arXiv 2005.122082020-05-22nlin.SI

短脉冲方程的孤子解

Soliton Resolution for the Short-pluse Equation

Yiling Yang, Engui Fan

备注：: 48 pages. arXiv admin note: text overlap with arXiv:1912.10358

摘要：

本文利用$\overline\partial$最速下降法研究了聚焦非线性短脉冲方程的Cauchy问题。\begin{align} &u_{xt}=u+\frac{1}{6}(u^3)_{xx}, \nonumber\\ &u(x,0)=u_0(x)\in H^{1,1}(R),\nonumber \end{align}，其中$H^{1,1}(R)$是加权Sobolev空间。由于谱变量z在WKI型Lax对中是同阶的，我们构造了SP方程在新尺度$(y,t)$下的解，而原尺度$(x,t)$则是用新尺度下的函数和Riemann-Hilbert问题的解给出的.在将$v_1\leq v_1 \in R^-$和$ξ=\frac{y}{t}<0$分层的新尺度$(y,t)$的任意固定时空锥中，计算了\begin{equation} C(y_1,y_2,v_1,v_2) = \left\lbrace (y,t) \in R^2|y=y_0+vt, y_0 \in[y_1,y_2]\text{, } v\in[v_1,v_2]\right\rbrace, \nonumber \end{equation}解的长时间渐近展开式，证明了孤子分辨猜想由三项组成：首阶项可以用$N(I)$ -孤子来刻画，当孤子通过锥时，其参数被定域孤子-孤子相互作用的总和所调制;第二个$t^{-1/2}$阶项来自连续谱上的孤子-辐射相互作用，直到来自$\overline\partial$方程的残差阶数$\mathcal{O}(|t|^{-1})$。我们的结果还表明，短脉冲方程的孤子解是渐近稳定的。

英文摘要：

In this paper, we study the Cauchy problem for the focusing nonlinear short-pluse equation by using $\overline\partial$ steepest descent method. \begin{align} &u_{xt}=u+\frac{1}{6}(u^3)_{xx}, \nonumber\\ &u(x,0)=u_0(x)\in H^{1,1}(R),\nonumber \end{align} where $H^{1,1}(R)$ is a weighted Sobolev space. Because the spectral variable z is the same order in the WKI-type Lax pair, we construct the solution of SP equation in the new scale $(y,t)$, whereas the original scale $(x,t)$ is given in terms of functions in the new scale and the solution of Riemann-Hilbert problem. In any fixed space-time cone of the new scale $(y,t)$ which stratify that $v_1\leq v_1 \in R^-$ and $ξ=\frac{y}{t}<0$, \begin{equation} C(y_1,y_2,v_1,v_2) = \left\lbrace (y,t) \in R^2|y=y_0+vt, y_0 \in[y_1,y_2]\text{, } v\in[v_1,v_2]\right\rbrace, \nonumber \end{equation} we compute the long time asymptotic expansion of the solution $u(x,t)$, which prove soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an $N(I)$-soliton whose parameters are modulated by a sum of localied soliton-soliton interactions as one moves through the cone; the second $t^{-1/2}$ order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order $\mathcal{O}(|t|^{-1})$ from a $\overline\partial$ equation. Our results also show that soliton solutions of short-pluse equation are asymptotically stable.

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