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

Abstract

The generalized sine-Gordon (sG) equation u tx = (1 + ν∂ 2 x ) sin u was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys.A: Math. Theor. 43 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with ν = −1. Here, we address the equation with ν = 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 Bäcklund transformation connecting solutions of the sG and generalized sG equations.

Smooth and singular multisoliton solutions of a modified Camassa–Holm equation with cubic nonlinearity and linear dispersion

“…Specifically, we seek soliton solutions which decay rapidly at infinity. We show that the system of bilinear equations deduced from the mCH equation is closely related to that of the generalized sG equation [18,19]. This fact helps us to construct soliton solutions of the mCH equation.…”

“…x, as was the case for the short-pulse [15] and generalized sG equations [18,19], unless we impose certain conditions on the parameters k j (j = 1, 2, .., N). To establish a criterion for obtaining single-valued (or smooth) functions, we require that the mapping (2.2) is one-toone which demands x y > 0.…”

“…We will show that it admits smooth soliton solutions like the bright solitons and breathers as well as the singular solitons. A direct method is employed to obtain solutions which mimics the construction of the soliton solutions of the generalized sine-Gordon (sG) equation [18,19], the dispersionless mCH equation [9] and the Novikov equation [20].…”

Parametric solutions of the generalized short pulse equations

“…The modified sG equation (4.18) is a completely integrable PDE, and its multisoliton solutions have been obtained in constructing solutions of the generalized sG equation [22]. The following proposition provides the parametric N-soliton solution.…”

A generalization of the coupled integrable dispersionless equations

“…The differences appear in the dependent variable transformation from bilinear equations to ordinary nonlinear equations, which lead to the generalized sG equation, coupled short pulse equation and coupled CID equations, respectively. Also, in –, hodograph transformation is used to deduce generalized sG and coupled SP equation, but CID equations is derived directly without such transformation.…”