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

1.非局部孤子方程解的新表示得到了Benjamin-Ono和非局部非线性薛定谔方程的孤子和周期波解的新表示。它们的推导是基于一个非线性代数方程组。本文所采用的方法不同于用逆散射法进行相应的推导。Camassa-Holm方程的多孤子解的参数化表示利用Hodograph变换构造了Camassa-Holm （CH）方程的多孤子解（n -孤子解）。与通常的孤子解的表示不同，它有一个参数表示。导出了解的大时间渐近性，得到了相移的计算公式。用与CH方程相似的方法，构造了Degaspeis-Procesi （DP）方程的单孤子解和双孤子解，并详细探讨了它们的性质。单孤子解的一个显著特征是振幅非线性地依赖于它的速度。采用零色散极限将峰解从孤子解中还原出来。推导了双孤子解的渐近形式，并给出了相移的相关公式。在随后的研究中，得到了DP方程的一般n孤子解，并检验了它的性质。

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.