May 21, 2026
Here classes of moving boundary problems of Stefan-type for both an established nonlinear evolution equation of cuspon theory and novel reciprocally linked solitonic equations are shown to be solvable via Painlevé II symmetry reduction.
Moving boundary problems of Stefan-type have had extensive application in continuum mechanics, notably in connection with change of phase in a non-linear heat conduction context and liquid transport through porous media in soil mechanics [1]–[3] and [4] (see also references cited therein). In modern soliton theory, classes of moving boundary problems for the canonical Dym equation [5] and reciprocal associated are derived in [6] which admit exact solution via Painlevé II symmetry reduction. This investigation was originally motivated, in part, by the classical Saffman-Taylor problem with surface tension [7]. The occurrence of the solitonic Dym equation in the Hele-Shaw theory was elucidated in [8]. Moving boundary problems for a range of solitonic equations have been shown to admit exact solution via Painlevé II symmetry reduction [9]–[14].
Reciprocal-type transformations had their origin in the derivation of invariance properties of conservation laws in homoentropic gas dynamics in [15] and were subsequently shown in [16] to constitute particular Bäcklund transformations [17], [18]. In modern soliton theory, reciprocal transformations were introduced in [19] and associated with admitted conservation laws. Therein, these were conjugated with the action of the classical Bianchi permutability theorem associated with invariance of the soliton system under a Bäcklund transformation. Thereby, multi-soliton solutions can be generated iteratively in an algorithmic manner. In [20], reciprocal transformations were applied to link the canonical AKNS and WKI inverse scattering schemes of [21] and [22], respectively. The linkage of certain classes of 1+1-dimensional solitonic hierarchies via reciprocal transformation has been detailed in [23]–[25]. Reciprocal transformations in 2+1-dimensions, as originally introduced in [26], have been applied to connect the Kadomtsev-Petviashvili, 2+1-dimensional Dym and modified Kadomtsev-Petviashvili solitonic hierarchies in [27].
In [28], a novel nonlinear evolution equation descriptive of certain cuspon and periodic cuspon phenomena was set down and notably, in particular, a Lax pair was derived. Here, classes of moving boundary problems of Stefan-type both for this cuspon equation and integrable extensions linked by reciprocal transformations are shown to be solvable via Painlevé II symmetry reduction.
The reciprocal transformation \[dx^* = v \, dx + ( - v_{xx} + 3v^2) \, dt, \quad t^* = t\] with compatibility condition for the canonical solitonic Korteweg-de Vries equation \[v_t - 6vv_x + v_{xxx} = 0\] with \(v = -1/m^*\) yields \[dx = m^* \, dx^* + \left( \frac{1}{2} \frac{\partial^2}{\partial{x^*}^2} \left( \frac{1}{m^{*2}} \right) - \frac{3}{m^* }\right) dt^*.\] The latter has compatibility condition \[m_t^* = \frac{1}{2} \left( \frac{1}{m^{*}} \right)_{x^*x^*x^*} - 3 \left( \frac{1}{m^*} \right)_{x^*},\] namely, the solitonic cuspon equation in [28]. In [13], exact solutions of a class of Korteweg-de Vries moving boundary problems has been solved via application of the Miura transformation \[v = u_x + u^2\] which connects the KdV equation (2.2) to the mKdV equation \[u_t - 6u^2u_x + u_{xxx} = 0.\] This important link may be derived in the context of a class of classical Bäcklund transformation due to to Clairin [17]. In [13], application was made of Painlevé II symmetry reduction to solve a class of KdV moving boundary problems of Stefan-type problems governed by the system \[v_t - 6vv_x + v_{xxx} = 0 ~~,~~ 0<x< S(t)= \gamma (t+a)^{1/3}~, t>0\] \[\left. \begin{array}{l} v_{xx} - 3v^2 = L_m S^i \dot{S} \\ v = P_m S^j \end{array} \right\} \text{ on } x = S(t), \quad t > 0\] \[(v_{xx} - 3v^2)|_{x=0} = H_0(t+a)^k, t > 0,\] \[S(0) = S_0~.\] The mKdV equation (2.6) maybe be shown to admit a Painlevé II symmetry reduction with \[u = (t+a)^p \Psi(\xi), \quad \xi ={ x\over {(t+a)^q}}.\] Thus, on substitution of the latter representation into (2.6) there results \[p \Psi - q \xi \Psi' - 6(t+a)^{2p-q+1} \Psi^2 \Psi' + (t+a)^{-3q+1} \Psi''' = 0\] whence \(p = -1/3, q = 1/3\) and \(\Psi(\xi)\) is governed by \[\Psi''' - 6\Psi^2 \Psi' - \frac{1}{3} (\xi \Psi)' = 0\] so that, on integration, \[\Psi'' - 2\Psi^3 - \frac{1}{3} \xi \Psi = \kappa^*~,~ \kappa^*\in{\mathbf{R}}.\] On introduction of the scalings \(\Psi = \delta w\), \(\xi = \epsilon z\), into the latter there results the classical Painlevé II equation \[w_{zz} = zw^3 + zw + \alpha~,\] with parameter. \(\alpha= \kappa^* \epsilon^2/ \delta\)
Under the Miura transformation (2.5), the class of solutions \[v = (t+a)^{-2/3} (\Psi'(\xi) + \Psi^2(\xi)) := (t+a)^{-2/3} \Lambda(\xi)\] of the KdV equation (2.2) is obtained wherein \(\Psi(\xi)\) is given by the scaled version (2.11) of the Painlevé II equation.
