Keywords: Time-dependent damped nonlinear wave equations, Lie point symmetry, Variational symmetry, Noether’s theorem, First Integral, Conservation law

1 Introduction↩︎

The aim of this paper is to study symmetries and conservation laws of the nonlinear wave equations \[\label{main} \mathcal{E}= \Box u+a(t)u_t+f(u)=0, \quad f''\neq 0\tag{1}\] where \(\Box =\partial_t^2-\Delta\) is the \((n+1)\)-dimensional wave operator, \(u(t,x)\) is the wave function with \((t,x)\in \mathbb{R}^{n+1}\), \(n\geq 2\) and \(f(u)\) is an arbitrary function.

Equation 1 is the Euler-Lagrange equation \(E_u(L)=\mathcal{E}\) with the Lagrangian \[\label{L} L=\mu(t)L_0, \quad L_0=\frac{1}{2}\big(u_t^2+|\nabla_{x} u|^2\big)-F(u), \quad F(u)=\int f(u)du\tag{2}\] where \(\mu>0\) satisfies \(\dot{\mu}=a(t)\mu\), \(\nabla_{x}\) and \(E_u\) are the spatial gradient and the Euler-lagrange operators, respectively. Quite recently, the same class in the one-dimensional case has been investigated in [1]. The authors of this paper use a multiplier approach to find conservation laws. Here, in the multi-dimensional case, we prefer to take the variational symmetry approach with the same goal in mind. Very recently, Noether symmetries and first integrals for the general class of Lagrangians \(L = \mu(t)L_0\), which models motion with general linear damping in a Riemannian space have been derived in Ref. [2].

Obviously, \(L_0\) in 2 is the Lagrangian of the undamped nonlinear wave equation obtained when \(a(t)\equiv 0\), \[\label{undamped} \Box u+f(u)=0.\tag{3}\] In a recent paper [3], we have analysed Lie point symmetries of more general classes of equations \[\begin{eqnarray} \Box u &=& G(x,u,\nabla u), \tag{4}\\ \Box u &=& G(x,u), \tag{5} \end{eqnarray}\] where \(x=(t,x)=(t,x_1,\ldots,x_n)\) and \(\nabla=(\partial_0,\nabla_x)=(\partial_t,\partial_{x_1},\ldots,\partial_{x_n})\) is the gradient operator in \(\mathbb{R}^{1,n}\). Throughout this paper, unless otherwise stated we will sum over repeated indices. Eq. 1 belongs to the first class 4 with the special choice \[G(x,u,\nabla u)=-a(t)u_t-f(u), \quad f''\ne 0,\] which is a nonlinear extension of the linear wave equation \[\label{linearwave} \Box u=0.\tag{6}\] A well-known fact about 6 is that it is invariant under the conformal symmetry group \(\mathsf{C}(1,n)\), isomorphic to the pseudo-orthogonal group \(\mathsf{O}(n+1,2)\).

For the undamped wave equation 3 , \(a(t)=0\), the energy \[\label{energy} E(u,\nabla u)=\int_{\mathbb{R}^n}\Big[\frac{1}{2}\big(u_t^2+|\nabla_{x} u|^2\big)+F(u)\Big]dx\tag{7}\] is a constant of the motion (conserved quantity). On the other hand, when \(a(t)\ne 0\), we have the relation \[\label{energy-decr} \frac{dE}{dt}=-a(t)\int_{\mathbb{R}^n}u_t^2dx \leq 0\tag{8}\] and the energy is monotonically decreasing for \(a>0\). Another conserved quantity is angular momentum (see below formula 34 ).

A number of interesting conservation laws of 6 were derived in [4] (see examples 4.15 and 4.36) based on Noether’s theorem. We shall make use of the same theorem to derive conservation laws of the class of 1 and its subclasses allowing maximal symmetry algebra.

2 Symmetries and Conservation Laws↩︎

A general element of a Lie symmetry algebra \(\mathfrak g\) will be a vector field of the form \[\label{vf} \mathbf{v}=\sum_{a=0}^{n}\xi_{a}\partial_{x_{a}}+\eta \partial_{u},\tag{9}\] where the coefficients \(\xi_{a}\) and \(\eta\) are functions of \((t,x,u)\) with the identification \(x_0=t\).

