A direct method for solving the generalized sineGordon equation II
Abstract
The generalized sineGordon (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 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 multivalued 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. At 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 September 2010
 DOI:
 10.1088/17518113/43/37/375201
 arXiv:
 arXiv:1007.1525
 Bibcode:
 2010JPhA...43K5201M
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 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) 105204