Inverse Scattering Transform For Coupled Nonlinear Equaitons

Integrable nonlinear equations and its soliton solutions is an important research area in nonlinear mathematical physics. The inverse scattering transform (IST) is one of the basic approach for solving these nonlinear equations. From the IST equation, we can derive the explicit expressions of the multi-soliton solutions. Moreover, the perturbation theory and Hamiltonian theory can be developed on the base of the IST theory.Some typical nonlinear equations, such as NLS equation, can be treated as a spe-cial case of a coupled nonlinear equation. Although they are integrable models, these coupled nonlinear equations have not been treated systematically and completely in the literature. The purpose of this dissertation is to extend the IST method to the coupled equations. It is of theoretical importance to the application of IST method to various type of nonlinear equations.We choose the coupled NLS equation with vanishing boundary conditions, cou-pled NLS equation with mixed boundary conditions (NLS+equation), and coupled discrete NLS equation (Ablowitz-Ladik equation) as the object of our research. We generalized the IST approach to these models. Finally, we find that, such a generaliza-tion is practicable, and we also get some new solutions.The IST for coupled nonlinear equations has some different points from their orig-inal one.1. The distribution of the zeros of the scattering data on the complex plane of spec-tral parameter is independent with each other, especially the location and number of zeros.2. Since the first Lax pair lost the symmetric form, some symmetric relations of the Jost functions does not hold now, thus the number of IST equation for solving must be doubled. We need to construct the IST equation twice.3. Due to the difference of the zeros, when deriving the multi-soliton solutions, the core matrix in the determinant calculation is no longer a square matrix, but a ar-bitrary matrix. This makes the calculation more complex. However, by using the rank analysis, together with the Cauchy-Binet formula, we prove the equivalenceof the denominator of the potential function, and give out the explicit expressionsof the solution.After deriving the multi-soliton solution, we demonstrated a group of simple so-lutions with little zeros, and pointed out the difference between the solutions of the original nonlinear equations.