The Lie symmetry algebra \(\mathfrak g=\mathop{\mathrm{c}}(1,n)\) of 6 has dimension \(\dim \mathfrak g=(n+2)(n+3)/2\) and is spanned by the vector fields \[\label{killing-vf} \begin{align} P_{a}&=\partial_{a},\\ J_{ab}&=x_{a}\partial_{b}-x_{b}\partial_{a},\\ D&=x^{a}\partial_{a} + \frac{(1-n)}{2}u\partial_u,\\ C_{a}&=2x_{a}D-(x_{\alpha}x^{\alpha})\partial_{a}, \end{align}\tag{10}\] where \(a, b= 0, 1, \dots, n\).

Preferably, \(C_a\) can be written as \[\label{conf-gen} \begin{align} C_0&=(t^2+r^2)\partial_t+2tx_l\partial_l+(1-n)tu\partial_u,\\ C_k&=2tx_k\partial_t+2x_k x_l \partial_l+(t^2-r^2)\partial_k+(1-n) x_k u \partial_u, \end{align}\tag{11}\] where \(r^2=|x|^2=\sum_{i=1}^{n}x_i^2\), \(k, l= 1, \dots, n\).

For arbitrary \(a(t)\) and \(f(u)\), the symmetry algebra of 1 is the Euclidean algebra \(\mathop{\mathrm{e}}(n)\) of translations and rotations of \(\mathbb{R}^n\). Its dimension is \(n(n+1)/2\) and infinitesimal symmetries are \[\label{sym-damped} P_{k}=\partial_{k}, \quad J_{kl}=x_{k}\partial_{l}-x_{l}\partial_{k}, \quad k<l, \quad k, l= 1, \dots, n.\tag{12}\] We note that the (equivalence) transformation \(u\to u+u_0\), \(u\to \lambda u\) takes Eq. 1 to the same form with \(f(u)\) changed. If we require invariance under the dilational symmetry \[\label{dilat} D=x^{\sigma}\partial_{\sigma} + d u\partial_u=t\partial_{t}+x_k \partial_{k}+d u\partial_{u},\tag{13}\] then we find that for \[\label{a-and-f} a(t)=\frac{m}{t}, \quad f(u)=f_0 u^p, \quad k\ne 1, \quad d=\frac{2}{1-k},\tag{14}\] where \(m\) and \(f_0\) are integration constants. Dilatational invariance under \[\label{D-2} D=t\partial_{t}+x_k \partial_{k} -\frac{2}{m} \partial_u\tag{15}\] leads to the exponential nonlinearity \[\label{f-exp} f(u)=f_0 e^{m u}.\tag{16}\]

Additional invariance under \[\label{Kk} C_k=2tx_k\partial_t+2x_k x_l \partial_l+(t^2-r^2)\partial_k+q x_k u \partial_u\tag{17}\] restricts \(p\) to be \[\label{k} p=\frac{n+3+m}{n-1+m}, \quad q=1-n-m, \quad m\ne 1-n.\tag{18}\] We hereby have corrected a misprint concerning the conformal factor \(q\) in our paper [3].

So we will be dealing with the damped PDE (partial differential equation) \[\label{damped-pde} \Box u+\frac{m}{t}u_t+f_0 u^p=0, \quad p=\frac{n+3+m}{n-1+m},\tag{19}\] or \[\label{damped-pde-exp} \Box u+\frac{m}{t}u_t+f_0 e^{m u}=0, \quad m\ne 0\tag{20}\] admitting a Lie symmetry algebra of dimension \((n+1)(n+2)/2\). In the case 20 , we replace \(C_k\) by \[\label{Kk-exp} C_k=2tx_k\partial_t+2x_k x_l \partial_l+(t^2-r^2)\partial_k-\frac{4}{m} x_k \partial_u, \quad m\ne 0, \quad k=1,2,\ldots, n.\tag{21}\]

\(f_0\) can be set to \(f_0=\pm 1\) using the above-mentioned equivalence transformation. We comment that when \(m=0\), Eq. 19 reduces to the well-known conformally-invariant nonlinear wave equation with Lie symmetry algebra (factoring out the symmetry representing the linear superposition principle) shared with the homogenous linear wave equation 6 . For an alternative derivation of 19 in the general case \(n\geq 2\), see Refs. [3], [5] or [6], [7] for \(n=2\). In the undamped case \(m=0\) with power non-linearity \(f(u)=\pm u^p\) where \(u=u(t,r)=u(t,|x|)\), a study of conservation laws was given in [8]. The interested reader can also find basic definitions, ideas and interesting results applied to a lot of PDEs in [9]–[11].

