修正 KdV 方程和三阶分散非线性薛定谔方程空间解析性半径的衰减

Renata O. Figueira, Mahendra Panthee
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引用次数: 0

摘要

我们考虑修正的 Korteweg-de Vries(mKdV)方程的初值问题(IVPs) $$\begin{aligned}\left\{ \begin{array}{l}\partial _t u+partial _x^3u+mu u^2partial _xu =0, quad x in \mathbb {R},\; t in \mathbb {R}, \ u(x,0) = u_0(x), end{array}\right.\end{aligned}$$where u is a real valued function and \(\mu =\pm 1\), and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short) $$\begin{aligned}.\left\{ \begin{array}{l}\partial _t v+i\alpha \partial _x^2v+i\beta \partial _x^3v+i\gamma |v|^2v = 0, \quad x\in \mathbb {R},\; t\in \mathbb {R}, \v(x,0) = v_0(x), \end{array}\right.\end{aligned}$ 其中 \(\alpha , \beta \) 和 \(\gamma \) 是实常数,v 是复值函数。在这两个问题中,初始数据 \(u_0\) 和 \(v_0\) 在 \(\mathbb {R}\) 上是解析的,并且在空间变量中具有均匀的解析半径 \(\sigma _0\) 。我们通过建立三线性估计的解析版本来证明这两个IVP对这样的数据都是局部良好求解的,并证明解的空间解析半径在某个生命期\(0<T_0\le 1\)之前保持不变\(\sigma _0\)。我们还考虑了当局部解在时间上全局扩展时空间解析性半径的演化,并证明对于任意时间\(T\ge T_0\),空间解析性半径从下往上受\(c T^{-\frac{4}{3}}\) 约束、(\(\mu=-1/)),对于失焦情况下的mKdV方程,由\(c T^{-(4+\varepsilon )}\), (\(\varepsilon >;0),用于 tNLS 方程。mKdV 方程的结果改进了博纳等人(Ann Inst Henri Poincaré 22:783-797, 2005)的结果,据我们所知,tNLS 方程的结果是新的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion

We consider the initial value problems (IVPs) for the modified Korteweg–de Vries (mKdV) equation

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t u+ \partial _x^3u+\mu u^2\partial _xu =0, \quad x\in \mathbb {R},\; t\in \mathbb {R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{aligned}$$

where u is a real valued function and \(\mu =\pm 1\), and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short)

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t v+i\alpha \partial _x^2v+\beta \partial _x^3v+i\gamma |v|^2v = 0, \quad x\in \mathbb {R},\; t\in \mathbb {R}, \\ v(x,0) = v_0(x), \end{array}\right. \end{aligned}$$

where \(\alpha , \beta \) and \(\gamma \) are real constants and v is a complex valued function. In both problems, the initial data \(u_0\) and \(v_0\) are analytic on \(\mathbb {R}\) and have uniform radius of analyticity \(\sigma _0\) in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same \(\sigma _0\) till some lifespan \(0<T_0\le 1\). We also consider the evolution of the radius of spatial analyticity \(\sigma (t)\) when the local solution extends globally in time and prove that for any time \(T\ge T_0\) it is bounded from below by \(c T^{-\frac{4}{3}}\), for the mKdV equation in the defocusing case (\(\mu = -1\)) and by \(c T^{-(4+\varepsilon )}\), \(\varepsilon >0\), for the tNLS equation. The result for the mKdV equation improves the one obtained in Bona et al. (Ann Inst Henri Poincaré 22:783–797, 2005) and, as far as we know, the result for the tNLS equation is the new one.

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