The KdV moving boundary conditions:
\[v_{xx} - 3v^2 = L_m S^i \dot{S} \text{ on } x = S(t)=\gamma (t+a)^{1/3}~, \quad t > 0~.\] Insertion of the relation (2.13) yields \[(t+a)^{-4/3} (\Lambda''(\gamma) - 3\Lambda^2(\gamma)) = \frac{1}{3} L_m \gamma^{4i/3} (t+a)^{(i-2)/3}\] whence \(i = -2\) together with \[L_m = 3\gamma^{-4i/3} [\Lambda''(\gamma) - 3\Lambda^2(\gamma)].\]
\[v = P_m S^i \dot{S} ~~~\text{ on }~~ x = S(t)=\gamma (t+a)^{1/3}~, \quad t > 0~.\] This requires \(j = -2\) and \[P_m = \gamma^2 \Lambda(\gamma) = \gamma^2 [\Psi'(\gamma) + \Psi^2(\gamma)].\]
\[(v_{xx} - 3v^2)\vert_{x=0} =H_0 (t+a)^k ~, \quad t > 0~\] This boundary condition yelds \(k=-4/3\) together with \[H_0=\Lambda''(0) -3 \Lambda^2(0)~.\]
The moving boundary problems for the cuspon equation (2.4) under the reciprocal transformation (2.1) with \(v= {1}/{m^*}\) become:
\[m_t^* = \frac{1}{2} \left( \frac{1}{m^{*2}} \right)_{x^*x^*x^*} - 3 \left( \frac{1}{m^*} \right)_{x^*}, \qquad x^*\vert_{x=0 }< x^* < x^*\vert_{x=S(t)} := S^*(t^*), \quad t^*>0.\]
\[\left. \begin{array}{l} \displaystyle{ \frac{1}{2} \left( \frac{1}{m^{*2}} \right)_{x^*x^*} - 3 \left( \frac{1}{m^*} \right) = m^* L_m S^i \dot{S}} \\[2ex] \displaystyle{ \frac{1}{m^*} = P_m S^j} \end{array} \right\} \qquad \text{on } x^* = S^*(t^*), \quad t^*>0\]
\[\left[ \frac{1}{2} \left( \frac{1}{m^{*2}} \right)_{x^*x^*} - 3 \left( \frac{1}{m^*} \right) \right] \Bigg\vert_{{x^*}\vert_{x=0}} = m^*\big|_{x^*\vert_{x=0}} H_0 (t+a)^k, \qquad t^*>0\]
\[S^*(0)=S_0^*.\] In the preceding, \(x^* = S^*(t^*)\) is the reciprocal moving boundary obtained by application of (2.1) so \(x = S(t) = \gamma(t+a)^{1/3}.\) Thus, \[\begin{array}{l} \displaystyle{ dx^* |_{x=S(t)} = \left[v dx - (v_{xx} - 3v^2) dt\right] |_{x=S(t)} }\\ \\ \displaystyle{\,\,\,\qquad\quad\quad= (P_m S^j \dot{S} + L_m S^i \dot{S}) dt} \end{array}\] wherein \(i=j=-2\). Accordingly, \[dx^* |_{x=S(t)} = \left( P_m +L_m\right) {(1/ 3 \gamma) } (t^*+a)^{-4/3} dt\] so that \[S^*(t^*) = \gamma^* (t^*+a)^{-1/3} + \delta^*~~,~~\gamma^* , \delta^* \in{\mathbf{R}}\] with \(\gamma^* = -\gamma^{-1}(P_m + L_m)\). The associated reciprocal initial boundary condition becomes: \[S^*(t^*) = \gamma^* a^{-1/3} + \delta^*~~.\] In addition, \[dx^* |_{x=0} = (-v_{xx} + 3v^2) dt |_{x=0}= H_0 (t^*+a)^k dt\] with \(k=-1/3\), whence \[x^* |_{x=0} = -3 H_0 (t^*+a)^{-1/3} + \epsilon^*~~,~~ \epsilon^* \in{\mathbf{R}}.\] Thus, the region \[x^*\vert_{x=0 }< x^* < x^*\vert_{x=S(t)}:= S^*(t^*)~,\] reciprocally associated with \(0 < x < S(t)\), is given by \[-3 H_0 (t^*+a)^{-1/3} + \epsilon^* < x^* < \gamma^* (t^*+a)^{-1/3} + \delta^*~.\]
It is remarked that moving boundary problems constrained by a pair of time-dependent boundaries occur, in particular, both in the context of resonant nonlinear Schrödinger boundary analysis [29] and certain Stefan-type problems in the context of nonlinear heat conduction incorporating a source term [30].
