Conservation laws of a nonlinear (n+1) wave equation

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Abstract

Conservation laws of the nonlinear (n+1) wave equation utt=div(f(u)gradu) involving an arbitrary function of the dependent variable, are obtained. This equation is not derivable from a variational principle. By writing the equation, which admits a partial Lagrangian, in the partial Euler–Lagrange form, partial Noether operators associated with the partial Lagrangian are obtained for all possible cases of the arbitrary function. Partial Noether operators are used via a formula in the construction of the conservation laws of the wave equation. If f(u) is an arbitrary function, we show that there is a finite number of conservation laws for n=1 and an infinite number of conservation laws for n2. None of the partial Noether operators is a Lie point symmetry of the equation. If f is constant, where all of the partial Noether operators are point symmetries of the equation, there is also an infinite number of conservation laws.

Introduction

The relationship between symmetries and conservation laws of differential equations has been a topic of great interest (see, e.g. [1], [2], [3], [4], [5], [6], [7]. A systematic way for the determination of conservation laws associated with variational symmetries for systems of Euler–Lagrange equations is indeed the famous Noether theorem [8], [9] (see also [1], [2], [3], [4], [5], [7], [10]. This theorem requires a Lagrangian. There are approaches that do not require a Lagrangian or even assume the existence of a Lagrangian for differential equations (DEs), e.g. scalar evolution equations (see, e.g. [5] and the recent paper [11]). Direct construction methods for multipliers and hence the conservation laws [12], Lagrangian approach for evolution equations [13] and formula for relationship between symmetries and conservation laws, irrespective of the existence of a Lagrangian of the system [7], have been investigated. Also, a basis of conservation laws was further investigated in [14] for DEs with and without Lagrangian formulation. Kara and Mahomed in [10] presented a new method to construct conservation laws of DEs via operators that are not necessarily symmetry generators of the underlying system. These partial Noether operators which are associated with partial Lagrangians are used via an explicit Noether-like formula in the construction of conservation laws of the system which need not be derivable from a variational principle. These systems are referred to as partial Euler–Lagrange equations with respect to partial Lagrangians. This approach provides a systematic way of obtaining conservation laws for systems which have partial Lagrangian formulations.

There has been much focus on the determination of conservation laws for various physical systems (see, e.g. [5], [11], [15]). In [11] a (2+1) evolution equation was considered for its conserved quantities using the direct method. Moreover there have been recent works on the (1+1) wave equation in [15] which contains two arbitrary functions. We provide a natural extension of [15], when one of the functions is zero, to the case of n space variables. For n=1 we recover the results of [15] in the general case.

The outline of the paper is as follows. In Section 2, we present salient points of the necessary theory. Then in Section 3, investigation on the existence of conservation laws of the nonlinear (2+1) wave equation for all cases of f(u) is carried out using the results of Section 2, and we derive new conservation laws for this equation. Then in Section 4, we generalize our work to the nonlinear (n+1) wave equation for all possibilities of the function f(u). Concluding remarks are given in the last section.

Section snippets

Operators and the partial Noether’s theorem

Consider the kth-order system of partial differential equations (PDEs) of n independent variables x=(x1,x2,,xn) and m dependent variables u=(u1,u2,,um)Eα(x,u,u(1),,u(k))=0,α=1,,m, where u(1),u(2),,u(k) denote the collections of all first, second, kth-order partial derivatives, i.e., uiα=Di(uα),uijα=DjDi(uα), …  respectively, with the total differentiation operator with respect to xi given by Di=xi+uiαuα+uijαujα+,i=1,,n, in which the summation convention is used.

The following

Application to nonlinear (2+1) wave equation

Our objective is to obtain all the first order conserved quantities of the nonlinear (2+1) wave equation utt(f(u)ux)x(f(u)uy)y=0. It should be remarked that this equation is not derivable from a variational principle. Here, we investigate conservation laws of the equation for all possible forms of f(u). We have looked at point type operators and we have restricted the gauge terms to be independent of derivatives. For simplicity, we denote the derivative of u w.r.t. the independent variables (u

Extension to nonlinear (n+1) wave equation

Our aim here is to obtain all the first-order conserved quantities of the following nonlinear (n+1) wave equation utt(f(u)uxi)xi=0,i=1,,n. It should be pointed out that this equation is not derivable from a variational principle. Here, we investigate conservation laws of the equation for all possible functions f(u). We have looked at point type operators and we have restricted the gauge terms to be independent of derivatives. Again, for simplicity, we denote the variables x1,,xn as xs and

Conclusion

New conservation laws are constructed for the nonlinear (n+1), n1 wave equation which is not derivable from a variational principle. We use the approach of [10]. For the equation containing an arbitrary function of the dependent variable, all possible cases are considered. When f(u) is an arbitrary function, we show that there is a finite number of conservation laws for n=1 which appear in [15] and an infinite number of conservation laws for n2. None of the partial Noether symmetry operators

Acknowledgements

We thank the Department of Mathematics and Statistics at the King Fahd University of Petroleum and Minerals, Saudi Arabia, for providing research facilities and FM also registers his gratitude to the Department for hospitality during the time this work was undertaken.

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