Our primary interest is to derive conservation laws of the damped nonlinear wave equation 19 utilizing variational structure of the equation. This will be done by means of possible variational symmetries.

A (local) conservation law is an expression of the form \[\label{conser-exp} \mathop{\mathrm{Div}}\tilde{I}=D_t I_0+\mathop{\mathrm{Div}}I=0, \quad \tilde{I}=(I_0,I)=(I_0,I_1,\ldots, I_n),\tag{22}\] which holds for all solutions of the PDE 1 . Here \(I_0\) is the conserved density, \(I=(I_1,\ldots, I_n)\) the associated (spatial) flux all of which are functions of \((t,x,u)\) and the derivatives of \(u\) with respect to the independent variables \((t,x)\in \mathbb{R}^{n+1}\). Here \(\mathop{\mathrm{Div}}I\) is defined to be the spatial total derivative of the \(n\)-tuple \(I\) with respect to \((t,x)\) \[\mathop{\mathrm{Div}}I=\sum_{j=1}^n D_{j} I_{j},\] where \(D_{j}\) is the total differential operator defined by \[D_{j}=\partial_{j}+u_{j}\partial_{u}+ u_{jk}\partial_{u_{k}}+\ldots\] For a more detailed discussion on this theme and related issues we refer to [4], [12], [13].

The following is the infinitesimal criterion of invariance for a first order Lagrangian \(L\): \[\label{inv-L} \mathrm{pr}^{(1)} \mathbf{v} (L)+L\mathop{\mathrm{Div}}{\xi}=\mathop{\mathrm{Div}}\tilde{B}=D_t B_0+\mathop{\mathrm{Div}}B,\tag{23}\] where \(\xi=(\xi_0,\xi_1,\ldots,\xi_n)\) and \(B\) is a \((n+1)\)-tuple of differential function in the jet space \(J(t,x,u,u_{a})\). The evolutionary form \(\mathbf{v}_{Q}=Q\partial_{u}\) of the vector field 9 permits us to express 23 \[\label{inv-L-evo} \mathrm{pr}^{(1)} \mathbf{v}_{Q} (L)=\mathop{\mathrm{Div}}B,\tag{24}\] where \(Q(t,x,u,u_a)\) is the characteristic function of the vector field \(\mathbf{v}\) of 9 and \(B\) is another \((n+1)\)-tuple differential function. In terms of \(Q(t,x,u,u_a)\) of the variational symmetries we can express the conservation laws in characteristic form \[\label{charac-form} \mathop{\mathrm{Div}}I =Q.E(L)=Q.\mathcal{E}=0,\tag{25}\] where \(\mathcal{E}=0\) is the wave equation defined by 1 . For a first order Lagrangian, the constants of motion or first integrals are given either by integrating by parts 25 or explicitly by the formula \[\label{cons-law-comp} I_a=\xi_a L+Q \frac{\partial L}{\partial u_a}=\mu(t)\left[\xi_a L_0+Q \frac{\partial L_0}{\partial u_a}\right], \quad a=0,1,\ldots, n.\tag{26}\]