In [31], a novel solitonic extension of the cuspon equation (2.4) was introduced, namely: \[m_t^* = \frac{1}{2} (m^{*-2})_{x^*x^*x^*} + \delta^* (m^{*-1})_{x^*} + \epsilon^* (m^{*-2})_{x^*}~~,~~ \delta^*, \epsilon^* \in{\mathbf{R}}.\] which is linked via a reciprocal transformation to the canonical solitonic Gardner equation \[v_\tau + 6v(1 - v)v_y + v_{yyy} = 0.\] The latter has diverse physical applications, notably in plasma physics, optical lattice theory, in addition to the analysis of nonlinear wave, propagation phenomena in both hydrodynamics and elastodynamics.
On application, to (3.2) of the reciprocal transformation \[dx^* = v \, dy - [v_{yy} + 3 v^2 - 2 v^3] d\tau, \quad t^* = \tau\] with \(v= { {1}/{m^{*}}}\), there results \[dy=m^*dx^*+\left[\frac{1}{2}\left(m^{*-2}\right)_{x^*x^*}+3m^{*-1}-2m^{*-2}\right]dt^*,\] with compatibility condition the extended cuspon equation (3.1) with parameters \(\delta^*= 3,~ \beta = -2\).
It was recently established in [31] that a class of nonlinear moving boundary problems for the Gardner equation (3.2) admits exact solutions via a Painlevé II symmetry reduction on application of a mKdV connection. This class was determined by the nonlinear system
\[v_\tau+6v(1-v)v_y+v_{yyy}=0, \qquad \frac{3\tau}{2}<y<\gamma(\tau+a)^{1/3}+\frac{3\tau}{2},\;\;\tau>0,\] \[\left. \begin{align} v_{yy}-2\left(v-\frac{1}{2}\right)^3&=L_m S^j\dot{S},\\ v-\frac{1}{2}&=P_m S^j \end{align} \right\} \quad\text{on}\quad y=\gamma(\tau+a)^{1/3}+\frac{3\tau}{2},\;\tau>0,\] and \[\left[v_{yy}-2\left(v-\frac{1}{2}\right)^3\right]_{y=3\tau/2}=H_0(\tau+a)^k, \qquad \tau>0,\] \[S(0)=S_0,\] wherein \(S(\tau)=\gamma(\tau+a)^{1/3}\).
The preceding system (3.5) was obtained in [31] by setting \[x=-\frac{3}{2}\tau+y,\qquad t=\tau,\qquad v=\frac{1}{2}+u,\] in the class of mKdV moving boundary problems \[u_t-6u^2u_x+u_{xxx}=0,\qquad 0<x<S(t),\;t>0,\] \[\left. \begin{align} u_{xx}-2u^3&=L_m S^j,\dot{S},\\ u&=P_m S^j, \end{align} \right\} \quad\text{on}\quad x=S(t),\;t>0,\] \[\left[u_{xx}-2u^3\right]_{x=0}=H_0(t+a)^k,\qquad t>0,\] \[S(0)=S_0.\]
This nonlinear system has been shown to admit exact solution via Painlevé II symmetry reduction [12]. In particular it admits the exact solution \[u= -\delta (t+a)^{-1/3}\phi'\left({{ x}\over{\epsilon(t+a)^{1/3}}}\right) \left[\phi\left({{x}\over{\epsilon(t+a)^{1/3}}}\right) \right]^{-1}\] where \(\phi\) is governed by an Airy equation.
A novel class of exactly solvable moving boundary problems for the extended solitonic cuspon equation results via the action of the reciprocal transformation (3.3) on the Gardner moving boundary system (3.5). The reciprocal extended cuspon moving boundary system inherits the property of admittance of exact solution via Painlevé II symmetry reduction.