\) is \(\mathrm{e}^{\mathrm{i}t\xi^2}\)?\(\xi>0\)).
Take test functions \(h\in\mathcal{S}_+(\mathbb{R})\) such that \(\hat{h}\) is compactly supported in \([a,b]\subset(0,\infty)\)?\(a>0\)?, and take \[f_\psi:=\int_0^\infty\psi(\lambda)\,m_-(\cdot,\lambda;u_0)\,\frac{\mathrm{d}\lambda}{2\pi},\quad\psi\in C_c^\infty(0,\infty),\;\operatorname{supp}\psi\subset[c,d]\;(c>0)\] to approximate \(f_c\).
Let \(u_n\in\mathcal{S}(\mathbb{R})\) converge strongly to \(u_0\) in \(H^{0,1}(\mathbb{R})\). For the smooth sequence \(u_n\), [31] proved that \[\label{eq:smooth95trans} J_t^n(x):=\Omega_t^{u_n}(f_\psi^n)(x)=\int_0^\infty\psi(\lambda)\,m_-^n(x,\lambda)\,\mathrm{e}^{\mathrm{i}t\lambda^2}\,\frac{\mathrm{d}\lambda}{2\pi},\tag{12}\] where \(f_\psi^n:=\int_0^\infty\psi(\lambda)\,m_-^n(\cdot,\lambda)\,\frac{\mathrm{d}\lambda}{2\pi}\). In the sequel, we extend this identity to \(H^{0,1}\).
Lemma 2. If \(u_n\to u_0\) in \(H^{0,1}(\mathbb{R})\), then for each \(\lambda>0\), \(m_-^n(\cdot,\lambda)\to m_-(\cdot,\lambda)\) uniformly in \(L^\infty_x(\mathbb{R})\).
Proof. By Section 7 of [15], we have \(m_-=(1-\mathrm{i}K_{u,\lambda})^{-1}(\mathrm{e}^{\mathrm{i}\lambda\cdot})\), where \(K_{u,\lambda}:L^\infty\to L^\infty\) is given by \(K_{u,\lambda}m(x)=\int_{-\infty}^x\mathrm{e}^{\mathrm{i}\lambda(x-y)}T_u m(y)\,\mathrm{d}y\). By Lemma 29 of [15], we have \(T_u m=g_{u,m}+\frac{c_{u,m}}{y-\mathrm{i}}\), where \(g_{u,m}\in L^1\) and \(c_{u,m}\in\mathbb{C}\) are controlled by \(\|u\|_{H^{0,1}}\|m\|_{L^\infty}\). Consequently, \[\|K_{u_n,\lambda}-K_{u_0,\lambda}\|_{\mathcal{B}(L^\infty)}\le C\|u_n-u_0\|_{H^{0,1}}\to0.\] By the bijectivity of \((1-\mathrm{i}K_{u_0,\lambda})\) on \(L^\infty\) (see Proposition 30 of [15]), we have \((1-\mathrm{i}K_{u_n,\lambda})^{-1}\to(1-\mathrm{i}K_{u_0,\lambda})^{-1}\), hence \(m_-^n\to m_-\) uniformly in \(L^\infty_x\). ◻
Lemma 3. For \(u_0\in H^{0,1}(\mathbb{R})\), in the sense of weak convergence in \(L^2_+(\mathbb{R})\), \[\label{eq:translation} \Omega_t^{u_0}(f_\psi)(x)=\int_0^\infty\psi(\lambda)\,m_-(x,\lambda)\,\mathrm{e}^{\mathrm{i}t\lambda^2}\,\frac{\mathrm{d}\lambda}{2\pi}.\tag{13}\]
Proof. By Lemma 2, \(m_-^n\) converges uniformly in \(L^\infty_x\), hence for any \(v\in C_c^\infty(\mathbb{R})\), \(\langle f_\psi^n,v\rangle\to\langle f_\psi,v\rangle\), i.e., \(f_\psi^n\rightharpoonup f_\psi\) weakly in \(L^2_+\). By the Distorted Plancherel Isometry, \(\|f_\psi^n\|_{L^2}=\|\psi\|_{L^2}/\sqrt{2\pi}=\|f_\psi\|_{L^2}\) is conserved. By the Radon-Riesz Theorem, weak convergence together with norm conservation implies \(\|f_\psi^n-f_\psi\|_{L^2_+}\to0\).
By Lemma 1, we obtain \(\|\Omega_t^{u_n}(f_\psi^n-f_\psi)\|_{L^2}\le\|f_\psi^n-f_\psi\|_{L^2}\to0\). Moreover, by [34], we have \(\Omega_t^{u_n}(f_\psi)(z)\to\Omega_t^{u_0}(f_\psi)(z)\) pointwise in \(\mathbb{C}_+\). Since the linear span of \(\{\frac{1}{x-\bar{z}}\}_{z\in\mathbb{C}_+}\) is dense in \(L^2_+\), by the Banach-Steinhaus Theorem, \((\Omega_t^{u_n}-\Omega_t^{u_0})f_\psi\rightharpoonup0\) in \(L^2_+\). In summary, \(\Omega_t^{u_n}(f_\psi^n)\rightharpoonup\Omega_t^{u_0}(f_\psi)\).
By the uniform convergence of \(m_-^n\), the right-hand side integral of 12 converges uniformly in \(L^\infty_x\) to \[J_t^0(x):=\int_0^\infty\psi(\lambda)m_-(x,\lambda)\mathrm{e}^{\mathrm{i}t\lambda^2}\frac{\mathrm{d}\lambda}{2\pi}.\] This means that the \(L^2_+\) weak limit of \(J_t^n\) coincides almost everywhere with its \(L^\infty\) limit, hence 13 holds. ◻
Substituting \(m_-(x,\lambda)=\mathrm{e}^{\mathrm{i}\lambda x}+\rho(x,\lambda)\) into 13 yields \(\mathcal{F}^{-1}(\psi(\xi)\mathrm{e}^{\mathrm{i}t\xi^2})=U_0(t)u_\infty^+[\psi]\), with the remainder \[\label{eq:I95rho} \mathcal{I}_\rho(t):=\langle\Omega_t^{u_0}(f_\psi)_{\mathrm{rem}},\,U_0(t)h\rangle =\int_a^b\mathrm{d}\xi\,\overline{\hat{h}(\xi)}\,\mathrm{e}^{-\mathrm{i}t\xi^2} \int_0^\infty\frac{\mathrm{d}\lambda}{2\pi}\,\psi(\lambda)\,\hat{\rho}(\xi,\lambda)\,\mathrm{e}^{\mathrm{i}t\lambda^2}.\tag{14}\] From \(\mathrm{e}^{-\mathrm{i}\lambda x}m_-(x,\lambda)\to1\;(x\to-\infty)\), the Volterra integral equation for \(\rho\) is \[\label{eq:volterra} \rho(x,\lambda)=\mathrm{i}\int_{-\infty}^x\mathrm{e}^{\mathrm{i}\lambda(x-y)}g_\lambda(y)\,\mathrm{d}y,\quad g_\lambda:=\Pi(u_0 m_-)\in L^2_x.\tag{15}\] Equivalently, the above is the convolution of \(g_\lambda\) with the function \(K(x)=\mathrm{i}H(x)\mathrm{e}^{\mathrm{i}\lambda x}\), where \(H\) is the Heaviside step function.
