Orbital stability of solitary waves for Kundu equation

Abstract

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987Grillakis et al. ( , 1990 cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c 3 + s 2 υ < 0, while Guo and Wu (1995) only considered the case 2c 3 + s 2 υ > 0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.

The Bifurcations of Traveling Wave Solutions of the Kundu Equation

“…Not only the existence of these solutions is proved, but also their concrete expressions are presented. Our work extends some previous results [10][11][12][13][14][15][16].…”

“…Remark 1. The traveling wave solutions ( ) ( = 3, 5, 6, 7, 9,11,14,15,18,19) are the same as the ones in [10][11][12][13][14], while the rest of the solutions are new, which cannot be found in the literature.…”

“…In this section, by using the above bifurcation phase portraits ( Figure 1) and (12) and (13), we will give the exact traveling wave solutions of (10) under the given parameter conditions shown in Section 2.…”

“…Bright/dark solitary wave solutions and triangular periodic wave solutions of (1a) and (1b) were obtained in [12]. Zhang et al [13] obtained exact solitary waves of (1a) and (1b) by using the undetermined coefficient method, and they also investigated orbital stability of solitary waves. Zhang [14] obtained various exact traveling wave solutions of (1a) and (1b) by the general solutions of a kind of subequation.…”

New abundant exact solutions for Kundu equation

“…[7] have studied orbital stability of solitary waves for Kundu equation. In this work, we obtained new solitary solutions, bell soliton solutions, homoclinic wave solutions and periodic-like solutions of Kudu equation [6] applying a new auxiliary equation method [3] .…”