Study on the asymptotic behavior of solutions for the nonlocal soliton equations in the zero dispersion limit

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1.New representation of the solutions for the nonlocal soliton equationsNew representations are obtained for the soliton and peiodic-wave solutions of the Benjamin-Ono and nonlocal nonlinear Schrodinger equations. Their derivation is based on a system of nonlinear algebraic equations. The method used here differs from the corresponding derivation by means of the inverse scattering method.2.Parametric representation of the multisoliton solution for the Camassa-Holm equationThe multisoliton solution (N-soliton solution) of the Camassa-Holm (CH) equation is constructed by means of the Hodograph transformation. Unlike the usual representation for the soliton solutions, it has a parametric representation. The large time asymptotic of the solution is derived and the formula for the phase shift is obtained.3.Multisoliton solutions of the Degasperis-Procesi equation and their peakon limitUsing the procedure similar to that used for the CH equation, the one- and two-soliton solutions of the Degaspeis-Procesi (DP) equation are constructed and their properties are explored in detail. A remarkable feature of the one-soliton solution is that the amplitude depends on its velocity nonlinearly. The peakon solution is reduced from the soliton solution by taking the zero dispersion limit. The asymptotic form of the two-soliton solution is also derived together with the associated formula for the phase shift. In a subsequent study, the general N-soliton solution is obtained for the DP equation and its property is examined.