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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Conservation laws for a strongly damped wave equation

Maria Luz Gandarias / Rafael de la Rosa / Maria Rosa
Published Online: 2017-05-04 | DOI: https://doi.org/10.1515/phys-2017-0033

Abstract

A strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. Classical symmetries, exact solutions and conservation laws are derived.

Keywords: Lie symmetries; Exact solutions; Partial differential equations; Conservation laws

PACS: 02.30.Jr; 02.20.Sv

1 Introduction

Strongly damped wave equations are of a great current interest, see Refs. [6, 810, 1315, 18, 20, 24, 25] and references therein. In [21] an initial-boundary value problem for a nonlinear wave equation in one space dimension with a nonlinear damping term |u|m–1ut and a source term of the form |u|p–1u with m, p > 1 has been considered. The authors analyzed the global well posedness of the initial-boundary value problem corresponding to uttuxx+|u|m1ut=|u|p1u.

It is well known [10, 18, 20, 21] that when the damping term |u|m–1ut is absent from the equation, then the source term |u|p–1u yields the solution to blow up in finite time. However, the interaction between the damping and source terms is often difficult to analyze. In [14] a new method of investigating the so-called quasilinear strongly-damped wave equations t2utΔxuΔxu+f(u)=x.ϕ.(xu)+g, in bounded 3D domains has been presented. This method allows us to establish the existence and uniqueness of energy solutions in some cases for the growth exponent of the non-linearity and f. In [10] for the strongly damped nonlinear wave equation uttΔutΔu+f(ut)+g(u)=h, with Dirichlet boundary conditions, where both the non-linearities f and g exhibit a critical growth, while h is a time-independent forcing term the existence of an exponential attractor of optimal regularity is proven.

One of the most studied equation is the case with only one nonlinearity is the following uttautxxuxx+g(u)=0,(1) which has been considered in [810, 13, 14].

It is known that conservation laws play a significant role in the solution process of an equation or a system of differential equations. Although not all of the conservation laws of partial differential equations (PDEs) may have physical interpretation they are essential in studying the integrability of the PDEs.

In [25], Anco and Bluman gave a general algorithmic method to find all conservations laws for evolution equations Many recent papers using this method have been published in [1, 11, 12]. After Ibragimov’s results several papers appeared concerned with self-adjointness and its applications to PDEs [22, 23]. In this work, we derive conservation laws by using the direct method of the multipliers [25]. In this paper, we consider the strongly damped wave equations of the form utt+f(u)utautxxuxx+g(u)=0,(2) where f(u) and g(u) are arbitrary non-zero functions and a > 0 is a given dissipation parameter. We provide a complete classification of all point symmetries and all low-order conservation laws admitted by this equation (2).

2 Classical symmetries

Lie classical method is based on the determination of the symmetry group of a differential equation, i.e., the largest group of transformations acting on dependent and independent variables of the equation so that maps solutions of the equation into other solutions.

In order to apply Lie classical method to equation (2) we consider the one-parameter Lie group of infinitesimal transformations in (x, t, u) given by x=x+ϵξ(x,t,u)+O(ϵ2),(3) t=t+ϵτ(x,t,u)+O(ϵ2),(4) u=u+ϵη(x,t,u)+O(ϵ2),(5) where is the group parameter. The symmetry group of equation (2) will be given by the set of vector fields of the form v=ξ(x,t,u)x+τ(x,t,u)t+η(x,t,u)u.(6)

Equation (2) admits a Lie point symmetry provided that pr(3)v(Δ)=0whenΔ=0, where Δ = utt + f(u)utautxxuxx + g(u), and pr(3)v is the third prolongation of the vector field (6) which is given by: pr(3)v=v+ζtut+ζttutt+ζxxuxx+ζtxxutxx,(7) where ζJ(x,t,u(3))=DJ(ητutξux)+τuJt+ξuJx,

with J = (j1, . . ., jk), 1 ≤ jk ≤ 2 and 1 ≤ k ≤ 3, and u(3) denotes the sets of partial derivatives up to third order [19]. Applying (7) we obtain a set of determining equations for the infinitesimals ξ(x, t, u), τ(x, t, u) and η(x, t, u). By solving the determining system we obtain the following results.

Case 1

The point symmetries admitted by a strongly damped wave equation (2) in the general case are generated by v1=x,v2=t.

Case 2

If f(u) = α and g(u) = βu + γ, with α, β and γ arbitrary constants, besides v1 and v2 we obtain 2 new generators: v3=uu,v4=q(x,t)u, where q(x, t) must satisfy equation (2).

Case 3

If f(u)=2aandg(u)=ua2+γ, with γ is an arbitrary constant, besides v1, v2 and v4, we obtain the following generator: v5=x2x+tttu+a2γau.

