On the asymptotic stability of $N$-soliton solutions of the three-wave resonant interaction equation

Abstract

The three-wave resonant interaction (three-wave) equation not only possesses $3\times 3$ matrix spectral problem, but also being absence of stationary phase points, which give rise to difficulty on the asymptotic analysis with stationary phase method or classical Deift-Zhou steepest descent method. In this paper, we study the long time asymptotics and asymptotic stability of $N$-soliton solutions of the initial value problem for the three-wave equation in the solitonic region \begin{align} &p_{ij,t}-n_{ij}p_{ij,x}+\sum_{k=1}^{3}(n_{kj}-n_{ik})p_{ik}p_{kj}=0, &p_{ij}(x, 0)=p_{ij,0}(x), \quad x \in \mathbb{R},\ t>0,\ i,j,k=1,2,3, \nonumber &for\ i\neq j,\ p_{ij}=-\bar{p}_{ji}, \ n_{ij}=-n_{ji}, \end{align} where $n_{ij}$ are constants. The study makes crucial use of the inverse scattering transform as well as of the $\overline\partial$ generalization of Deift-Zhou steepest descent method for oscillatory Riemann-Hilbert (RH) problems. Based on the spectral analysis of the Lax pair associated with the three-wave equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem. Further we derive the leading order approximation to the solution $p_{ij}(x, t)$ for the three-wave equation in the solitonic region of any fixed space-time cone. The asymptotic expansion can be characterized with an $N(I)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region; the residual error order $\mathcal{O}(t^{-1})$ from a $\overline\partial$ equation. Our results provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for three-wave equation.