As was also observed in Ref. [1], for the Lagrangian \(L=\mu L_0\) given by 2 , the action of \(L\) on the first prolongation of \(\mathbf{v}_{Q}\) is \[\label{pr-vf} \mathrm{pr}^{(1)} \mathbf{v}_{Q} (L)=-\mu Q \mathcal{E}+D_t(\mu u_t Q)+\mathop{\mathrm{Div}}(\mu Q \nabla u)\tag{27}\] and the invariance condition 24 takes the form \[\label{multip-form} \mathrm{pr}^{(1)} \mathbf{v}_{Q} (L)=-\mu Q \mathcal{E}+D_t(B_0-\mu u_t Q)+\mathop{\mathrm{Div}}(B-\mu Q \nabla_x u).\tag{28}\] This shows that if \(\mathbf{v}_{Q}\) generates a variational symmetry then \(\tilde{Q}=\mu Q\) is a multiplier for the corresponding conservation laws, namely, \[\label{first-int} \mu Q. \mathcal{E}=\mathop{\mathrm{Div}}\tilde{I}=D_t I_0+\mathop{\mathrm{Div}}I=0,\tag{29}\] and \(E_u(\mu Q. \mathcal{E})\) vanishes identically.

Integrating the conservation laws 29 over a bounded domain \(\Omega\subset \mathbb{R}^n\), using divergence theorem under the assumption that the flux term (an \(n\)-tuple of functions of \((t,x,u)\) and first order derivatives of \(u\)) \(I\) tends to zero as \(x\) approaches the boundary \(\partial \Omega\), or if \(\Omega=\mathbb{R}^n\) as \(|x|\to \infty\) we can construct a constant of the motion of 1 in the integral form \[\label{const-motion} \int_{\Omega} I_0 \; dx=\text{const.}\tag{30}\]

2.1 Conservation laws for 1↩︎

We now turn to the derivation of conservation laws for the original class 1 for any function \(a(t)\), and start with Noether symmetries corresponding to the Euclidean algebra \(\mathop{\mathrm{e}}(n)\). The characteristic function for \(P_k\) is \(Q=u_k\) (up to sign), \(k=1,2,\ldots, n\) and integration by parts of the equation \[\label{cons-Pk} \mathop{\mathrm{Div}}\tilde{I}=\mu u_k \mathcal{E}=D_t(\mu u_k u_t)+\mu D_j\left[u_k u_j+(\frac{1}{2}\left(u_t^2-\left|\nabla_x u \right|^2-2F(u)\right)\delta_{kj}\right].\tag{31}\] So we have the conserved density and the current components of the linear momentum conservation \[\label{cons-Pk-comp} I_0^{k}=\mu u_k u_t, \quad I_j^{k}=\mu(u_k u_j+ L \delta_{kj}), \quad \mu=\exp{\int a(t)dt}, \quad j,k=1,2,\ldots, n.\tag{32}\]

A similar manipulation gives the angular momentum conservation for the characteristic function \(Q=x_k u_l-x_l u_k\), \(k<l\) of the geometrical rotations \(J_{kl}\) \[\label{angular} \mu (x_k u_l-x_l u_k) \mathcal{E}=D_t(\mu Q u_t)+\mu D_j[x_l \Phi_{jk}-x_k \Phi_{jl}+(\delta_{jl}x_k-\delta_{kj} x_l)F],\tag{33}\] where the symmetric function \(\Phi_{jk}\) is defined as \[\Phi_{jk}=u_j u_k-\frac{1}{2}\left|\nabla_x u \right|^2 \delta_{jk}.\] From this we see that the components of the conservation laws in the form \[\label{angular-comp} I_0^{kl}=\mu (x_k u_l-x_l u_k) u_t, \quad I_j^{kl}=\mu [x_l \Phi_{jk}-x_k \Phi_{jl}+(\delta_{jl}x_k-\delta_{kj} x_l)F]=0, \quad j,k,l=1,2,\ldots, n.\tag{34}\]