In the class of tempered distributions \(\mathcal{S}'(\mathbb{R})\), \(\mathcal{F}(H)(\xi)=\pi\delta(\xi)-\mathrm{i}\,\mathrm{P.V.}\frac{1}{\xi}\), hence \[\mathcal{F}(H(x)\mathrm{e}^{\mathrm{i}\lambda x})(\xi)=\pi\delta(\xi-\lambda)-\mathrm{i}\,\mathrm{P.V.}\frac{1}{\xi-\lambda}.\] This means \(\widehat{K}(\xi)=\mathrm{P.V.}\frac{1}{\xi-\lambda}+\mathrm{i}\pi\delta(\xi-\lambda)\). Consequently, \[\label{eq:pole95dist} \hat{\rho}(\xi,\lambda)=\widehat{K}(\xi)\,\hat{g}_\lambda(\xi) =\Big(\mathrm{P.V.}\frac{1}{\xi-\lambda}+\mathrm{i}\pi\delta(\xi-\lambda)\Big)\hat{g}_\lambda(\xi).\tag{16}\]
We examine the regularity of \(\hat{g}_\lambda(\xi)=\mathbf{1}_{\xi>0}\widehat{u_0 m_-}(\xi)\) with respect to \(\xi\). We have \[\partial_\xi\hat{g}_\lambda(\xi)=\mathbf{1}_{\xi>0}\partial_\xi\widehat{u_0 m_-}(\xi)+\widehat{u_0 m_-}(0^+)\delta_0(\xi) =\Pi\mathcal{F}(-\mathrm{i}x u_0 m_-)(\xi)+\widehat{u_0 m_-}(0^+)\delta_0(\xi).\] Since \(\langle x\rangle u_0\in L^2\), we have \(x u_0 m_-\in L^2_x\), hence the first term above belongs to \(L^2_\xi\). Define \(\Phi(\xi,\lambda):=\overline{\hat{h}(\xi)}\psi(\lambda)\hat{g}_\lambda(\xi)\). Since \(\hat{h}\) is compactly supported in \([a,b]\subset(0,\infty)\), we have \(\overline{\hat{h}(0)}\equiv0\). When differentiating \(\Phi\), the \(\delta_0(\xi)\) term vanishes, hence \(\partial_\xi\Phi(\cdot,\lambda)\in L^2_\xi\). By the one-dimensional Sobolev embedding \(H^1\hookrightarrow C^{1/2}\), \(\Phi(\cdot,\lambda)\) possesses uniform Hölder-\(1/2\) continuity with respect to \(\xi\). The uniformity with respect to \(\lambda\in[c,d]\) is guaranteed by Section 7 of [15].
Fix \(\xi\in[a,b]\). Substitute 16 into 14 and treat each term separately. The \(\delta(\xi-\lambda)\) term, upon integration over \(\lambda\), gives \[\label{eq:delta95term} \mathcal{I}_\delta(\xi)=\frac{1}{2\pi}\psi(\xi)\big(+\mathrm{i}\pi\hat{g}_\xi(\xi)\big)=+\frac{\mathrm{i}}{2}\Phi(\xi,\xi).\tag{17}\] On the other hand, the principal value integral is \[\mathcal{I}_{PV}(t,\xi)=\frac{1}{2\pi}\,\mathrm{e}^{-\mathrm{i}t\xi^2}\,\mathrm{P.V.}\!\int_0^\infty\frac{\Phi(\xi,\lambda)}{\xi-\lambda}\,\mathrm{e}^{\mathrm{i}t\lambda^2}\,\mathrm{d}\lambda.\] Let \(u=\xi-\lambda\), yielding \[\mathcal{I}_{PV}(t,\xi)=\frac{1}{2\pi}\,\mathrm{P.V.}\!\int_{-\infty}^\xi\frac{\Phi(\xi,\xi-u)}{u}\,\mathrm{e}^{-\mathrm{i}t(2\xi u-u^2)}\,\mathrm{d}u.\] Introduce the change of variables \[\label{eq:v95subst} v:=2\xi u-u^2,\tag{18}\] and define \(G_\xi(v):=\Phi(\xi,\xi-u(v))\cdot\frac{2\xi-u(v)}{2(\xi-u(v))}\), so that \(G_\xi(0)=\Phi(\xi,\xi)\). We then obtain \[\label{eq:PV95transformed} \mathcal{I}_{PV}(t,\xi)=\frac{1}{2\pi}\,\mathrm{P.V.}\!\int_{-\infty}^{\xi^2-c^2}\frac{G_\xi(v)}{v}\,\mathrm{e}^{-\mathrm{i}tv}\,\mathrm{d}v.\tag{19}\] Let \(A(\xi):=\xi^2-c^2\), and we distinguish two cases. If \(A(\xi)>0\), using \(\mathrm{P.V.}\!\int_{-\infty}^{\infty}\frac{\mathrm{e}^{-\mathrm{i}tv}}{v}\,\mathrm{d}v=-\mathrm{i}\pi\) (\(t>0\)), decompose 19 as \[\begin{align} \mathcal{I}_{PV}(t,\xi)&=\frac{1}{2\pi}\int_{-\infty}^{A(\xi)}\frac{G_\xi(v)-G_\xi(0)}{v}\,\mathrm{e}^{-\mathrm{i}tv}\,\mathrm{d}v\\ &\quad+\frac{G_\xi(0)}{2\pi}\Big(\mathrm{P.V.}\!\int_{-\infty}^{\infty}\frac{\mathrm{e}^{-\mathrm{i}tv}}{v}\,\mathrm{d}v-\int_{A(\xi)}^\infty\frac{\mathrm{e}^{-\mathrm{i}tv}}{v}\,\mathrm{d}v\Big). \end{align}\] In the first term above, \(\frac{G_\xi(v)-G_\xi(0)}{v}\) belongs to \(L^1_v\) by Hölder-\(1/2\) continuity, and tends to \(0\) as \(t\to\infty\) by the Riemann-Lebesgue Lemma. In the second term above, the principal value integral equals \(-\mathrm{i}\pi\), and since \(A(\xi)>0\), the integration interval \([A(\xi),\infty)\) does not contain the singularity; integration by parts shows that it decays at the rate \(\mathcal{O}(t^{-1})\). Hence \[\lim_{t\to\infty}\mathcal{I}_{PV}(t,\xi)=\frac{G_\xi(0)}{2\pi}(-\mathrm{i}\pi)=-\frac{\mathrm{i}}{2}\Phi(\xi,\xi).\] If \(A(\xi)\le0\), the integral tends to \(0\) as \(t\to\infty\) by the Riemann-Lebesgue Lemma. In this case \(\xi\le c\) and \(\operatorname{supp}\psi=[c,d]\), so \(\psi(\xi)=0\) and \(\Phi(\xi,\xi)=0\). In summary, for any \(\xi\in[a,b]\), \[\label{eq:PV95limit} \lim_{t\to\infty}\mathcal{I}_{PV}(t,\xi)=-\frac{\mathrm{i}}{2}\Phi(\xi,\xi).\tag{20}\] From 17 and 20 , we obtain \[\label{eq:cancellation} \lim_{t\to\infty}\big(\mathcal{I}_\delta(\xi)+\mathcal{I}_{PV}(t,\xi)\big)=+\frac{\mathrm{i}}{2}\Phi(\xi,\xi)-\frac{\mathrm{i}}{2}\Phi(\xi,\xi)=0.\tag{21}\] One sees that each term in the above decomposition is uniformly controlled by an \(L^1_\xi\) function independent of \(t\). By Lebesgue Theorem, \[\lim_{t\to\infty}\mathcal{I}_\rho(t)=\int_a^b\lim_{t\to\infty}\big(\mathcal{I}_\delta(\xi)+\mathcal{I}_{PV}(t,\xi)\big)\,\mathrm{d}\xi=0.\] Therefore \(\langle\Omega_t^{u_0}(f_\psi),U_0(t)h\rangle\to\langle u_\infty^+[\psi],h\rangle\) holds for all test functions, and the weak limit 11 is established. Together with 10 , the Radon-Riesz Theorem yields the \(L^2\) strong convergence \[\lim_{t\to\infty}\big\|\Omega_t^{u_0}(f_c)-U_0(t)u_\infty^+\big\|_{L^2}=0.\] Adding the multisoliton from the discrete spectrum component together with \(u=2\operatorname{Re}(\Pi u)\), we obtain the conclusion of Theorem 3.
