## 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.

Show Summary Details# Conservation laws for a strongly damped wave equation

#### Open Access

## Abstract

## 1 Introduction

## 2 Classical symmetries

#### Case 1

#### Case 2

#### Case 3

## 3 Symmetry reductions and exact solutions

#### Reduction 1

#### Reduction 2

#### Reduction 3

## 4 Conservation laws

## 5 Conclusions

## References

## About the article

## Citing Articles

*Results in Physics*, 2018*Open Physics*, 2018, Volume 16, Number 1, Page 211*Physica A: Statistical Mechanics and its Applications*, 2018

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Editor-in-Chief: Seidel, Sally

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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

Strongly damped wave equations are of a great current interest, see Refs. [6, 8–10, 13–15, 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}^{–1}*u _{t}* and a source term of the form |

It is well known [10, 18, 20, 21] that when the damping term |*u*|^{m}^{–1}*u _{t}* is absent from the equation, then the source term |

One of the most studied equation is the case with only one nonlinearity is the following $${u}_{tt}-a{u}_{txx}-{u}_{xx}+g(u)=0,$$(1) which has been considered in [8–10, 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 [2–5], 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 [2–5]. In this paper, we consider the strongly damped wave equations of the form
$${u}_{tt}+f(u){u}_{t}-a{u}_{txx}-{u}_{xx}+g(u)=0,$$(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).

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
$$\begin{array}{ccc}{x}^{\ast}& =& x+\u03f5\phantom{\rule{thinmathspace}{0ex}}\xi (x,t,u)+O({\u03f5}^{2}),\end{array}$$(3)
$$\begin{array}{ccc}{t}^{\ast}& =& t+\u03f5\phantom{\rule{thinmathspace}{0ex}}\tau (x,t,u)+O({\u03f5}^{2}),\end{array}$$(4)
$$\begin{array}{ccc}{u}^{\ast}& =& u+\u03f5\phantom{\rule{thinmathspace}{0ex}}\eta (x,t,u)+O({\u03f5}^{2}),\end{array}$$(5) where *∊* is the group parameter. The symmetry group of equation (2) will be given by the set of vector fields of the form
$$\mathbf{v}=\xi (x,t,u){\mathrm{\partial}}_{x}+\tau (x,t,u){\mathrm{\partial}}_{t}+\eta (x,t,u){\mathrm{\partial}}_{u}.$$(6)

Equation (2) admits a Lie point symmetry provided that
$$\begin{array}{c}p{r}^{(3)}\mathbf{v}(\mathit{\Delta})=0\phantom{\rule{2em}{0ex}}\text{when}\phantom{\rule{2em}{0ex}}\mathit{\Delta}=0,\end{array}$$ where *Δ* = *u _{tt}* +

with *J* = (*j*_{1}, . . ., *j _{k}*), 1 ≤

The point symmetries admitted by a strongly damped wave equation (2) in the general case are generated by $${\mathbf{v}}_{\mathbf{1}}={\mathrm{\partial}}_{x},\phantom{\rule{2em}{0ex}}{\mathbf{v}}_{\mathbf{2}}={\mathrm{\partial}}_{t}.$$

If *f*(*u*) = *α* and *g*(*u*) = *βu* + *γ*, with *α*, *β* and *γ* arbitrary constants, besides **v _{1}** and

If $f(u)=\frac{2}{a}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}g(u)=\frac{u}{{a}^{2}}+\gamma ,$ with *γ* is an arbitrary constant, besides **v _{1}**,

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).

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}* +

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
$$\begin{array}{cc}f(u)={\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}u+{\mathit{k}}_{\mathit{2}},& g(u)={\mathit{k}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{u}^{2}+{\mathit{k}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}u+{\mathit{k}}_{\mathit{5}}\end{array}$$ and we search for exact solutions of (9) with
$$\begin{array}{cc}f(h)={\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}h+{\mathit{k}}_{\mathit{2}},& g(h)={\mathit{k}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{h}^{2}+{\mathit{k}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}h+{\mathit{k}}_{\mathit{5}}.\end{array}$$

We are now considering as simplest equation $${\left({y}_{z}\right)}^{2}-{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{y}^{4}-{\mathit{k}}_{\mathit{2}}\phantom{\rule{thinmathspace}{0ex}}{y}^{3}-{\mathit{k}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{y}^{2}-{\mathit{k}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}y-{\mathit{k}}_{\mathit{5}}=0.$$(10)

The general solution of equation (10) with *a*_{1} ≠ 0 can be written in terms of the Jacobi elliptic functions *sn*(*kz*, *p*).

