论文标题
短置方程的孤子分辨率
Soliton Resolution for the Short-pluse Equation
论文作者
论文摘要
在本文中，我们通过使用$ \ overline \ partial $陡峭下降方法来研究cauchy的问题，用于焦点非线性短板方程。 \ begin {align}＆u_ {xt} = u++\ frac {1} {6}（u^3）_ {xx}，\ nonumber \\＆u（x，0）= u_0（x，x）= u_0（x）\加权的索博尔空间。由于频谱变量z在WKI型LAX对中是相同的顺序，因此我们在新的量表$（y，t）$中构建SP方程的解决方案，而原始的比例$（x，t）$是按新量表和Riemann-Hilbert问题的解决方案提供的。在任何固定的时空圆锥中，新的比例$（y，t）$，将$ v_1 \ leq v_1 \ in r^ - $和$ξ= \ frac {y} {y} {t} {t} <0 $，\ begin \ begin \ begin {equination} r^2 | y = y_0+vt，y_0 \ in [y_1，y_2] \ text {，} v \ in [v_1，v_1，v_2] \ right \ rbrace，\ nonumber \ end \ end {qore {equication}我们要计算the解决方案$ u（x，t）的长时间的noce undose $ u（x，x，t resiT resity res intive：以$ n（i）$ - soliton为特征，其参数被一个局部孤子 - 索尼顿相互作用的总和调制，因为一个人通过锥体移动；第二个$ t^{ - 1/2} $订单项来自连续频谱上的单人辐射互动，直至剩余错误订单$ \ MATHCAL {o}（| t |^{ - 1}）$来自$ \ overline \ partial $方程。我们的结果还表明，短板方程的孤子溶液在渐近稳定。
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.