Theorem 5 follows by a similar argument as in the proof of Theorem 3, combined with the Gagliardo-Nirenberg inequality.
For the Lax operator, denote \(R_\lambda := (L_{u_0}-\lambda I)^{-1}\), where \(\lambda\notin\sigma(L_{u_0})\). For each negative eigenvalue \(\lambda<0\) of \(L_{u_0}\) and corresponding eigenfunction \(\varphi\), we have \[\label{eq46comm} (L_{u_0}-\lambda)(X^*\varphi) = -i\varphi + \frac{i}{2\pi}\,I_+(\varphi)\,\Pi u_0.\tag{22}\] This implies \[\label{eq46global95res} R_\lambda R_0(z) - R_0(z) R_\lambda = R_\lambda R_0(z)\,C_0\,R_0(z) R_\lambda,\tag{23}\] where \[\label{eq46C0} C_0 := i I_{L^2_+} - \frac{i}{2\pi}\,\Pi u_0\otimes I_+^*.\tag{24}\] This holds for all \(\lambda\notin\sigma(L_{u_0})\) and \(\operatorname{Im} z>0\).
Let \((\lambda_k)_{k\ge1}\) be the negative eigenvalues of \(L_{u_0}\), arranged in increasing order, and let \(\varphi_k\) be the corresponding normalized orthogonal eigenfunctions. Define \[\label{eq46Hd} \mathcal{H}_d := \overline{\operatorname{span}\{\varphi_k\}_{k\ge1}} \subset L^2_+(\mathbb{R}), \qquad \mathcal{H}_c := \mathcal{H}_d^\perp.\tag{25}\] Denote by \(P_d\) and \(P_c:=I-P_d\) the orthogonal projections onto \(\mathcal{H}_d\) and \(\mathcal{H}_c\), respectively, and define \[\label{eq46LdLc} L_d := L_{u_0}|_{\mathcal{H}_d}, \qquad L_c := L_{u_0}|_{\mathcal{H}_c}.\tag{26}\] The assumption \(\sum_{k\ge1}|\lambda_k|<\infty\) implies \[\label{eq46Ld95trace} L_d \in \mathfrak{S}_1(\mathcal{H}_d).\tag{27}\] In particular, \(L_d\) is bounded with \(\|L_d\|\le |\lambda_1|\), \(L_c\) is self-adjoint on \(\mathcal{H}_c\), and \(\|\Pi u_0\|_{L^2}^2 = \sum_k 2\pi|\lambda_k| = \sum_k |\langle\Pi u_0,\varphi_k\rangle|^2\), hence \(\Pi u_0\in\mathcal{H}_d\).
Define \[\label{eq46W95def} W(z) := P_c\,R_0(z)\,P_d : \mathcal{H}_d \longrightarrow \mathcal{H}_c, \qquad \operatorname{Im} z > 0.\tag{28}\] For each \(\operatorname{Im} z>0\), \(W(z)\) is a bounded operator, and \(z\mapsto W(z)\) is analytic in the upper half-plane. In the sequel, we prove that \(W(z)\equiv0\) is equivalent to \(\mathcal{H}_d\) being invariant under \(R_0(z)\).
Multiply identity 23 on the left by \(P_c\) and on the right by \(P_d\). Using \(P_c R_\lambda = R_\lambda^c P_c\) and \(R_\lambda P_d = P_d R_\lambda^d\), we obtain \[\label{eq46projected} R_\lambda^c\,W(z) - W(z)\,R_\lambda^d = R_\lambda^c\, \underbrace{P_c R_0(z) C_0 R_0(z) P_d}_{=:K(z)}\, R_\lambda^d.\tag{29}\] Expanding \(K(z)\) yields \[\begin{align} K(z) &= i P_c R_0(z)^2 P_d - \frac{i}{2\pi} P_c R_0(z)(\Pi u_0\otimes I_+^*) R_0(z) P_d. \label{eq46K95expand} \end{align}\tag{30}\] For \(i P_c R_0(z)^2 P_d\) we have \[\begin{align} P_c R_0(z)^2 P_d &= P_c R_0(z)(P_c+P_d) R_0(z) P_d \nonumber\\ &= (P_c R_0(z) P_c)(P_c R_0(z) P_d) + (P_c R_0(z) P_d)(P_d R_0(z) P_d) \nonumber\\ &=: R_c(z) W(z) + W(z) R_d(z), \label{eq46R295decomp} \end{align}\tag{31}\] where \(R_c(z):=P_c R_0(z) P_c\) and \(R_d(z):=P_d R_0(z) P_d\). For \(\frac{i}{2\pi} P_c R_0(z)(\Pi u_0\otimes I_+^*) R_0(z) P_d\), using \(P_c\Pi u_0 = 0\) we obtain \[\begin{align} P_c R_0(z)(\Pi u_0\otimes I_+^*) R_0(z) P_d &= P_c R_0(z) P_d \Pi u_0 \otimes I_+^* R_0(z) P_d \nonumber\\ &= W(z) \Pi u_0 \otimes I_+^* R_0(z) P_d. \label{eq46rank195term} \end{align}\tag{32}\] Substituting, we obtain \[\begin{align} K(z) = i R_c(z) W(z) + W(z)\Bigl( i R_d(z) - \frac{i}{2\pi} \Pi u_0 \otimes I_+^* R_0(z) P_d \Bigr). \label{eq46K95final} \end{align}\tag{33}\] Substituting back into the projected identity, multiplying on the left by \(L_c-\lambda\) and on the right by \(L_d-\lambda\), we obtain \[\begin{align} W(z) L_d - L_c W(z) = i R_c(z) W(z) + W(z)\Bigl( i R_d(z) - \frac{i}{2\pi} \Pi u_0 \otimes I_+^* R_0(z) P_d \Bigr). \label{eq46master} \end{align}\tag{34}\] Rearranging terms yields the following Sylvester equation \[\label{eq46Sylvester} A(z)\,W(z) - W(z)\,B(z) = 0, \qquad \operatorname{Im} z > 0,\tag{35}\] where \[\begin{align} A(z) &:= L_c + i P_c R_0(z) P_c, \tag{36}\\ B(z) &:= L_d - i P_d R_0(z) P_d + \frac{i}{2\pi} \Pi u_0 \otimes I_+^* R_0(z) P_d. \tag{37} \end{align}\] Thus, \(A(z)\) is a closed operator on \(\mathcal{H}_c\), and \(B(z)\) is a bounded operator on \(\mathcal{H}_d\).
