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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# 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.

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 $utt−uxx+|u|m−1ut=|u|p−1u.$

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 $∂t2u−∂tΔxu−Δxu+f(u)=∇x.ϕ′.(∇xu)+g,$ 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 $utt−Δut−Δu+f(ut)+g(u)=h,$ 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 $utt−autxx−uxx+g(u)=0,$(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 $utt+f(u)ut−autxx−uxx+g(u)=0,$(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: $pr(3)v=v+ζt∂ut+ζtt∂utt+ζxx∂uxx+ζtxx∂utxx,$(7) where $ζJ(x,t,u(3))=DJ(η−τut−ξux)+τuJt+ξuJx,$

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: $v3=u∂u,v4=q(x,t)∂u,$ where q(x, t) must satisfy equation (2).

#### Case 3

If $f\left(u\right)=\frac{2}{a}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}g\left(u\right)=\frac{u}{{a}^{2}}+\gamma ,$ with γ is an arbitrary constant, besides v1, v2 and v4, we obtain the following generator: $v5=x2∂x+t∂t−tu+a2γa∂u.$

## 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 $z=x−λt,u=h(z),$(8) and the following reduced ODE $achzzz+c2hzz−hzz−cf(h)hz+g(h)=0.$(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 $yz2−k1y4−k2y3−k3y2−k4y−k5=0.$(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 $achzzz+c2hzz−hzz−c(k1h+k2)hz+k3h2+k4h+k5=0$(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 $a1=0,k2=3aa2,k2=aa3,k3=32a2(1−c2),k4=a3(1−c2),k5=12a4(1−c2).$(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 $k1=−12a,k2=4a,k3=6(c2−1),k4=4(1−c2),k5=0,$ equation (11) admits the solution h = sech2(z) and (2) admit a soliton solution $u(x,t)=sech2(x−ct)$(13)

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

Figure 1

Soliton solution (13) with c = 1.

#### Reduction 2

For f(u) = α and g(u) = ßu+γ, taking $q\left(x,t\right)=-\frac{\gamma }{\beta },$ by using generator $v=∂t−ua+γaβ∂u.$

We get the invariant solution $z=x,u=e−tah(z)−γβ,$(14) where h(z) must satisfy $a2β−aα+1h(z)=0.$(15)

If we take $\alpha =\frac{{a}^{2}\beta +1}{a}$ equation (15) is trivially satisfied. Thus, undoing transformation (14), we obtain a solution of equation (2) for $f\left(u\right)=\frac{{a}^{2}\beta +1}{a}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}g\left(u\right)=\beta u+\gamma$ which is given by $ux,t=u=e−tah(x)−γβ.$

#### Reduction 3

Finally, for $f\left(u\right)=\frac{2}{a}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}g\left(u\right)=\frac{u}{{a}^{2}}+\gamma ,$ by using generator v5 the corresponding invariant solution is written in the form $z=x2t,u=e−tah(z)−a2γ,$(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 $hz=c2+∫2−a2c1aze−z4a+c0(z)−c0πE12−zaze−z4a4a2z−zadz,$

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 $DtC~1+DxC~2=utt+f(u)ut−autxx−uxx+g(u)Λ,$(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 $δδu((utt+f(u)ut−autxx−uxx+g(u))Λ)=0.$(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: $Q1=etk2+xk1k2−1k2a−k2,$(22a) $Q2=etk2−xk1k2−1k2a−k2,$(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) $Q4=e12−tac+tk1+tβ+cx,$(24a) $Q5=e12−tac+tk1+tβ−cx,$(24b) $Q6=e12−tac+tk1−tβ+cx,$(24c) $Q7=e12−tac+tk1−tβ−cx,$(24d) $f(u)=k1,g(u)=k2uwithβ=a2c2−2ak1c+k12−4k2+4c,k1,k2,cconstants;$(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: $T1=ut−ak1u−ua−k2etk2+xk1k2−1k2a−k2k1u+k212k3u2+k4u+k5etk2+xk1k2−1k2a−k2,$(25a) $X1=−aut+uk1k2−1k2a−k2etk2+xk1k2−1k2a−k2autx+uxetk2+xk1k2−1k2a−k2;$(25b) $T2=ut−ak1u−ua−k2etk2−xk1k2−1k2a−k2k1u+k212k3u2+k4u+k5etk2−xk1k2−1k2a−k2,$(26a) $X2=aut+uk1k2−1k2a−k2etk2−xk1k2−1k2a−k2−autx+uxetk2−xk1k2−1k2a−k2;$(26b) $T3=etag(u)ψ(x)a−ψ(x)xxua+ψ(x)ut$(27a) $X3=etaaut+uψ(x)x−ψ(x)autx+ux;$(27b) $T4=−12e12t−ac+k1+β+cxacu+βu−k1u−2ut,$(28a) $X4=e12t−ac+k1+β+cxaut+uc−autx−ux;$(28b) $T5=−12e12t−ac+k1+β−cxacu+βu−k1u−2ut,$(29a) $X5=−e12t−ac+k1+β−cxaut+uc+autx+ux;$(29b) $T6=−12e12t−ac+k1−β+cxacu−βu−k1u−2ut,$(30a) $X6=e12t−ac+k1−β+cxaut+uc−autx−ux;$(30b) $T7=−12e12t−ac+k1−β−cxacu−βu−k1u−2ut,$(31a) $X7=−e12t−ac+k1−β−cxaut+uc+autx+ux;$(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.

