Symmetry reductions, exact solutions, and conservation laws of the generalized Zakharov equations

Abstract

In this paper, the generalized Zakharov equations, which describe interactions between high- and low-frequency waves in plasma physics are studied from the perspective of Lie symmetry analysis and conservation laws. Based on some subalgebras of symmetries, several reductions and numerous new exact solutions are obtained. All of these solutions represent modified traveling waves. The obtained solutions include expressions involving Airy functions, Bessel functions, Whittaker functions, and generalized hypergeometric functions. Previously unknown conservation laws are constructed for the generalized Zakharov equations using the direct method. Profiles are presented for some of these new solutions.

Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons

“…The results again shows that the conservation theorem presented by Ibragimov along with some of its extensions are shown to comprise a special case of a conservation identity (18) for the SA method.…”

“…In general, for a given PDE system (1) expressed in a standard Cauchy-Kovalevskaya form, all non-trivial conservation laws arise from linear combinations of the PDE system (1) with multipliers that hold non-trivial divergence expressions [4,5,9,18]. In this paper, we suppose that (1) is expressed in a standard Cauchy-Kovalevskaya form.…”

Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation

“…In this part, we derive the optimal system of sub-algebras [4] of the vector fields in Equations (5)- (8). To do this, we begin by noting that each X i (i = 1, 2, 3, 4) yields an adjoint representation Ad(exp( X i ))X j , defined by [4] …”

“…In the last few decades, many scientists and mathematicians have extensively studied the dynamic behaviors of several PDEs using different concepts. Symmetry analyses have been used to study the PDEs [2][3][4][5][6]. It has been found that some new solutions to PDEs can be obtained from the old ones through symmetry transformations [2].…”

“…The main aim of this work is to derive the Lie point symmetries [2,4], the optimal system of one-dimensional sub-algebras, invariant solutions, and the CL of the equation by invoking the conservation theorem due to Ibragimov [7,8]. The invariant solutions are derived by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction process using the Riccati Bernoulli sub-ODE (RBSO) [19].…”