We now evaluate 35 along the imaginary axis \(z=i\eta\). We first prove that for sufficiently large \(\eta\), \(\sigma(A(i\eta))\) and \(\sigma(B(i\eta))\) can be separated. Since the spectrum of \(A(i\eta)\) can be expressed via \(A(i\eta) = L_c + i R_c(i\eta)\) with \(L_c\ge 0\) and \(\|R_c(i\eta)\|\le 1/\eta\), for \(f\in\mathcal{D}(L_c)\) we have \[\label{eq46A95numerical} \operatorname{Re}\langle A(i\eta)f,f\rangle = \langle L_c f,f\rangle + \operatorname{Re}\langle i R_c(i\eta)f,f\rangle \ge 0 - \frac{1}{\eta}\|f\|^2 = -\frac{1}{\eta}\|f\|^2.\tag{38}\] Consequently, \[\label{eq46A95spectrum} \sigma(A(i\eta)) \subset \{\zeta\in\mathbb{C}: \operatorname{Re}\zeta \ge -1/\eta\}.\tag{39}\] On the other hand, the spectrum of \(B(i\eta)\) can be expressed via \(B(i\eta) = L_d + \Delta(i\eta)\), where \[\label{eq46Delta} \Delta(i\eta) := -i R_d(i\eta) + \frac{i}{2\pi} \Pi u_0 \otimes I_+^* R_0(i\eta) P_d.\tag{40}\] For \(f\in\mathcal{H}_d\) (\(\|f\|=1\)), let \(g=R_0(i\eta)f\). From \(-X^*R_0=I-i\eta R_0\) we obtain \[\begin{align} \frac{1}{4\pi}|I_+(g)|^2 &= \operatorname{Im}\langle -X^* g, g\rangle = \operatorname{Im}\langle f - i\eta R_0 f, R_0 f\rangle \nonumber\\ &\le \|f\|\,\|R_0 f\| \le \frac{1}{\eta}, \label{eq46Iplus95bound} \end{align}\tag{41}\] hence \(|I_+(R_0(i\eta)f)| \le 2\sqrt{\pi/\eta}\), and \[\label{eq46functional95bound} \|I_+^* R_0(i\eta) P_d\|_{\mathcal{H}_d\to\mathbb{C}} \le 2\sqrt{\frac{\pi}{\eta}}.\tag{42}\] Therefore, \[\label{eq46Delta95bound} \|\Delta(i\eta)\| \le \frac{1}{\eta} + \frac{\|\Pi u_0\|}{\sqrt{\pi\eta}}.\tag{43}\] For sufficiently large \(\eta\), we have \(\|\Delta(i\eta)\|\le |\lambda_1|/3\), which implies \[\label{eq46B95spectrum} \sigma(B(i\eta)) \subset \{\zeta\in\mathbb{C}: \operatorname{Re}\zeta \le \lambda_1 + |\lambda_1|/3 = 2\lambda_1/3 < 0\}.\tag{44}\] Thus, choose \(\eta_0\) sufficiently large so that \(1/\eta_0 < |\lambda_1|/3\); then for all \(\eta\ge\eta_0\) we have \[\label{eq46separation} \sigma(A(i\eta)) \subset \{\operatorname{Re}\zeta \ge -1/\eta\} \subset \{\operatorname{Re}\zeta > \lambda_1/2\}, \qquad \sigma(B(i\eta)) \subset \{\operatorname{Re}\zeta \le 2\lambda_1/3 < \lambda_1/2\}.\tag{45}\] The two spectra are separated by the vertical line \(\operatorname{Re}\zeta = \lambda_1/2\). Since \(\sigma(A(i\eta))\) and \(\sigma(B(i\eta))\) are disjoint, the Sylvester equation \(AW - WB = 0\) yields \(W=0\) by Rosenblum’s theorem. Take a contour \(\Gamma\) enclosing \(\sigma(B(i\eta))\) and lying entirely within \(\rho(A(i\eta))\). From \(AW=WB\) we obtain, for \(\zeta\in\Gamma\), \[\label{eq46resolvent95sylvester} (A(i\eta)-\zeta)^{-1} W(i\eta) = W(i\eta) (B(i\eta)-\zeta)^{-1}.\tag{46}\] Integrating both sides along \(\Gamma\) yields \[\label{AB} \frac{1}{2\pi i}\oint_\Gamma (A-\zeta)^{-1} W\,d\zeta = \frac{1}{2\pi i}\oint_\Gamma W (B-\zeta)^{-1}\,d\zeta.\tag{47}\] Since \(\Gamma\) contains no spectrum of \(A\), \((A-\zeta)^{-1}\) is analytic on and inside \(\Gamma\), hence the left-hand side of 47 equals \(0\cdot W = 0\). For the right-hand side of 47 , \(\Gamma\) encloses the entire spectrum of the bounded operator \(B\), so the integral equals the Riesz projection \(I_{\mathcal{H}_d}\). The right-hand side becomes \(W\cdot I = W\). Thus \(0 = W(i\eta)\), i.e., \(W(i\eta)=0\) for all \(\eta\ge\eta_0\).
The map \(z\mapsto W(z)\) is analytic on the connected open set \(\{\operatorname{Im} z>0\}\) and vanishes identically on \(z=i\eta\) (\(\eta\ge\eta_0\)), which implies \(W(z) \equiv 0,\enspace \operatorname{Im} z > 0.\) Hence \(P_c R_0(z) P_d = 0\) for all \(\operatorname{Im} z>0\), i.e., \(\mathcal{H}_d\) is invariant under \(R_0(z)\). Since \[\label{eq46Rz95restricted} R(z) := R_0(z)\big|_{\mathcal{H}_d} : \mathcal{H}_d \longrightarrow \mathcal{H}_d, \qquad \operatorname{Im} z > 0,\tag{48}\] is a globally bounded operator on \(\mathcal{H}_d\) with \(\|R(z)\|\le (\operatorname{Im} z)^{-1}\), it follows that \(\mathcal{H}_d\) is invariant under \(R_\lambda\), and we denote \(R_\lambda^d := R_\lambda|_{\mathcal{H}_d} = (L_d-\lambda I)^{-1}\). Since \(\mathcal{H}_d\) is invariant under both \(R_\lambda\) and \(R_0(z)\), we obtain \[\label{eq46restricted95res} R_\lambda^d R(z) - R(z) R_\lambda^d = R_\lambda^d R(z)\,C\,R(z) R_\lambda^d,\tag{49}\] where \(C := C_0|_{\mathcal{H}_d} = i I_{\mathcal{H}_d} - \frac{i}{2\pi} \Pi u_0\otimes I_+^*\).