Now we wish to determine if the subclasses 19 and 20 admit variational symmetries. Our main tool is the infinitesimal criterion 23 (necessary and sufficient for a connected group of transformations to be a symmetry of a variational problem). Let us start with the dilational generator \[\label{D-d} D=t\partial_{t}+x_k \partial_{k}+d u\partial_{u}, \quad d\ne 0\tag{35}\] and compute the left side of 23 as \[\mathrm{pr}^{(1)} D (L)+(n+1)L=(t \dot{\mu}+(n+1)\mu+2\mu(d-1))L_0-\mu(d u F'-2(d-1)F)\] using \[\mathop{\mathrm{Div}}\xi=n+1, \quad \mathrm{pr}^{(1)} D=D+(d-1)u_i \partial_{u_i}.\] The coefficients of \(L_0\) and \(\mu\ne 0\) in this relation imply \[t \dot{\mu}+(n+1)\mu+2\mu(d-1)=\mu[t a(t)+(n+1)+2(d-1)]=0, \quad d u F'-2(d-1)F=0.\] The first relation suggests that we must have \(t a=m=\rm constant\) and \(\mu=t^m\) so we find the dilational factor \(d=(1-m-n)/2\). The second one in terms of \(f\) is \[d u f'-(d-2)f=0,\] which has the solution \[\label{max-power-f} f=f_0 u^p, \quad p=\frac{d-2}{d}=\frac{n+3+m}{n-1+m},\tag{36}\] exactly as in the form 19 . Consequently, this result confirms that \(D\) with \(d=(1-m-n)/2\) is a variational symmetry for arbitrary \(m\) and \(f(u)\) as in 36 . Taking \[Q=d u-t u_t-x_j u_j, \quad L_0=\frac{1}{2}\left(u_t^2-\left|\nabla_x u \right|^2\right)-\frac{f_0}{p+1}u^{p+1}, \quad p=\frac{n+3+m}{n-1+m}\] from 26 it follows the following conserved currents \[\label{dil-currents} I_0=t^m[t L_0+Q u_t], \quad I_j=t^m[x_j L_0+Q u_j].\tag{37}\]

The same calculation for 15 gives a dimension-dependent constraint on \(m\), \(m=1-n\). Hence, only for these values of \(m\), conservation laws can be obtained from 26 .

We can repeat the same steps for the conformal symmetry generators \(C_k\) with the conformal factor \(q\), that is ab initio left free. We take into account the relations \[\mathop{\mathrm{Div}}\xi=2(n+1)x_k, \quad \mathrm{pr}^{(1)} C_k=C_k+\tau_{k,0}\partial_{u_t}+\tau_{k,l} \partial_{u_l},\] where the coefficients of the first prolongation are functions of \((t,x,u,u_a)\) defined by \[\tau_{k,l}=D_{l}Q_k+u_{jl}\xi_{j}=D_{l} \eta_k -u_{j}D_{l} \xi_j, \quad j,k=1,2,\ldots,n.\] If we take the coefficients \(\xi\) and \(\eta\) of \(C_k\) as \[\xi_{k,0}=2t x_k, \quad \xi_{k,l}=2x_k x_l+(t^2-r^2)\delta_{kl}, \quad \eta_k=q x_k u\] and use the relations \[2tx_k\dot{\mu}=2m \mu x_k, \quad u_t\tau_{k,0}=(q-2)x_k u_t^2-2t u_t u_k, \quad -u_l \tau_{k,l}=-quu_k-(q-2)x_k\left|\nabla_x u \right|^2+2tu_t u_k\] we obtain \[\begin{gather} \label{Ckvar} \mathrm{pr}^{(1)} C_k(L)+2(n+1)x_k L= \\ -q \mu u u_k+\mu(q+m+n-1)[u_t^2-\left|\nabla_x u \right|^2]-\mu[q u F'+2(m+n+1)F]x_k. \end{gather}\tag{38}\] The left side of this expression is a total derivative \(D_k(-q t^m u^2)/2\) if we choose \[q=1-m-n, \quad qu f'+[q+2(m+n+1)]f=0, \quad f=F'.\] The second relation shows that \(f=f_0 u^p\), \(p=(n+3+m)/(n-1+m)\) and the conformal transformations 17 are also variational (more precisely divergence) symmetries. The conserved currents of the corresponding conservation law from formula 26 are \[\label{consv-law-conf} I_{k,0}=t^m(2 x_k L_0-Q_k u_k), \quad I_{k,j}=t^m[L_0 \xi_{k,j}-Q_k u_j+\frac{q}{2}u^2 \delta_{jk}], \quad j,k=1,2,\ldots, n.\tag{39}\] In 26 , \(\xi_{k,j}\) denotes the horizontal coefficients of the \(k\)-th components of \(C_k\) and \(Q_k\) is the characteristic function of \(C_k\) in 17 .

