Abstract
In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method.
Highlights
IntroductionSince the spectral problem for Satsuma equation (SSE) (4) is 3 × 3 and possesses deep reduction, the high order Darboux transformation (DT) can not be obtained directly
Kodama and Hasegawa [15, 16] proposed a higher-order nonlinear Schrodinger (NLS) equation: iuT + α1uXX + α2|u|2u + i(β1uXXX + β2(|u|2u)X + β3(|u|2)X u) = 0 (1)to model the propagation and interaction of the ultrashort pulses in the subpicosecond or femtosecond regime, where u represents the slowly varying envelope of the electric field X and T are, respectively, the normalized distance along the direction of the propagation and retarded time, α1, α2, β1, β2 and β3 are the real parameters with respect to the group velocity dispersion, self-phase modulation, third-order dispersion, self-steepening and stimulated Raman scattering, respectively.In general, the equation (1) is not completely integrable
It is well known that, the Darboux transformation (DT) is a powerful method to construct the exact solutions for the integrable equations
Summary
Since the spectral problem for SSE (4) is 3 × 3 and possesses deep reduction, the high order DT can not be obtained directly. We construct the solitonic solution for both the focusing and defocusing SSE on the NVBC systematically. For the focusing case, when the spectral parameter is located in the image axis, we can obtain the soliton solution and resonant soliton solution, periodical solution, half periodical solution, rational. Based on the limit technique, we give the generalized DT which could be used to derive the high order solutions. To obtain the elementary DT for SSE (4), we merely need to find the deep reduction for the DT of coupled NLSE. To derive the DT for system (5), we need to use the symmetry relation
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