Boussinesq solitary-wave as a multiple-time solution of the Korteweg–de Vries hierarchy

Abstract

We study the Boussinesq equation from the point of view of a multiple-time reductive perturbation method. As a consequence of the elimination of the secular producing terms through the use of the Korteweg-de Vries hierarchy, we show that the solitary-wave of the Boussinesq equation is a solitary-wave satisfying simultaneously all equations of the Korteweg-de Vries hierarchy, each one in an appropriate slow time variable.

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

“…It is customary to combine (24) and (32) to yields a higher-order KdV equation but the precise form of this higher-order KdV equation is not unique [23,24,25]. The problem of non-uniqueness does not occur in the method we are using in our analytical work, where the system of reduced amplitude equations (24) and (32) is uniquely determined for the Cauchy problem associated with the regularized Boussinesq equation (13).…”

“…The bound (34) of Theorem 1 generalizes the result of Theorem 7 in [7] obtained in the context of Boussinesq systems. However, if the authors of [7] restrict their consideration to the zero mean value for the initial data (case (ii') in Theorem 7) and show that the nonzero mean values do not produce good long-wave approximations (cases (ii) and (iii) in Theorem 7), we show that the mean value in the initial data for U | t=0 = F can be naturally incorporated in the justification analysis by modifying the velocity term of the uncoupled KdV equations (24). .…”

“…Before closing the introduction, we shall also discuss the reductive perturbation schemes, which were used to obtain the integrable KdV hierarchy from the shallow water-wave and Boussinesq equations [23,24,25]. It was later shown in [17,22] that there are obstacles to the asymptotic integrability of the original physical equations reduced to the integrable KdV hierarchy in the sense that the formal asymptotic expansions become non-uniform at higher orders of .…”

Integrable theory of the perturbation equations

“…Recently it has been observed by Kraenkel et al. that a multiple time scale expansion may relate the solutions of long surface water-waves and the Boussinesq equation to KdV integrable hierarchy [15] [16] .…”

The Modified Reductive Perturbation Method as Applied to the Boussinesq Equation

“…The results of the present work and of those given in [12] and [13] proved that the "modified reductive perturbation method", presented by us, is the most simple and effective one. The present problem has been studied by Kraenkel et al [14] but the method they used is quite complicated compared to ours. The presented method is rather simple and it is based on the idea of balancing higher-order nonlinearities with higher-order dispersive effects.…”