Finally, we check the formula 23 for \(C_k\) given in 21 , \(f(u)=f_0 e^{m u}\) and find \[\label{check-inv-conf-exp} \mathrm{pr}^{(1)} C_k(L)+2(n+1)x_k L=\frac{4}{m} t^m u_k+t^{m}(m+n-1)x_k [u_t^2-\left|\nabla_x u \right|^2].\tag{40}\] Again we can express the right side of 40 as a total derivative putting \(m=1-n\) in the form \[D_k B, \quad B=\frac{4}{m}t^m u.\] The corresponding dimension-dependent conserved currents are \[\label{comp-exp} I_0^k=t^{1-n}(2 x_k L_0-Q u_k), \quad I_j^k=t^{1-n}[L_0 \xi_{k,j}-Q_k u_j-\frac{4}{1-n}u \delta_{jk}], \quad j,k=1,2,\ldots, n.\tag{41}\]

For any global solution to the damped wave equation 1 decaying sufficiently rapidly as \(r=|x|\to \infty\), the spatial integrals of the conserved densities \(I_0\) found above provide us with constants of the motion of 1 .

We recall that in the undamped case 3 , for any \(f(u)\), Eq. is invariant under the Poincaré group \(\mathop{\mathrm{P}}(1,n)\) which is a semidirect product of the Lorentz group with the group of spacetime translations: \(\mathop{\mathrm{G}}=\mathop{\mathrm{P}}(1,n)=\mathbb{R}^{1,n}\rtimes \mathop{\mathrm{O}}(1,n)\). This group of transformations are variational symmetries and has dimension \(\dim {\mathop{\mathrm{G}}}=(n+1)(n+2)/2\). All conservation laws associated to them can be found by 26 . The symmetry group for the special power linearity \(f(u)=f_0u^p\), \(p=(n+3)/(n-1)\) is even larger, the conformal group \(\mathop{\mathrm{C}}(1,n)\). In this case, there are \(n+2\) more conservation laws obtainable from 26 . One of them is of course the conservation of energy given by 7 with multiplier \(Q=u_t\). The remaining symmetries consist of one more conformal and all Lorentz generators given by \[\label{C044Kk} C_0=(t^2+r^2)\partial_t+2tx_l\partial_l+(1-n)tu\partial_u, \quad K_k=J_{k0}=t\partial_{k}+x_k \partial_{t}, \quad k=1,2,\ldots,n\tag{42}\] are also variational.

The fact that in the one-dimensional case there exists a simple change of dependent variable \[\label{trans-u} u(t,x)=\mu(t)^{-1/2}v(t,x)\tag{43}\] respecting the \(u\)-dependence of the nonlinear interaction \(f(u)\) and transforming the damped equation 1 to the undamped form 3 , namely removing the damping term \(a(t)\) was studied in Ref. [1]. This happens when \(a(t)\) satisfies \[\label{a-ode} \frac{\dot{a}}{2}+\frac{a^2}{4}+\frac{\kappa}{2}\int a(t)dt=\sigma_0=\text{const.},\tag{44}\] and \(f(u)\), which has no explicit time dependence, is a logarithmic interaction \[\label{log-u} f(u)=(\sigma+\kappa \ln |u|)u.\tag{45}\] \(f(u)\) is transformed to \(g(v)=(\sigma-\sigma_0+\kappa \ln |v|)v\). This means, by transformation, the constant \(\sigma\) shifts as \(\sigma\to \sigma-\sigma_0\). The same result holds in the multidimensional case. The general solution of 44 is not elementary. It can be expressed in terms of Lambert function. The elementary solutions exist for instance when \(\kappa=0\), but in this case \(g(v)\) becomes linear. In this case, all known conservation law results as mentioned above paragraph for the undamped PDE.

We conclude with the observation that, for the damping term \(a(t)=m/t\) (\(\mu=t^m\)) frequently used here, the left side of 44 is \[m\left[ \frac{1}{4}t^{-2}(m-2)+\frac{\kappa}{2}\ln |t|\right], \quad m\ne 0\] which can never be a constant unless \(m=2\), \(\kappa=0\) for which the logarithmic nonlinearity would not be present.

References↩︎