Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Local Well-posedness of the Coupled KdV-KdV Systems on $\mathbb{R}$ (1812.08261v3)

Published 19 Dec 2018 in math.AP

Abstract: Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation [ u_t +uu_x +u_{xxx}=0, \quad \left. u \right |{t=0}=u{0} ] in the space $H{s} (\mathbb{R})$ (or $H{s} (\mathbb{T})$), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems [\left{\begin{array}{rcl} u_t+a_{1}u_{xxx} &=& c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x},\ v_t+a_{2}v_{xxx}&=& c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x},\ \left. (u,v)\right |{t=0} &=& (u{0},v_{0}) \end{array}\right.] in the space $\mathcal{H}s (\mathbb{R}) := Hs (\mathbb{R})\times Hs (\mathbb{R})$. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few. In this paper we look for those values of $s\in \mathbb{R}$ for which the cKdV systems are well-posed in $\mathcal{H}s (\mathbb{R})$. Our findings enable us to provide a complete classification for the cKdV systems in terms of the analytical well-posedness in $\mathcal{H}s (\mathbb{R})$ based on its coefficients $a_i$, $c_{ij}$ and $d_{ij}$ for $i,j=1,2$. The key ingredients in the proofs are the bilinear estimates under the Fourier restriction space norms. There are four types of the bilinear estimates that need to be investigated. Sharp results are established for all of them. In contrast to the lone critical index $-\frac{3}{4}$ for the single KdV equation, the critical indexes for the cKdV systems are $-\frac{13}{12}$, $-\frac{3}{4}$, $0$ and $\frac{3}{4}$. As a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index $s{*}\in{-\frac{13}{12},\,-\frac{3}{4},\,0,\,\frac{3}{4}}$ such that any system in this class is locally analytically well-posed if $s>s{*}$ while the bilinear estimate fails if $s<s{*}$.

Summary

We haven't generated a summary for this paper yet.