Since \(L_d\) is bounded, for \(|\lambda| > \|L_d\|\), \(R_\lambda^d\) admits a norm-convergent Neumann series \[\label{eq46Neumann} R_\lambda^d = -\frac{1}{\lambda} I - \frac{1}{\lambda^2} L_d - \frac{1}{\lambda^3} L_d^2 - \cdots,\tag{50}\] Substituting this into 49 , the left-hand side becomes \[\begin{align} R_\lambda^d R(z) - R(z) R_\lambda^d &= \Bigl(-\frac{1}{\lambda}I - \frac{1}{\lambda^2}L_d + \mathcal{O}(\lambda^{-3})\Bigr) R(z) - R(z)\Bigl(-\frac{1}{\lambda}I - \frac{1}{\lambda^2}L_d + \mathcal{O}(\lambda^{-3})\Bigr) \nonumber\\ &= -\frac{1}{\lambda^2}[L_d, R(z)] + \mathcal{O}(\lambda^{-3}). \label{eq46LHS95expand} \end{align}\tag{51}\] The right-hand side becomes \[\begin{align} R_\lambda^d R(z) C R(z) R_\lambda^d &= \Bigl(-\frac{1}{\lambda}I + \mathcal{O}(\lambda^{-2})\Bigr) R(z) C R(z) \Bigl(-\frac{1}{\lambda}I + \mathcal{O}(\lambda^{-2})\Bigr) \nonumber\\ &= \frac{1}{\lambda^2} R(z) C R(z) + \mathcal{O}(\lambda^{-3}). \label{eq46RHS95expand} \end{align}\tag{52}\] This identity holds for all \(|\lambda|>\|L_d\|\) (\(\lambda\notin\sigma(L_d)\)). Equating the \(\lambda^{-2}\) coefficients of the Laurent series yields \[\label{eq46commutator95identity} -[L_d, R(z)] = R(z) C R(z).\tag{53}\] Expanding \(C\) gives \[\label{eq46commutator95expanded} [L_d, R(z)] = -i R(z)^2 + \frac{i}{2\pi} R(z)(\Pi u_0\otimes I_+^*) R(z).\tag{54}\] Since \(L_d\in\mathfrak{S}_1(\mathcal{H}_d)\), we have \([L_d, R(z)]\in\mathfrak{S}_1(\mathcal{H}_d)\). Because \(R(z)(\Pi u_0\otimes I_+^*) R(z) = (R(z)\Pi u_0)\otimes(I_+^*\circ R(z)^*)\) is a rank-one operator, it belongs to \(\mathfrak{S}_1\), and one readily sees that their difference is also trace class, i.e., \[\label{eq46R295trace95class} R(z)^2 \in \mathfrak{S}_1(\mathcal{H}_d).\tag{55}\] In particular, the operator \(M:=-X^*|_{\mathcal{H}_d}\) has purely discrete spectrum, consisting of finitely many eigenvalues.
Let \(N\ge 1\) and let \(P_N\) be the orthogonal projection onto \(\operatorname{span}\{\varphi_1,\dots,\varphi_N\}\). Since \(\{\varphi_k\}\) are eigenfunctions of \(L_d\), \(P_N L_d = L_d P_N\). Multiplying the commutator identity on both sides by \(P_N\) \[\label{eq46PN95commutator} P_N[L_d, R(z)]P_N = -i P_N R(z)^2 P_N + \frac{i}{2\pi} P_N R(z)(\Pi u_0\otimes I_+^*) R(z) P_N.\tag{56}\] Using \(P_N L_d = L_d P_N\), the left-hand side simplifies to \[\begin{align} P_N[L_d, R(z)]P_N &= P_N L_d R(z) P_N - P_N R(z) L_d P_N \nonumber\\ &= L_d P_N R(z) P_N - P_N R(z) P_N L_d \nonumber\\ &= [L_d P_N, P_N R(z) P_N]. \label{eq46PN95comm95simplified} \end{align}\tag{57}\] Since \[\label{eq46PN95trace} 0 = -i\operatorname{Tr}(P_N R(z)^2 P_N) + \frac{i}{2\pi} \operatorname{Tr}\bigl( P_N R(z)(\Pi u_0\otimes I_+^*) R(z) P_N \bigr)\tag{58}\] and \[\begin{align} \operatorname{Tr}\bigl( P_N R(z)(\Pi u_0\otimes I_+^*) R(z) P_N \bigr) &= \operatorname{Tr}\bigl( (P_N R(z)\Pi u_0) \otimes (I_+^*\circ R(z) P_N) \bigr) \nonumber\\ &= I_+\bigl( R(z) P_N R(z) \Pi u_0 \bigr), \label{eq46rank95one95trace} \end{align}\tag{59}\] we obtain \[\label{eq46trace95identity95N} \operatorname{Tr}(P_N R(z)^2 P_N) = \frac{1}{2\pi}\, I_+\bigl( R(z) P_N R(z) \Pi u_0 \bigr).\tag{60}\] This implies \[\label{eq46trace95norm95convergence} P_N R(z)^2 P_N \xrightarrow{\mathfrak{S}_1} R(z)^2 \qquad (N\to\infty),\tag{61}\] hence \(\operatorname{Tr}(P_N R(z)^2 P_N) \to \operatorname{Tr}(R(z)^2)\). For the right-hand side of 49 , set \(v_N := R(z) P_N R(z) \Pi u_0\). Since \(R(z):\mathcal{H}_d\to\mathcal{D}(M)\subset\mathcal{D}(X^*)\) is a bounded map into \(\mathcal{D}(X^*)\), and \(P_N R(z)\Pi u_0\to R(z)\Pi u_0\) strongly in \(\mathcal{H}_d\), it follows that \(v_N\to R(z)^2\Pi u_0\) in the topology of \(\mathcal{D}(X^*)\). Since \(I_+\) is continuous on \(\mathcal{D}(X^*)\), we have \(I_+(v_N)\to I_+(R(z)^2\Pi u_0)\). Passing to the limit \(N\to\infty\) yields \[\label{eq46exact95trace} \operatorname{Tr}(R(z)^2) = \frac{1}{2\pi}\, I_+\bigl( R(z)^2 \Pi u_0 \bigr), \qquad \operatorname{Im} z > 0.\tag{62}\]