3 Symmetry reductions and exact solutions

In this section, we use the Lie point symmetries admitted by (2) to obtain group-invariant solutions, then by using the simplest equation method due to Kudriashov we derive some exact solutions for (2).

Reduction 1

In order to find travelling wave solutions for equation (2) we consider that f(u) and g(u) are arbitrary functions, by using the generator X = λ∂x + t we obtain the similarity variables and similarity solution z=xλt,u=h(z),(8) and the following reduced ODE achzzz+c2hzzhzzcf(h)hz+g(h)=0.(9)

In [16] Kudryashov introduced the simplest equation method in order to look for exact solutions of nonlinear differential equations. In [17] it was pointed out that many of the so called “new travelling wave solutions” could be derived from the solutions of a simple nonlinear ordinary differential equation. In this paper looking for travelling waves solutions of (2) and observing that (2) can be written in conserved form for f(u) linear and g(u) quadratic we set f(u)=k1u+k2,g(u)=k3u2+k4u+k5 and we search for exact solutions of (9) with f(h)=k1h+k2,g(h)=k3h2+k4h+k5.

We are now considering as simplest equation yz2k1y4k2y3k3y2k4yk5=0.(10)

The general solution of equation (10) with a1 ≠ 0 can be written in terms of the Jacobi elliptic functions sn(kz, p).

The general solution of equation (10) with a1 = 0 can be written in terms of the Jacobi elliptic function sn2(kz, p) or via de Weierstrass elliptic function. We now look for exact solutions h(z) of achzzz+c2hzzhzzc(k1h+k2)hz+k3h2+k4h+k5=0(11) such that h(z) satisfies (10). By substituting yz2=a1y4+a2y3+a3y2+a4y+a5 and yzz=2a1y3+3a2y22+a3y+a42 into (11) we obtain that the following conditions must be satisfied a1=0,k2=3aa2,k2=aa3,k3=32a2(1c2),k4=a3(1c2),k5=12a4(1c2).(12)

We obtain the following:

Theorem Equation (11) admits any solution of equation (10) with a1 = 0 if f(h)=k1h+k2,g(h)=k3h2+k4h+k5 where ki, i = 0, ... 5 are given by (12).

As a particular case for k1=12a,k2=4a,k3=6(c21),k4=4(1c2),k5=0, equation (11) admits the solution h = sech2(z) and (2) admit a soliton solution u(x,t)=sech2(xct)(13)

In Figure 1, we plot solution u(x, t) (13), which describes a soliton.

Soliton solution (13) with c = 1.
Figure 1

Soliton solution (13) with c = 1.

Reduction 2

For f(u) = α and g(u) = ßu+γ, taking q(x,t)=γβ, by using generator v=tua+γaβu.

We get the invariant solution z=x,u=etah(z)γβ,(14) where h(z) must satisfy a2βaα+1h(z)=0.(15)

If we take α=a2β+1a equation (15) is trivially satisfied. Thus, undoing transformation (14), we obtain a solution of equation (2) for f(u)=a2β+1aandg(u)=βu+γ which is given by ux,t=u=etah(x)γβ.

Reduction 3

Finally, for f(u)=2aandg(u)=ua2+γ, by using generator v5 the corresponding invariant solution is written in the form z=x2t,u=etah(z)a2γ,(16)

where h(z) must satisfy 4az2h+z2+10azh+2z+ah=0.(17)

Integrating once with respect to z we obtain 4az2h+2a+zh+c1=0,(18)

where c0 is a constant of integration. For this reduction, ODE (18) is a second order equation whose solution is hz=c2+2a2c1azez4a+c0(z)c0πE12zazez4a4a2zzadz,

where c1, c2 are arbitrary constants and E represents the error function. Therefore, undoing transformation (16), we obtain a solution of equation (2).

4 Conservation laws

We will obtain conservation laws for the strongly damped wave equation (2) by applying the general multiplier method Ref. [25].

A local conservation law of equation (2) is a continuity equation DtC1+DxC2=0,(19) that holds for the whole set of solutions u(x, t), where the conserved density C1 and the spatial flux C2 are functions of x, t, u, and derivatives of u. Here Dt, Dx denote total derivatives with respect to t and x respectively. The pair of expressions (C1, C2) is called a conserved current.

Two local conservation laws are considered to be equivalent [5] if they differ by a trivial conservation law C1 = DXΘ, C2 = –DtΘ, where C1 and C2 are evaluated on the set of solutions of equation (2), and Θ is some function of x, t, u, and derivatives of u.

