The Lyapunov stability of the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation

The Lyapunov stability is established for the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono (BO) equation. We characterize the N-soliton profiles as critical points of certain Lyapunov functional. By using several results derived by the inverse scattering transform of the BO equation, we demonstarate the convexity of the Lyapunov functional when evaluated at the N-soliton profiles. From this fact, we deduce that the N-soliton solutions are energetically stable.Comment: To appear in Journa of Mathematical Physic

A direct method for solving the generalized sine-Gordon equation II

The generalized sine-Gordon (sG) equation $u_{tx}=(1+\nu\partial_x^2)\sin\,u$ was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with $\nu=-1$. Here, we address the equation with $\nu=1$. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. Last, we provide a recipe for deriving an infinite number of conservation laws by using a novel B\"acklund transformation connecting solutions of the sG and generalized sG equations.Comment: To appear in J. Phys. A: Math. Theor. The first part of this paper has been published in J. Phys. A: Math. Theor. 43 (2010) 10520