Soliton resolution for the three-wave resonant interaction equation

Yiling Yang, Engui Fan

Published 2021-01-10Version 1

In this paper, we apply $\overline\partial$ steepest descent method to study the long time asymptotic behavior for the Cauchy of the three-wave resonant interaction equation \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,\\ &for\ i\neq j,\ p_{ij}=-\bar{p}_{ji}, \ n_{ij}=-n_{ji}, \end{align} where $n_{ij}$ are constants. It is shown that the solution of the Cauchy problem can be characterized via a the solution of Riemann-Hilbert problem. In any fixed space-time cone \begin{equation} C(x_1,x_2,v_1,v_2) = \left\lbrace (x,t)\in \mathbb{R}^2 : x=x_0+vt, x_0\in[x_1,x_2]\text{, }v\in[v_1,v_2]\right\rbrace, \end{equation} we further compute the long time asymptotic expansion of the solution $u(x,t)$, which implies soliton resolution conjecture and 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.