We begin by observing that equation (2) has a Cauchy-Kovaleskaya form. Consequently, the results in Ref.[3, 4] show that all non-trivial conservation laws arise from multipliers. Specifically, when we move off of the set of solutions of equation (2), every non-trivial local conservation law (19) is equivalent to one that can be expressed in the characteristic form DtC~1+DxC~2=utt+f(u)utautxxuxx+g(u)Λ,(20)

where Λ(x, t, u, ux, ut, ...) is a multiplier, and where (1, 2) differs from (C1, C2) by a trivial conserved current. On the set of solutions u(x, t) of equation (2), the characteristic form (20) reduces to the conservation law (19).

In general, a function Λ(x, t, u, ux, ut, ...) is a multiplier if it is non-singular on the set of solutions u(x, t) of equation (2), and if its product with equation (2) is a divergence expression with respect to t and x. There is a one-to-one correspondence between non-trivial multipliers and non-trivial conservation laws in characteristic form.

The determining equation to obtain all multipliers is δδu((utt+f(u)utautxxuxx+g(u))Λ)=0.(21)

This equation must hold off of the set of solutions of equation (2). Once the multipliers are found, the corresponding non-trivial conservation laws are obtained either by using a homotopy formula Ref. [24] or by integrating the characteristic equation (20) Ref. [7].

We will now find all multipliers Λ(x,t,u,ut,ux,utx,uxx), and obtain corresponding non-trivial conservation laws. The determining equation (21) splits with respect to the variables utt, uttt, uttx, utxx, uxxx, utttx, uttxx, utxxx, uxxxx. This yields a linear determining system for Λ(x, t, u, ut, ux, utx, uxx) which can be solved by the same algorithmic method used to solve the determining equation for infinitesimal symmetries. By using Maple we solve this determining system subject to the classification conditions. After solving the determining system we do not obtain any multiplier for general f(u) and g(u)

All special cases for which additional multipliers (2) are admitted consist of: Q1=etk2+xk1k21k2ak2,(22a) Q2=etk2xk1k21k2ak2,(22b) f(u)=k1+k2(k3u+k4),g(u)=12k3u2+k4u+k5,k1,k2,k3,k4,k5constants;(22c) Q3=ψxeta,ψ(x)arbitrary,(23a) f(u)=ag(u)+1a,g(u)arbitrary;(23b) Q4=e12tac+tk1+tβ+cx,(24a) Q5=e12tac+tk1+tβcx,(24b) Q6=e12tac+tk1tβ+cx,(24c) Q7=e12tac+tk1tβcx,(24d) f(u)=k1,g(u)=k2uwithβ=a2c22ak1c+k124k2+4c,k1,k2,cconstants;(24e)

For each solution Q, a corresponding conserved density and flux can be derived by integration of the divergence identity (20) [5, 7]. The admitted conservation laws in each case are given by: T1=utak1uuak2etk2+xk1k21k2ak2k1u+k212k3u2+k4u+k5etk2+xk1k21k2ak2,(25a) X1=aut+uk1k21k2ak2etk2+xk1k21k2ak2autx+uxetk2+xk1k21k2ak2;(25b) T2=utak1uuak2etk2xk1k21k2ak2k1u+k212k3u2+k4u+k5etk2xk1k21k2ak2,(26a) X2=aut+uk1k21k2ak2etk2xk1k21k2ak2autx+uxetk2xk1k21k2ak2;(26b) T3=etag(u)ψ(x)aψ(x)xxua+ψ(x)ut(27a) X3=etaaut+uψ(x)xψ(x)autx+ux;(27b) T4=12e12tac+k1+β+cxacu+βuk1u2ut,(28a) X4=e12tac+k1+β+cxaut+ucautxux;(28b) T5=12e12tac+k1+βcxacu+βuk1u2ut,(29a) X5=e12tac+k1+βcxaut+uc+autx+ux;(29b) T6=12e12tac+k1β+cxacuβuk1u2ut,(30a) X6=e12tac+k1β+cxaut+ucautxux;(30b) T7=12e12tac+k1βcxacuβuk1u2ut,(31a) X7=e12tac+k1βcxaut+uc+autx+ux;(31b)

5 Conclusions

In this paper, we have considered a strongly damped wave equations with two arbitrary non-zero functions f(u) and g(u) and a positive constant a. We have provided a complete classification of all point symmetries and all low-order conservation laws admitted by this equation. Equation (9) when f(u) is a linear function and g(u) is a quadratic function has abundant exact solutions that can be expressed in terms of the Jacobi elliptic functions and consequently the equation (9) has abundant exact solutions that can be expressed in terms of trigonometric and hyperbolic functions. Hence (2) has a plenty of periodic waves, solitary waves, etc. These solutions are derived from the solutions of a simple nonlinear ordinary differential equation.

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About the article

Accepted: 2017-02-17

Received: 2017-01-16

Published Online: 2017-05-04


Citation Information: Open Physics, Volume 15, Issue 1, Pages 300–305, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0033.

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© 2017 M. Luz Gandarias et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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