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Conservation laws for a strongly damped wave equation

  • Maria Luz Gandarias EMAIL logo , Rafael de la Rosa and Maria Rosa
Published/Copyright: May 4, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

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, 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–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

(1)utt−autxx−uxx+g(u)=0,

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

(2)utt+f(u)ut−autxx−uxx+g(u)=0,

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

(3)x∗=x+ϵξ(x,t,u)+O(ϵ2),
(4)t∗=t+ϵτ(x,t,u)+O(ϵ2),
(5)u∗=u+ϵη(x,t,u)+O(ϵ2),

where ∊ is the group parameter. The symmetry group of equation (2) will be given by the set of vector fields of the form

(6)v=ξ(x,t,u)∂x+τ(x,t,u)∂t+η(x,t,u)∂u.

Equation (2) admits a Lie point symmetry provided that

pr(3)v(Δ)=0whenΔ=0,

where Δ = utt + f(u)ut – autxx –uxx + g(u), and pr(3)v is the third prolongation of the vector field (6) which is given by:

(7)pr(3)v=v+ζt∂ut+ζtt∂utt+ζxx∂uxx+ζtxx∂utxx,

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(u)=2aandg(u)=ua2+γ, 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

(8)z=x−λt,u=h(z),

and the following reduced ODE

(9)achzzz+c2hzz−hzz−cf(h)hz+g(h)=0.

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

(10)yz2−k1y4−k2y3−k3y2−k4y−k5=0.

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

(11)achzzz+c2hzz−hzz−c(k1h+k2)hz+k3h2+k4h+k5=0

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

(12)a1=0,k2=3aa2,k2=aa3,k3=32a2(1−c2),k4=a3(1−c2),k5=12a4(1−c2).

We obtain the following:

TheoremEquation (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
(13)u(x,t)=sech2(x−ct)

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

Figure 1 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=∂t−ua+γaβ∂u.

We get the invariant solution

(14)z=x,u=e−tah(z)−γβ,

where h(z) must satisfy

(15)a2β−aα+1h(z)=0.

If we take α=a2β+1aequation (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=e−tah(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

(16)z=x2t,u=e−tah(z)−a2γ,

where h(z) must satisfy

(17)4az2h‴+z2+10azh″+2z+ah′=0.

Integrating once with respect to z we obtain

(18)4az2h″+2a+zh′+c1=0,

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. [2–5].

A local conservation law of equation (2) is a continuity equation

(19)DtC1+DxC2=0,

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

(20)DtC~1+DxC~2=utt+f(u)ut−autxx−uxx+g(u)Λ,

where Λ(x, t, u, ux, ut, ...) is a multiplier, and where (C̃1, C̃2) 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

(21)δδu((utt+f(u)ut−autxx−uxx+g(u))Λ)=0.

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

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:

(22a)Q1=etk2+xk1k2−1k2a−k2,
(22b)Q2=etk2−xk1k2−1k2a−k2,
(22c)f(u)=k1+k2(k3u+k4),g(u)=12k3u2+k4u+k5,k1,k2,k3,k4,k5constants;
(23a)Q3=ψxeta,ψ(x)arbitrary,
(23b)f(u)=ag′(u)+1a,g(u)arbitrary;
(24a)Q4=e12−tac+tk1+tβ+cx,
(24b)Q5=e12−tac+tk1+tβ−cx,
(24c)Q6=e12−tac+tk1−tβ+cx,
(24d)Q7=e12−tac+tk1−tβ−cx,
(24e)f(u)=k1,g(u)=k2uwithβ=a2c2−2ak1c+k12−4k2+4c,k1,k2,cconstants;

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:

(25a)T1=ut−ak1u−ua−k2etk2+xk1k2−1k2a−k2k1u+k212k3u2+k4u+k5etk2+xk1k2−1k2a−k2,
(25b)X1=−aut+uk1k2−1k2a−k2etk2+xk1k2−1k2a−k2autx+uxetk2+xk1k2−1k2a−k2;
(26a)T2=ut−ak1u−ua−k2etk2−xk1k2−1k2a−k2k1u+k212k3u2+k4u+k5etk2−xk1k2−1k2a−k2,
(26b)X2=aut+uk1k2−1k2a−k2etk2−xk1k2−1k2a−k2−autx+uxetk2−xk1k2−1k2a−k2;
(27a)T3=etag(u)ψ(x)a−ψ(x)xxua+ψ(x)ut
(27b)X3=etaaut+uψ(x)x−ψ(x)autx+ux;
(28a)T4=−12e12t−ac+k1+β+cxacu+βu−k1u−2ut,
(28b)X4=e12t−ac+k1+β+cxaut+uc−autx−ux;
(29a)T5=−12e12t−ac+k1+β−cxacu+βu−k1u−2ut,
(29b)X5=−e12t−ac+k1+β−cxaut+uc+autx+ux;
(30a)T6=−12e12t−ac+k1−β+cxacu−βu−k1u−2ut,
(30b)X6=e12t−ac+k1−β+cxaut+uc−autx−ux;
(31a)T7=−12e12t−ac+k1−β−cxacu−βu−k1u−2ut,
(31b)X7=−e12t−ac+k1−β−cxaut+uc+autx+ux;

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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Received: 2017-1-16
Accepted: 2017-2-17
Published Online: 2017-5-4

© 2017 M. Luz Gandarias et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0033
Keywords for this article
Lie symmetries; Exact solutions; Partial differential equations; Conservation laws
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  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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