Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation

Abstract

Summary. Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equationFor p > 5, the equation has both stable and unstable solitary-wave solutions, according to the theory of Souganidis and Strauss. Using a high-order accurate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among other conclusions, we find that unstable solitary waves may evolve into several, stable solitary waves and that positive initial data need not feature solitary waves at all in its long-time asymptotics.

Orbital stability of negative solitary waves

“…As also observed in [8], if p ≥ 2 is odd, −Φ c (x − ct) is also a solution of the generalized BBM equation. This follows immediately from the fact that −u also satisfies Eq.…”

“…This result was proved by Souganidis and Strauss [16] using the general theory of Albert, Bona, Grillakis, Souganidis, Shatah and Strauss laid down in [9,11], and pioneered by Boussinesq, Benjamin and coworkers [4,7,10,15]. For a thorough review of the results, and a numerical study of the stability of positive solitary waves, the reader may refer [8].…”

Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations

“…These values have been obtained by means of the fourth-order accurate Eqs. (19) and (20) which also require the inversion of two diagonally-dominant tridiagonal matrices for the determination of F i and G i , subject to the same conditions as the ones specified above at i ¼ 1 and i ¼ N. Therefore, the compact method presented in this section is global because F i and G i are calculated from Eqs. (19) and (20) and is expected to be more computationally costly than a second-order accurate discretization of the spatial derivatives of u and v. Hereafter, we shall refer to the compact method with h ¼ 1 2 presented in this section as C244.…”

“…The GRLW equation has been studied both analytically by means of a large variety of approximation, perturbation, variational, Adomian's decomposition, etc., methods [13][14][15][16] and numerically by means of explicit and implicit procedures [17], energy-preserving (conservative) finite difference methods [18], methods of lines with Fourier-pseudospectral approximations for periodic boundary conditions [19], third-order accurate Runge-Kutta techniques [20], iterative finite difference methods [21], Galerkin, Petrov-Galerkin and cubic and quadartic B-splines techniques [22][23][24][25][26][27], meshless techniques [28], sinc-collocation procedures [29], etc.…”

Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

“…Some of the previous works on the GRLW equation include an implicit second-order accurate and stable energy preserving finite difference method based on the use of central difference equations for the time and space derivatives [2], a method of lines based on the discretization of the spatial derivatives by means of Fourier pseudo-spectral approximations [3], a Fourier spectral method for the initial value problem of the GRLW equation [4], and a linearized implicit pseudo-spectral method [5]. Moreover numerical techniques such as finite difference methods [6][7][8][9][10][11], a spectral method [12], finite element methods based on the least square principle [13][14][15], finite element methods based on Galerkin and collocation principles [16][17][18][19][20][21][22][23][24], the Petrov-Galerkin method [25], the radial basis function collocation method [26,27], the Sinc-collocation method [28], the collocation method with quintic B-splines [29] and the cubic B-spline finite element method [30,31] have been devised for finding numerical solutions of special kinds of GRLW equations.…”