Define the analytic function on the upper half-plane \[\label{eq46F95def} F(z) := \frac{1}{2\pi}\, I_+\bigl( R(z) \Pi u_0 \bigr).\tag{63}\] Using \(\frac{d}{dz} R(z) = -R(z)^2\), \[\label{eq46F95derivative} F'(z) = -\frac{1}{2\pi}\, I_+\bigl( R(z)^2 \Pi u_0 \bigr) = -\operatorname{Tr}(R(z)^2).\tag{64}\] Denote the eigenvalues of \(M\) by \(\{-p_m\}_{m=1}^\infty\). Since \(M\) is the restriction of \(-X^*\), its numerical range lies in the upper half-plane, hence \(\operatorname{Im} p_m > 0\). Because \(R(z)^2 = (M+zI)^{-2}\) has eigenvalues \((z+p_m)^{-2}\), Lidskii’s trace theorem gives \[\label{eq46Lidskii} \operatorname{Tr}(R(z)^2) = \sum_{m=1}^\infty \frac{1}{(z+p_m)^2}.\tag{65}\] Substituting into \(F'(z)\) and integrating from \(+i\infty\) to \(z\) along the upper half-plane yields \[\label{eq46F95integrated} F(z)= F(+i\infty) - \int_{+i\infty}^z F'(\zeta)\,d\zeta = \sum_{m=1}^\infty \frac{1}{z+p_m}.\tag{66}\] Since \[\label{eq46explicit95formula} \Pi u_0(z) = \frac{1}{2\pi i}\, I_+\bigl( (X^*-z)^{-1} \Pi u_0 \bigr) = \frac{1}{2\pi i}\, I_+\bigl( -(M+z)^{-1} \Pi u_0 \bigr) = -\frac{1}{2\pi i}\, I_+\bigl( R(z) \Pi u_0 \bigr),\tag{67}\] we have \[\label{eq46Piu95meromorphic} \Pi u_0(z) = i\,F(z) = i\sum_{m=1}^\infty \frac{1}{z+p_m}, \qquad \operatorname{Im} z > 0.\tag{68}\] Thus \(\Pi u_0\) is a meromorphic function in the upper half-plane with simple poles at \(-p_m\), each of residue \(i\). Consequently, \[\label{eq46u095from95Pi} u_0(x) = 2\,\operatorname{Re}\bigl( \Pi u_0(x) \bigr) = 2\,\operatorname{Re}\Bigl( i\sum_{m=1}^\infty \frac{1}{x+p_m} \Bigr) = \sum_{m=1}^\infty \frac{2\operatorname{Im} p_m}{|x+p_m|^2}, \qquad x\in\mathbb{R}.\tag{69}\] Computing the \(L^2\) norm of \(\Pi u_0\) via the residue theorem yields \[\|\Pi u_0\|_{L^2}^2 = \int_\mathbb{R} \Bigl| i\sum_{j=1}^\infty \frac{1}{x+p_j} \Bigr|^2\,dx = \sum_{j,k=1}^\infty \frac{2\pi i}{p_j - \overline{p_k}} = 2\pi \sum_{j,k=1}^\infty \frac{\operatorname{Im} p_j + \operatorname{Im} p_k}{(\operatorname{Re} p_j - \operatorname{Re} p_k)^2 + (\operatorname{Im} p_j + \operatorname{Im} p_k)^2}.\] Applying Tonelli’s theorem together with the hypothesis \(\|\Pi u_0\|_{L^2}^2<\infty\), we obtain \[\label{eq46double95sum95finite} \sum_{j=1}^\infty \sum_{k=1}^\infty \frac{\operatorname{Im} p_j}{(\operatorname{Re} p_j - \operatorname{Re} p_k)^2 + (\operatorname{Im} p_j + \operatorname{Im} p_k)^2} < \infty.\tag{70}\] Note that \(|p_j - \overline{p_k}|^2 = (\operatorname{Re} p_j - \operatorname{Re} p_k)^2 + (\operatorname{Im} p_j + \operatorname{Im} p_k)^2\). This completes the proof of Proposition 7.
Under the assumptions of Theorem 9, the orthogonal projection of the initial data onto the Hardy space can be expressed as \(\Pi u_0(x) = i \sum_{j=1}^\infty \frac{1}{x + p_j}\), which leads to \[\label{5jifen} \|\Pi u_0\|_{L^2(\mathbb{R})}^2 = \int_{-\infty}^\infty \left( i \sum_{j=1}^\infty \frac{1}{x + p_j} \right) \left( -i \sum_{k=1}^\infty \frac{1}{x + \overline{p_k}} \right) \mathrm{d}x.\tag{71}\] For the integral \(\int_{-\infty}^\infty \frac{1}{(x+p_j)(x+\overline{p_k})} \mathrm{d}x\), by closing the contour in the upper half-plane and applying the residue theorem, the contour encloses only the pole \(z = -\overline{p_k}\). Thus, we calculate \[\int_{-\infty}^\infty \frac{1}{(x+p_j)(x+\overline{p_k})} \mathrm{d}x = 2\pi i \cdot \operatorname{Res}_{z = -\overline{p_k}} \left[ \frac{1}{(z + p_j)(z + \overline{p_k})} \right] = \frac{2\pi i}{p_j - \overline{p_k}}.\] Substituting this back into 71 and using the symmetry of the indices \((j,k)\), we obtain \[\|\Pi u_0\|_{L^2}^2 = \frac{1}{2} \sum_{j=1}^\infty \sum_{k=1}^\infty \left( \frac{2\pi i}{p_j - \overline{p_k}} + \frac{2\pi i}{p_k - \overline{p_j}} \right) = 4\pi \sum_{j=1}^\infty \sum_{k=1}^\infty \frac{\mathrm{Im}(p_j)}{|p_j - \overline{p_k}|^2}.\]
Based on the resolvent projection formula derived from the explicit formula \[\Pi u(t, x) = \frac{1}{2\pi i} I_+ \left( (X^* - 2t L_{u_0} - x)^{-1} \Pi u_0 \right),\] by expanding it under the orthonormal eigenbasis \(\{\varphi_k\}_{k=1}^\infty\) of the Lax operator, the explicit formula for the solution can be expressed in the following form: \[\label{form3} \Pi u(t, x) = i \sum_{j=1}^\infty \sum_{k=1}^\infty W_{jk} [D(t, x) + E]^{-1}_{jk},\tag{72}\] where the matrices are defined as follows:
Diagonal matrix: \(D_{kk}(t, x) = x - c_k t + x_k^\infty + i y_k^\infty\); \(D_{jk} = 0\) for \(j \neq k\).
Off-diagonal matrix: \(E_{kk} = 0\); \(E_{jk} = \frac{2i (y_k^\infty)^{3/2} (y_j^\infty)^{1/2}}{y_k^\infty - y_j^\infty} e^{i(\theta_j - \theta_k)}\) for \(j \neq k\).
Coefficient matrix: \(W_{jk} = \sqrt{\frac{y_j^\infty}{y_k^\infty}} e^{i(\theta_k - \theta_j)}\).
