Bifurcations and new exact travelling wave solutions for the Gerdjikov–Ivanov equation

In this paper, the Gerdjikov–Ivanov equation is investigated by using the bifurcation theory and the method of phase portraits analysis. The existence of every kind of travelling waves is proved, in some conditions, exact parametric representations of above travelling waves in explicit form are obtained.

[1]  Tao Geng,et al.  A new application of Riccati equation to some nonlinear evolution equations , 2008 .

[2]  Engui Fan,et al.  Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure , 2001 .

[3]  Engui Fan,et al.  Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation , 2000 .

[4]  Engui Fan,et al.  Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation , 2000 .

[5]  R. S. Johnson.  On the modulation of water waves in the neighbourhood of kh ≈ 1.363 , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  Yao Long,et al.  Bifurcations of travelling wave solutions for a general Sine–Gordon equation , 2006 .

[7]  Yuji Kodama,et al.  Optical solitons in a monomode fiber , 1985 .

[8]  M. A. Abdou,et al.  New explicit and exact traveling wave solutions for two nonlinear evolution equations , 2008 .

[9]  A. Kundu.  Exact solutions to higher-order nonlinear equations through gauge transformation , 1987 .

[10]  Peter A. Clarkson,et al.  Exact solutions of the multidimensional derivative nonlinear Schrodinger equation for many-body systems of criticality , 1990 .

[11]  Engui Fan,et al.  Variable separation and algebro-geometric solutions of the Gerdjikov–Ivanov equation , 2004 .

[12]  Jinliang Zhang,et al.  Exact solutions to a class of nonlinear Schrödinger-type equations , 2006 .

[13]  Li Jibin,et al.  Bifurcations of travelling wave solutions for a variant of Camassa-Holm equation , 2008 .

[14]  E. Fan,et al.  A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations , 2001 .

[15]  Saburo Kakei,et al.  SOLUTIONS OF A DERIVATIVE NONLINEAR SCHRÖDINGER HIERARCHY AND ITS SIMILARITY REDUCTION , 2005, Glasgow Mathematical Journal.

[16]  Integrable discretizations of derivative nonlinear Schrödinger equations , 2001, nlin/0105053.