Abstract
In this paper, we study the exact solitary wave solutions and periodic wave solutions of the S-S equation and give the relationships between solutions and the Hamilton energy of their amplitudes. First, on the basis of the theory of dynamical system, we make qualitative analysis on the amplitudes of solutions. Then, by using undetermined hypothesis method, the first integral method, and the appropriate transformation, two bell-shaped solitary wave solutions and six exact periodic wave solutions are obtained. Furthermore, we discuss the evolutionary relationships between these solutions and find that the appearance of these solutions for the S-S equation is essentially determined by the value which the Hamilton energy takes. Finally, we give some diagrams which show the changing process from the periodic wave solutions to the solitary wave solutions when the Hamilton energy changes.
Highlights
+ |z|2 z was proposed by Sasa and Satsuma in studying the integrability of the Schrodinger equation with higher-order nonlinear terms, and it is a typical equation which describes the propagation of short pulses in an optical fiber [1,2,3,4]. e 1-soliton solutions and the N-soliton solutions of equation (1) were given by Sasa and Satsuma in [1]
Ey have worked out some solitary wave solutions and periodic wave solutions of the form z(X, T) u(ξ)ei(k1X− c1T), ξ X − cT, (2)
We point out that the amplitudes uP±1(ξ) of the periodic wave solutions zP±1(ξ) for equation (1) correspond to the periodic trajectories, which are contained in the homoclinic trajectory centered on P1 and P3 in the global phase portrait (Figure 1(a)), respectively
Summary
We point out that the amplitudes uP±1(ξ) of the periodic wave solutions zP±1(ξ) for equation (1) correspond to the periodic trajectories, which are contained in the homoclinic trajectory centered on P1 and P3 in the global phase portrait (Figure 1(a)), respectively.
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