We introduce a positive integer \(N \ge 1\) and truncate the above matrices into finite-dimensional submatrices of the first \(N\) solitons. Let \(\mathcal{E}_{\mathrm{tail}}^{(N)}\) be the sum from the \((N+1)\)-th soliton to the infinite solitons, namely \(\mathcal{E}_{\mathrm{tail}}^{(N)}=\sum_{k=N+1}^{\infty}{R_{p_{k}^{\infty}}\left( x-c_{p_{k}^{\infty}}t \right)}\), which leads to \[\label{gaopin} \|\mathcal{E}_{\mathrm{tail}}^{(N)}(t, \cdot)\|_{L^\infty} \le \sum_{k=N+1}^\infty 4|\lambda_k|\to 0, \quad N\to\infty.\tag{73}\]
For the truncated \(N \times N\) submatrices, consider \(\left( D_{N}^{-1}E_N \right) ^2\). We have \[\left[ \left( D_{N}^{-1}E_N \right) ^2 \right] _{jk}=\sum_{l=1}^N{\left( D_{N}^{-1}E_N \right) _{jl}\left( D_{N}^{-1}E_N \right) _{lk}}=\sum_{l=1}^N{D_{jj}^{-1}E_{jl}D_{ll}^{-1}E_{lk}}.\] Since the diagonal elements of \(E_N\) are all \(0\), it follows that each component on the right-hand side of the above equation contains a coefficient of the form \(D_{jj}^{-1}(t,x) D_{ll}^{-1}(t,x)\). Given that \(\left| D_{jj} \right|+\left| D_{ll} \right|\geqslant \left| c_j-c_l \right|t\), this implies \(\frac{1}{\left| D_{jj}D_{ll} \right|}\leqslant \mathcal{O} \left( \frac{1}{t} \right).\) Therefore, when the time \(t\) is sufficiently large, the spectral radius of \(D^{-1}_{N}E_N\) is less than \(1\). Consequently, we have \[\label{niuman} [D_N + E_N]^{-1} = D_N^{-1} - D_N^{-1}E_N D_N^{-1} + D_N^{-1}E_N D_N^{-1}E_N D_N^{-1} - \dots.\tag{74}\]
Substituting the \(0\)-th order term \(D_N^{-1}\) of the expansion into 72 , we obtain \(\Pi u^{\left( 0 \right)}\left( t,x \right) =i\sum_{k=1}^N{W_{kk}D_{kk}^{-1}}\). Since \(W_{kk}=1\), we arrive at \[\label{0jie} \Pi u^{\left( 0 \right)}\left( t,x \right) =\sum_{k=1}^N{\frac{2y_{k}^{\infty}}{\left( x-c_{p^\infty_k}t+x_{k}^{\infty} \right)^2 +\left( y_{k}^{\infty} \right) ^2}}=u_{\mathrm{sol}}^{N}\left( t,x \right).\tag{75}\]
Similarly, substituting the \(1\)-st order term of the expansion 74 into 72 yields \[\label{1jie} \mathcal{E}^{(1)}_N(t,x) = -i \sum_{j=1}^N \sum_{k=1}^N \underbrace{\big( W_{jk} E_{jk} \big)}_{:=A_{jk}} D_{jj}^{-1}(t,x) D_{kk}^{-1}(t,x),\tag{76}\] where \(A_{jk}\) is given by \[A_{jk} = \left( \sqrt{\frac{y_j^\infty}{y_k^\infty}} e^{i(\theta_k - \theta_j)} \right) \left( \frac{2i (y_k^\infty)^{3/2} (y_j^\infty)^{1/2}}{y_k^\infty - y_j^\infty} e^{i(\theta_j - \theta_k)} \right) = \frac{2i y_j^\infty y_k^\infty}{y_k^\infty - y_j^\infty}.\] This implies \(A_{kj} = -A_{jk}\). Since \[\sum_{j,k} A_{jk} D_{jj}^{-1} D_{kk}^{-1} = \sum_{k,j} A_{kj} D_{kk}^{-1} D_{jj}^{-1} = \sum_{j,k} (-A_{jk}) D_{jj}^{-1} D_{kk}^{-1},\] we obtain \(\mathcal{E}^{(1)}_N(t,x) = 0\).
Substituting the \(2\)-nd order term of the expansion 74 into 72 , we obtain \[\mathcal{E}^{(2)}_N(t,x) = i \sum_{\substack{j, k, l \le N \\ j \neq l, k \neq l}} W_{jk} E_{jl} E_{lk} \frac{1}{D_j D_l D_k}.\] Taking the absolute value of the numerator and substituting \(y_m^\infty = \frac{1}{2|\lambda_m|}\) along with the identity \(|y_a^\infty - y_b^\infty| = 2 y_a^\infty y_b^\infty |\lambda_a - \lambda_b|\), it follows that: \[|W_{jk} E_{jl} E_{lk}| = \frac{4 y_j^\infty (y_l^\infty)^2 y_k^\infty}{|y_l^\infty - y_j^\infty| \cdot |y_k^\infty - y_l^\infty|} = \frac{1}{|\lambda_j - \lambda_l| \cdot |\lambda_k - \lambda_l|}.\] For the denominator, each single term satisfies \(|D_k| \ge \mathrm{Im}(D_{kk}) = y_k^\infty = \frac{1}{2|\lambda_k|}\). For any \(j \neq l\), employing the triangle inequality yields \[|D_j| + |D_l| \ge \big| (x - c_j t + x_j^\infty) - (x - c_l t + x_l^\infty) \big| \ge |c_j - c_l|t - |x_j^\infty - x_l^\infty|.\] Substituting \(|c_j - c_l| = 2|\lambda_j - \lambda_l|\), when the time \(t\) is sufficiently large, we have \(|D_j| + |D_l| \ge |\lambda_j - \lambda_l|t\). Using the inequality \(|D_j D_l| \ge \frac{1}{2} \min(|D_j|, |D_l|) (|D_j| + |D_l|)\) and the fact that \(\min(|D_j|, |D_l|) \ge y_1^\infty = \frac{1}{2|\lambda_1|}\), we obtain: \[\frac{1}{|D_j D_l|} \le \frac{2}{y_1^\infty ( |D_j| + |D_l| )} \le \frac{4|\lambda_1|}{|\lambda_j - \lambda_l| t}.\] According to the above estimation process, we arrive at \[\label{erjie} \|\mathcal{E}_{\mathrm{N}}^{(2)}(t, \cdot)\|_{L^\infty} \le \frac{C'}{t} \sum_{\substack{j, k, l \le N \\ j \neq l, k \neq l}} \frac{|\lambda_k|}{|\lambda_j - \lambda_l|^2 \cdot |\lambda_k - \lambda_l|}.\tag{77}\]
For the third term and all subsequent expansion terms in 74 , let \(A := D_N^{-1} E_N\). We have already proved that the norm of \(A^2\) is controlled by \(C_N t^{-1}\), where \(C_N\) is a constant depending on the truncation \(N\). This implies that the errors \(\mathcal{E}_{\mathrm{N}}^{(j)}, \enspace j > 2\), obtained by substituting the third and all subsequent expansion terms of 74 into 72 , are all controlled by \(C_N t^{-1}\).
