Applications Of Riemann-Hilbert Approach To Solve Soliton Equations

Soliton appears as a result of the balance between the nonlinearity and the dispersion. The soliton theory, especially searching for soliton solutions of soliton equations with important physical background, has attracted many researchers due to its significant role in various fields such as water waves, nonlinear fiber optics, plasma physics, fluid mechanics, etc. It is also known that the methods to obtain soliton solutions for soliton equations include inverse scattering transform, Darboux transformation, B¨acklund transformation, Hirota bilinear method,Wronskian technique, and others.In this thesis, our main aim is to extend the Riemann-Hilbert approach to the derivations of multi-soliton solutions for five soliton equations. These equations include the coupled m Kd V equation, the generalized Sasa-Satsuma equation, the coupled Sasa-Satsuma equation, the coupled Gerdjikov-Ivanov equation, and the long wave-short wave equation, which are all associated with higher order spectral problems. The Lax pair of the coupled Sasa-Satsuma equation involves a 5 × 5matrix spectral problem, while the others are associated with 3 × 3 ones.By performing spectral analysis of the Lax pairs of the soliton equations,we construct the sectionally analytic matrix functions, from which RiemannHilbert problems are formulated for these equations. Then by solving special Riemann-Hilbert problems corresponding to the reflectionless cases, we systematically derive multi-soliton solutions for the five soliton equations, respectively.In particular, some interesting three-dimensional plots are given to describe the corresponding soliton characteristics. In addition, in the end of each chapter we express the obtained multi-soliton solution formulae as the ratios of determinants,which are more compact and convenient for symbolic computations.