The general solution of equation (10) with *a*_{1} = 0 can be written in terms of the Jacobi elliptic function *sn*^{2}(*kz*, *p*) or via de Weierstrass elliptic function. We now look for exact solutions *h*(*z*) of
$$a\phantom{\rule{thinmathspace}{0ex}}c\phantom{\rule{thinmathspace}{0ex}}{h}_{zzz}+{c}^{2}{h}_{zz}-{h}_{zz}-c({\mathit{k}}_{\mathit{1}}h+{\mathit{k}}_{\mathit{2}}){h}_{z}+{\mathit{k}}_{\mathit{3}}{h}^{2}+{\mathit{k}}_{\mathit{4}}h+{\mathit{k}}_{\mathit{5}}=0$$(11) such that *h*(*z*) satisfies (10). By substituting
$${y}_{z}^{2}={\mathit{a}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{y}^{4}+{\mathit{a}}_{\mathit{2}}\phantom{\rule{thinmathspace}{0ex}}{y}^{3}+{\mathit{a}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{y}^{2}+{\mathit{a}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}y+{\mathit{a}}_{\mathit{5}}$$ and
$${y}_{zz}=2\phantom{\rule{thinmathspace}{0ex}}{\mathit{a}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{y}^{3}+\frac{3\phantom{\rule{thinmathspace}{0ex}}{\mathit{a}}_{\mathit{2}}\phantom{\rule{thinmathspace}{0ex}}{y}^{2}}{2}+{\mathit{a}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}y+\frac{{\mathit{a}}_{\mathit{4}}}{2}$$ into (11) we obtain that the following conditions must be satisfied
$$\begin{array}{lll}{a}_{1}=0,& {k}_{2}=3\phantom{\rule{thinmathspace}{0ex}}a\phantom{\rule{thinmathspace}{0ex}}{a}_{2},& {k}_{2}=a\phantom{\rule{thinmathspace}{0ex}}{a}_{3},\\ {k}_{3}=\frac{3}{2}{a}_{2}(1-{c}^{2}),& {k}_{4}={a}_{3}(1-{c}^{2}),& {k}_{5}=\frac{1}{2}{a}_{4}(1-{c}^{2}).& \end{array}$$(12)

We obtain the following:

**Theorem** Equation (11) admits any solution of equation (10) with *a*_{1} = 0 if
$$\begin{array}{cc}f(h)={\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}h+{\mathit{k}}_{\mathit{2}},& g(h)={\mathit{k}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{h}^{2}+{\mathit{k}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}h+{\mathit{k}}_{\mathit{5}}\end{array}$$ where *k _{i}*,

As a particular case for
$$\begin{array}{lll}{k}_{1}=-12a,& {k}_{2}=4a,& {k}_{3}=6({c}^{2}-1),\\ {k}_{4}=4(1-{c}^{2}),& {k}_{5}=0,& \end{array}$$ equation (11) admits the solution *h* = *sech*^{2}(*z*) and (2) admit a soliton solution
$$u(x,t)=sec{h}^{2}(x-ct)$$(13)

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

For *f*(*u*) = *α* and *g*(*u*) = *ßu*+*γ*, taking $q(x,t)=-\frac{\gamma}{\beta},$ by using generator
$$\begin{array}{c}\mathbf{v}={\mathrm{\partial}}_{t}-\left(\frac{u}{a}+\frac{\gamma}{a\beta}\right){\mathrm{\partial}}_{u}.\end{array}$$

We get the invariant solution
$$z=x,\phantom{\rule{2em}{0ex}}u={e}^{\frac{-t}{a}}h(z)-\frac{\gamma}{\beta},$$(14) where *h*(*z*) must satisfy
$$\left({a}^{2}\beta -a\alpha +1\right)h(z)=0.$$(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(u)=\frac{{a}^{2}\beta +1}{a}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}g(u)=\beta u+\gamma $ which is given by $$\begin{array}{c}u\left(x,t\right)=u={e}^{\frac{-t}{a}}h(x)-\frac{\gamma}{\beta}.\end{array}$$

