This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate ca...This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate case of two solitons and ＂ghost＂ solitons, etc. Co-moving coordinate frames are employed in asymptotic analysis.展开更多
One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equatio...One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equations of Cauchy integral satisfy the Lax equations. Such a basic problem still exists in the procedure of deriving the dark soliton solutions of the NLS equation in normal dispersion with non-vanishing boundary conditions through the inverse scattering transform. In this paper, a pair of Jost solutions with same analytic properties are composed to be a 2 × 2 matrix and then another pair are introduced to be its right inverse confirmed by the Liouville theorem. As they are both 2 × 2 matrices, the right inverse should be the left inverse too, based upon which it is not difficult to show that these Jost solutions satisfy both the first and second Lax equations. As a result of compatibility condition, the dark soliton solutions definitely satisfy the NLS equation in normal dispersion with non-vanishing boundary conditions.展开更多
基金The project supported by National Natural Science Foundation of China under Grant Nos.10371070 and 10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers
文摘This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate case of two solitons and ＂ghost＂ solitons, etc. Co-moving coordinate frames are employed in asymptotic analysis.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10474076 and 10375041
文摘One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equations of Cauchy integral satisfy the Lax equations. Such a basic problem still exists in the procedure of deriving the dark soliton solutions of the NLS equation in normal dispersion with non-vanishing boundary conditions through the inverse scattering transform. In this paper, a pair of Jost solutions with same analytic properties are composed to be a 2 × 2 matrix and then another pair are introduced to be its right inverse confirmed by the Liouville theorem. As they are both 2 × 2 matrices, the right inverse should be the left inverse too, based upon which it is not difficult to show that these Jost solutions satisfy both the first and second Lax equations. As a result of compatibility condition, the dark soliton solutions definitely satisfy the NLS equation in normal dispersion with non-vanishing boundary conditions.