## References

• [1]

Adem K.R., Khalique C.M., Symmetry analysis and conservation laws of a generalized two-dimensional nonlinear KP-MEW equation, Mathematical Problems in Engineering, 2015, 2015, Article ID 805763.

• [2]

Anco S.C., Bluman G., Direct construction of conservation laws from field equations, Phys. Rev. Lett., 1997, 78, 2869–2873.

• [3]

Anco S.C., Bluman G., Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications, Eur. J. Appl. Math., 2002, 13, 545–566. Google Scholar

• [4]

Anco S.C., Bluman G., Direct construction method for conservation laws for partial differential equations Part II: General treatment, Euro. Jnl of Applied Mathematics, 2002, 41, 567–585. Google Scholar

• [5]

Anco S.C., Generalization of Noether’s theorem in modern form to non-variational partial differential equations, To appear in Fields Institute Communications: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. arXiv: mathph/1605.08734 Google Scholar

• [6]

Arima R., Hasegawa Y., On global solutions for mixed problem of a semi-linear differential equation, Proc. Jpn. Acad., 1963, 39, 721–725.

• [7]

Bluman G.W., Cheviakov A., Anco S.C., Applications of symmetry methods to partial differential equations, New York: Springer, 2009. Google Scholar

• [8]

Carvalho A., Cholewa J., Attractors for strongly damped wave equations with critical nonlinearities, Pac. J. Math., 2002, 207, 287–310.

• [9]

Carvalho A., Cholewa J., Dlotko T., Strongly damped wave problems: Bootstrapping and regularity of solutions, J. Differ. Equations, 2008, 244(9), 2310–2333.

• [10]

Dell’Oro F., Pata V., Strongly damped wave equations with critical nonlinearities, Nonlinear Anal., 2012, 75(14), 5723–5735.

• [11]

Gandarias M.L., Khalique CM., Symmetries solutions and conservation laws of a class of nonlinear dispersive wave equations, Communications in Nonlinear Science and Numerical Simulation, 2016, 32, 114–121.

• [12]

Gandarias M.L., Bruzón M.S., Rosa M., Symmetries and conservation laws for some compacton equation, Mathematical Problems in Engineering, 2015, 2015, Article ID 430823.

• [13]

Ghidaglia J., Marzocchi A., Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 1991, 22, 879–895.

• [14]

Kalantarov V.K., Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 1986, 152, 50–54. Google Scholar

• [15]

Khanmamedov A., Strongly damped wave equation with exponential nonlinearities, J. Math. Anal. Appl., 2014, 419, 663–687.

• [16]

Kudryashov N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons Fractals, 2005, 24, 1217–1231.

• [17]

Kudryashov N.A., Loguinova N.B., Extended simplest equation method for nonlinear differential equations, Applied Mathematics and Computation, 2008, 205, 396–402.

• [18]

Li H., Zhou S., Yin F., Global periodic attractor for strongly damped wave equations with time-periodic driving force, J. Math. Phys., 2004, 45, 3462–3467.

• [19]

Olver P., Applications of Lie groups to differential equations, Springer-Verlag, 1993. Google Scholar

• [20]

Pata V., Squassina M., On the strongly damped wave equation, Commun. Math. Phys., 2005, 253, 511–533.

• [21]

Rammaha M., Strei T., Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Am. Math. Soc, 2002, 354(9), 3621–3637.

• [22]

Tracinà R., Bruzón M.S., Gandarias M.L., On the nonlinear self-adjointness of a class of fourth-order evolution equations, Applied Mathematics and Computation, 2016, 275, 299–304.

• [23]

Tracinà R., Freire I. L., Torrisi M., Nonlinear self-adjointness of a class of third order nonlinear dispersive equations, Communications in Nonlinear Science and Numerical Simulation, 2016, 32, 225–233.

• [24]

Yang M., Sun C., Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity, Trans. Am. Math. Soc., 2009, 361(2), 1069–1101. Google Scholar

• [25]

Webb G., Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Can. J. Math., 1980, 32, 631–643.

Accepted: 2017-02-17

Published Online: 2017-05-04

Citation Information: Open Physics, Volume 15, Issue 1, Pages 300–305, ISSN (Online) 2391-5471,

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