Finally, for $f(u)=\frac{2}{a}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}g(u)=\frac{u}{{a}^{2}}+\gamma ,$ by using generator **v _{5}** the corresponding invariant solution is written in the form
$$z=\frac{{x}^{2}}{t},\phantom{\rule{2em}{0ex}}u={e}^{\frac{-t}{a}}h(z)-{a}^{2}\gamma ,$$(16)

where *h*(*z*) must satisfy
$$\begin{array}{l}4a{z}^{2}{h}^{\u2034}+\left({z}^{2}+10az\right){h}^{\u2033}+2\left(z+a\right){h}^{\prime}=0.\end{array}$$(17)

Integrating once with respect to *z* we obtain
$$\begin{array}{l}4a{z}^{2}{h}^{\u2033}+\left(2a+z\right){h}^{\prime}+{c}_{1}=\mathrm{0,}\end{array}$$(18)

where *c*_{0} is a constant of integration. For this reduction, ODE (18) is a second order equation whose solution is
$$\begin{array}{l}h\left(z\right)={c}_{2}+\\ \int \frac{2\sqrt{-a}\left(2{c}_{1}az{e}^{-\frac{z}{4a}}+{c}_{0}\sqrt{(}z)\right)-{c}_{0}\sqrt{\pi}E\left(\frac{1}{2}\sqrt{-\frac{z}{a}}z\right){e}^{-\frac{z}{4a}}}{4{a}^{2}z\sqrt{-\frac{z}{a}}}dz,\end{array}$$

where *c*_{1}, *c*_{2} are arbitrary constants and *E* represents the error function. Therefore, undoing transformation (16), we obtain a solution of equation (2).

We will obtain conservation laws for the strongly damped wave equation (2) by applying the general multiplier method Ref. [2–5].

A local conservation law of equation (2) is a continuity equation
$${D}_{t}{C}^{1}+{D}_{x}{C}^{2}=0,$$(19) that holds for the whole set of solutions *u*(*x*, *t*), where the conserved density *C*^{1} and the spatial flux *C*^{2} are functions of *x*, *t*, *u*, and derivatives of *u*. Here *D _{t}*,

Two local conservation laws are considered to be equivalent [5] if they differ by a trivial conservation law *C*^{1} = *D _{X}Θ*,

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 $${D}_{t}{\stackrel{~}{C}}^{1}+{D}_{x}{\stackrel{~}{C}}^{2}=\left({u}_{tt}+f(u){u}_{t}-a{u}_{txx}-{u}_{xx}+g(u)\right)\Lambda ,$$(20)

where *Λ*(*x*, *t*, *u*, *u _{x}*,

In general, a function *Λ*(*x*, *t*, *u*, *u _{x}*,

The determining equation to obtain all multipliers is $$\frac{\delta}{\delta u}(({u}_{tt}+f(u){u}_{t}-a{u}_{txx}-{u}_{xx}+g(u))\Lambda )=0.$$(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. [2–4] or by integrating the characteristic equation (20) Ref. [7].

We will now find all multipliers
$$\Lambda (x,t,u,{u}_{t},{u}_{x},{u}_{tx},{u}_{xx}),$$ and obtain corresponding non-trivial conservation laws. The determining equation (21) splits with respect to the variables *u _{tt}*,