In summary, the error between the exact solution \(u(t, x)\) and \(u_{\mathrm{sol}}(t, x)\) is controlled by the sum of the error 73 and the error 77 . Since the truncation parameter \(N \ge 1\) is arbitrary, for any \(t > 0\), we obtain \[\label{zongwucha} \left\| u(t, \cdot) - u_{\mathrm{sol}}(t, \cdot) \right\|_{L^\infty(\mathbb{R})} \le C \inf_{N \ge 1} \left\{ \sum_{k=N+1}^\infty |\lambda_k| + \frac{1}{t} \sum_{\substack{j, k, l \le N \\ j \neq l, k \neq l}} \frac{|\lambda_k|}{|\lambda_j - \lambda_l|^2 \cdot |\lambda_k - \lambda_l|} \right\}.\tag{78}\]
Let \(\mathcal{E}_{\mathrm{stat}}(N)=\sum_{k=N+1}^\infty |\lambda_k|\) and \[\mathcal{E}_{\mathrm{dyn}}(N)=\sum_{l=1}^N \sum_{\substack{j=1 \\ j \neq l}}^N \sum_{\substack{k=1 \\ k \neq l}}^N \frac{|\lambda_k|}{|\lambda_j - \lambda_l|^2 \cdot |\lambda_k - \lambda_l|}.\]
Under the condition \(|\lambda_k| \le C_1 k^{-\alpha}\) with \(\alpha > 1\), we have \[\label{proof95static} \mathcal{E}_{\mathrm{stat}}(N) = \sum_{k=N+1}^\infty |\lambda_k| \le C_1 \sum_{k=N+1}^\infty k^{-\alpha} \le C_1 \int_N^\infty x^{-\alpha} \,\mathrm{d}x = \frac{C_1}{\alpha - 1} N^{1-\alpha},\tag{79}\] which implies that \(\mathcal{E}_{\mathrm{stat}}(N) = \mathcal{O}\left( N^{1-\alpha} \right)\).
For \(\mathcal{E}_{\mathrm{dyn}}(N)\), we have \[\mathcal{E}_{\mathrm{dyn}}(N) = \sum_{l=1}^N \left( \sum_{\substack{j=1 \\ j \neq l}}^N \frac{1}{|\lambda_j - \lambda_l|^2} \right) \left( \sum_{\substack{k=1 \\ k \neq l}}^N \frac{|\lambda_k|}{|\lambda_k - \lambda_l|} \right) := \sum_{l=1}^N A_l \cdot B_l.\]
Based on the condition \(|\lambda_j - \lambda_l| \ge C_2 |j - l| \max(j,l)^{-\alpha-1}\), the following estimate for \(A_l\) holds: \[\frac{1}{|\lambda_j - \lambda_l|^2} \le \frac{\max(j,l)^{2\alpha+2}}{C_2^2 |j-l|^2}.\] By using \(\sum_{m=1}^\infty m^{-2} = \pi^2/6\), we further obtain the following estimate: \[\label{Alguji} A_l \le \frac{N^{2\alpha+2}}{C_2^2} \sum_{\substack{j=1 \\ j \neq l}}^N \frac{1}{|j-l|^2} \le \frac{N^{2\alpha+2}}{C_2^2} \left( 2 \sum_{m=1}^\infty \frac{1}{m^2} \right) = \frac{\pi^2}{3 C_2^2} N^{2\alpha+2} = \mathcal{O}\left( N^{2\alpha+2} \right).\tag{80}\]
For \(B_l = \sum_{k \neq l} \frac{|\lambda_k|}{|\lambda_k - \lambda_l|}\), we discuss the two cases \(k > l\) and \(k < l\). When \(k > l\), we have \(|\lambda_k| < |\lambda_l|\) and \(\max(k,l) = k\). In this case, the denominator is \(|\lambda_k - \lambda_l| = |\lambda_l| - |\lambda_k| \ge C_2 (k-l) k^{-\alpha-1}\). Scaling the numerator as \(|\lambda_k| \le C_1 k^{-\alpha}\), we obtain \(\frac{|\lambda _k|}{|\lambda _k-\lambda _l|}\le \frac{C_1}{C_2}\frac{N}{k-l}\). Summing over \(k\), we have \[\label{qingkaung1} \sum_{k=l+1}^N \frac{C_1 N}{C_2 (k-l)} = \frac{C_1 N}{C_2} \sum_{m=1}^{N-l} \frac{1}{m} \le \frac{C_1 N}{C_2} \big( 1 + \ln(N-l) \big) = \mathcal{O}(N \ln N).\tag{81}\] When \(k < l\), we have \(|\lambda_k| > |\lambda_l|\) and \(\max(k,l) = l\). Since \(\frac{|\lambda _k|}{|\lambda _k-\lambda _l|}=1+\frac{|\lambda _l|}{|\lambda _k|-|\lambda _l|},\) substituting \(|\lambda_l| \le C_1 l^{-\alpha}\) and \(|\lambda_k - \lambda_l| \ge C_2 (l-k) l^{-\alpha-1}\) obtained from the condition 6 into the above expression yields \(\frac{|\lambda _l|}{|\lambda _k|-|\lambda _l|}\le \frac{C_1}{C_2}\frac{N}{l-k}.\) Summing over \(k\), we have \[\label{qingkaung2} \sum_{k=1}^{l-1} \left( 1 + \frac{C_1 N}{C_2 (l-k)} \right) \le N + \frac{C_1 N}{C_2} \sum_{m=1}^{l-1} \frac{1}{m} = \mathcal{O}(N \ln N).\tag{82}\] Combining 81 and 82 , we obtain that for any \(l \le N\), \(B_l = \mathcal{O}(N \ln N)\) holds.
Substituting the estimates of \(A_l\) and \(B_l\) into \(\mathcal{E}_{\mathrm{dyn}}(N)\), we obtain \[\label{proof95dynamic} \mathcal{E}_{\mathrm{dyn}}(N) = \sum_{l=1}^N A_l B_l \le \sum_{l=1}^N \mathcal{O}(N^{2\alpha+2}) \cdot \mathcal{O}(N \ln N) \le \mathcal{O}\left( N^{2\alpha+5} \right).\tag{83}\]
Substituting the estimate 79 for \(\mathcal{E}_{\mathrm{stat}}(N)\) and the estimate 83 for \(\mathcal{E}_{\mathrm{dyn}}(N)\) back into 78 and setting \(t = N^{3\alpha+4}\) , we have \[\left\| u(t, \cdot) - u_{\mathrm{sol}}(t, \cdot) \right\|_{L^\infty} = \mathcal{O}\left( \left( t^{\frac{1}{3\alpha+4}} \right)^{1-\alpha} \right) = \mathcal{O}\left( t^{-\frac{\alpha-1}{3\alpha+4}} \right).\]
\(^{*}\)Corresponding authors(sftian@cumt.edu.cn, shoufu2006@126.com (S.F. Tian)).
\(^\dagger\)This author is contributed equally as the first author.↩︎