Stability Analysis for Travelling Wave Solutions of the Olver and Fifth-Order KdV Equations

Abstract

The Olver equation is governing a unidirectional model for describing long and small amplitude waves in shallow water waves. The solitary wave solutions of the Olver and fifth-order KdV equations can be obtained by using extended tanh and sech-tanh methods. The present results are describing the generation and evolution of such waves, their interactions, and their stability. Moreover, the methods can be applied to a wide class of nonlinear evolution equations. All solutions are exact and stable and have applications in physics.

Boundary value problem for the nonlinear analogues of the Boussinesq equation: Numerical solution and its stability

“…In order to see the sustainability of the system, the solution must be tested in terms of stability, a general description of which was given by Lyapunov [24] as the tendency of the system to reestablish its equilibrium state. More specifically, for the case of solitary and travelling wave solutions of the initial boundary value problems, stability condition was usually analyzed via energy considerations [25,26,27,28,9]. According to Benjamin [25], solitary waves of constant momentum must be local energy minimizers for stability.…”

“…In addition, Grillakis et al [28] studied the stability of solitary waves for symmetric systems and used the condition E ′′ (φ )≥0, where E is the energy and φ is the solitary wave solution. Seadawy et al [9] found the travelling wave solutions for small amplitude waves in shallow waters supposing ζ =χ − kt where χ is the position in the direction of propagation and k is the propagation speed of the wave. Defining the Hamiltonian system for the momentum as ν, they expressed the sufficient condition for stability as ∂ ν/∂ k>0 [9,10,11,12,13].…”

“…Seadawy et al found the solitary wave solutions for Zufiria's high-order Boussinesq equation using a series expansion method [8]. They showed that this approach can be used to find the analytical solitary wave solutions for several types of partial differential equations [9,10,11,12,13,14,15]. Another interesting paper was published by Moutsopoulos who presented an analytical solution for the case of a sudden change in the depth of the water by matching the Adomian decomposition method and Tolikas's polynomial expression for upstream and downstream regions, respectively [16,17].…”

H4-Solutions for the Olver–Benney equation

“…From a mathematical point of view, under suitable assumptions on , q, , , , , the existence of the travelling waves solutions for (1.6) is proven in [14,20], while a method to find exact solutions of (1.6) is given in [13]. Instead, in [19], the local wellposedness of the Cauchy problem of (1.1) is proven.…”

New exact solutions for Kudryashov–Sinelshchikov equation

“…For example, the homogeneous balance method [21], the tanh method [22], the Jacobi elliptic function expansion [23][24][25][26], the truncated Painlevé expansion [27], differential quadrature method [28], Hirota bilinear method [29], Darboux transformations [30], the trial equation method [31]. Seadawy et al [32] proposed the sech-tanh method to solve the Olver equation and the fifth-order KdV equation and obtained traveling wave solutions; in [33][34][35][36][37][38] was introduced a method called the G G -expansion method and one obtained a traveling solution for the four well established nonlinear evolution equations. In [5], the authors obtained traveling wave solutions for the generalized Camassa-Holm equation by polynomial expansion methods.…”