All special cases for which additional multipliers (2) are admitted consist of: $$\begin{array}{c}{Q}_{1}={\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}+\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}},\end{array}$$(22a) $$\begin{array}{c}{Q}_{2}={\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}-\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}},\end{array}$$(22b) $$\begin{array}{rl}& f(u)={k}_{1}+{k}_{2}({k}_{3}u+{k}_{4}),\phantom{\rule{1em}{0ex}}g(u)=\frac{1}{2}{k}_{3}{u}^{2}+{k}_{4}u+{k}_{5},\\ & {k}_{1},{k}_{2},{k}_{3},{k}_{4},{k}_{5}\phantom{\rule{1em}{0ex}}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{s};\end{array}$$(22c) $$\begin{array}{c}{Q}_{3}=\psi \left(x\right){\mathrm{e}}^{\frac{t}{a}},\phantom{\rule{1em}{0ex}}\psi (x)\phantom{\rule{thinmathspace}{0ex}}\text{arbitrary},\end{array}$$(23a) $$\begin{array}{c}f(u)=a{g}^{\prime}(u)+\frac{1}{a},\phantom{\rule{1em}{0ex}}g(u)\phantom{\rule{thinmathspace}{0ex}}\text{arbitrary};\end{array}$$(23b) $$\begin{array}{c}{Q}_{4}={\mathrm{e}}^{\frac{1}{2}\left(-ta\mathit{c}+t{\mathit{k}}_{\mathit{1}}+t\beta +\sqrt{\mathit{c}}x\right)},\end{array}$$(24a) $$\begin{array}{c}{Q}_{5}={\mathrm{e}}^{\frac{1}{2}\left(-ta\mathit{c}+t{\mathit{k}}_{\mathit{1}}+t\beta -\sqrt{\mathit{c}}x\right)},\end{array}$$(24b) $$\begin{array}{c}{Q}_{6}={\mathrm{e}}^{\frac{1}{2}\left(-ta\mathit{c}+t{\mathit{k}}_{\mathit{1}}-t\beta +\sqrt{\mathit{c}}x\right)},\end{array}$$(24c) $$\begin{array}{c}{Q}_{7}={\mathrm{e}}^{\frac{1}{2}\left(-ta\mathit{c}+t{\mathit{k}}_{\mathit{1}}-t\beta -\sqrt{\mathit{c}}x\right)},\end{array}$$(24d) $$\begin{array}{rl}& f(u)={k}_{1},\phantom{\rule{1em}{0ex}}g(u)={k}_{2}u\\ & \text{with}\phantom{\rule{1em}{0ex}}\beta =\sqrt{{a}^{2}{\mathit{c}}^{2}-2\phantom{\rule{thinmathspace}{0ex}}a{\mathit{k}}_{\mathit{1}}\mathit{c}+{\mathit{k}}_{\mathit{1}}^{\mathit{2}}-4\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}+4\phantom{\rule{thinmathspace}{0ex}}\mathit{c}},\\ & {k}_{1},{k}_{2},c\phantom{\rule{1em}{0ex}}\text{constants};\end{array}$$(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:
$$\begin{array}{ll}{T}_{1}=& \left({u}_{t}-\frac{a{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}-{u}_{}}{a-{\mathit{k}}_{\mathit{2}}}\right){\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}+\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}}\\ & \left({\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}+{\mathit{k}}_{\mathit{2}}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{1}{2}{\mathit{k}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{{u}_{}}^{2}+{\mathit{k}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}+{\mathit{k}}_{\mathit{5}}\right)\right){\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}+\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}},\end{array}$$(25a)
$$\begin{array}{cc}{X}_{1}=& \frac{-\left(a{u}_{t}+{u}_{}\right)\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}\left(a-{\mathit{k}}_{\mathit{2}}\right)}}{\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}+\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}}\\ & \left(a{u}_{tx}+{u}_{x}\right){\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}+\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}};\end{array}$$(25b)
$$\begin{array}{ll}{T}_{2}=& \left({u}_{t}-\frac{a{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}-{u}_{}}{a-{\mathit{k}}_{\mathit{2}}}\right){\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}-\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}}\\ & \left({\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}+{\mathit{k}}_{\mathit{2}}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{1}{2}{\mathit{k}}_{\mathit{3}}\phantom{\rule{thinmathspace}{0ex}}{{u}_{}}^{2}+{\mathit{k}}_{\mathit{4}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}+{\mathit{k}}_{\mathit{5}}\right)\right){\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}-\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}},\end{array}$$(26a)
$$\begin{array}{cc}{X}_{2}& =\frac{\left(a{u}_{t}+{u}_{}\right)\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}\left(a-{\mathit{k}}_{\mathit{2}}\right)}}{\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}-\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}}\\ & -\left(a{u}_{tx}+{u}_{x}\right){\mathrm{e}}^{\frac{t}{{\mathit{k}}_{\mathit{2}}}-\frac{x\sqrt{{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{k}}_{\mathit{2}}-1}}{\sqrt{{\mathit{k}}_{\mathit{2}}}\sqrt{a-{\mathit{k}}_{\mathit{2}}}}};\end{array}$$(26b)
$$\begin{array}{c}{T}_{3}={\mathrm{e}}^{\frac{t}{a}}\left(g(u)\psi (x)a-\psi (x{)}_{xx}{u}_{}a+\psi (x){u}_{t}\right)\end{array}$$(27a)
$$\begin{array}{c}{X}_{3}={\mathrm{e}}^{\frac{t}{a}}\left(\left(a{u}_{t}+{u}_{}\right)\psi (x{)}_{x}-\psi (x)\left(a{u}_{tx}+{u}_{x}\right)\right);\end{array}$$(27b)
$$\begin{array}{c}{T}_{4}=-\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}+\beta \right)+\sqrt{\mathit{c}}x}\left(a\mathit{c}{u}_{}+\beta {u}_{}-{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}-2\phantom{\rule{thinmathspace}{0ex}}{u}_{t}\right),\end{array}$$(28a)
$$\begin{array}{cc}{X}_{4}& ={\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}+\beta \right)+\sqrt{c}x}\left(\left(a{u}_{t}+{u}_{}\right)\sqrt{\mathit{c}}-a{u}_{tx}-{u}_{x}\right);\end{array}$$(28b)
$$\begin{array}{c}{T}_{5}=-\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}+\beta \right)-\sqrt{\mathit{c}}x}\left(a\mathit{c}{u}_{}+\beta {u}_{}-{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}-2\phantom{\rule{thinmathspace}{0ex}}{u}_{t}\right),\end{array}$$(29a)
$$\begin{array}{cc}{X}_{5}& =-{\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}+\beta \right)-\sqrt{c}x}\left(\left(a{u}_{t}+{u}_{}\right)\sqrt{\mathit{c}}+a{u}_{tx}\right.\\ & \left.+{u}_{x}\right);\end{array}$$(29b)
$$\begin{array}{c}{T}_{6}=-\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}-\beta \right)+\sqrt{\mathit{c}}x}\left(a\mathit{c}{u}_{}-\beta {u}_{}-{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}-2\phantom{\rule{thinmathspace}{0ex}}{u}_{t}\right),\end{array}$$(30a)
$$\begin{array}{c}{X}_{6}={\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}-\beta \right)+\sqrt{c}x}\left(\left(a{u}_{t}+{u}_{}\right)\sqrt{\mathit{c}}-a{u}_{tx}-{u}_{x}\right);\end{array}$$(30b)
$$\begin{array}{cc}{T}_{7}& =-\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}-\beta \right)-\sqrt{\mathit{c}}x}\left(a\mathit{c}{u}_{}-\beta {u}_{}-{\mathit{k}}_{\mathit{1}}\phantom{\rule{thinmathspace}{0ex}}{u}_{}-2\phantom{\rule{thinmathspace}{0ex}}{u}_{t}\right),\end{array}$$(31a)
$$\begin{array}{cc}{X}_{7}& =-{\mathrm{e}}^{\frac{1}{2}t\left(-ac+{k}_{1}-\beta \right)-\sqrt{c}x}\left(\left(a{u}_{t}+{u}_{}\right)\sqrt{\mathit{c}}+a{u}_{tx}+{u}_{x}\right);\end{array}$$(31b)

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.

- [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. Web of ScienceGoogle Scholar

- [2]
Anco S.C., Bluman G., Direct construction of conservation laws from field equations, Phys. Rev. Lett., 1997, 78, 2869–2873. CrossrefGoogle Scholar

- [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. CrossrefGoogle Scholar

- [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. CrossrefGoogle Scholar

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

- [10]
Dell’Oro F., Pata V., Strongly damped wave equations with critical nonlinearities, Nonlinear Anal., 2012, 75(14), 5723–5735. CrossrefWeb of ScienceGoogle Scholar

- [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. Web of ScienceCrossrefGoogle Scholar

- [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. Web of ScienceGoogle Scholar

- [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. CrossrefGoogle Scholar

- [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. CrossrefGoogle Scholar

- [16]
Kudryashov N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons Fractals, 2005, 24, 1217–1231. CrossrefGoogle Scholar

- [17]
Kudryashov N.A., Loguinova N.B., Extended simplest equation method for nonlinear differential equations, Applied Mathematics and Computation, 2008, 205, 396–402. Web of ScienceCrossrefGoogle Scholar

- [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. CrossrefGoogle Scholar

- [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. CrossrefGoogle Scholar

- [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. CrossrefGoogle Scholar

- [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. CrossrefWeb of ScienceGoogle Scholar

- [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. CrossrefWeb of ScienceGoogle Scholar

- [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. CrossrefGoogle Scholar

**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.

© 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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