Local Existence and Uniqueness of Spatially Quasi-Periodic Solutions to the Generalized KdV Equation

Abstract

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the generalized KdV equation (gKdV for short) on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying {\bf the higher dimensional discrete convolution operation for several functions}: \[\underbrace{c\times\cdots\times c}_{\mathfrak p~\text{times}}~(\text{total distance}):=\sum_{\substack{\clubsuit_1,\cdots,\clubsuit_{\mathfrak p}\in\mathbb Z^\nu\\ \clubsuit_1+\cdots+\clubsuit_{\mathfrak p}=~\text{total distance}}}\prod_{j=1}^{\mathfrak p}c(\clubsuit_j).\] In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental ({\color{red}i.e., a} Cauchy sequence). We first give a detailed discussion of the proof of the existence and uniqueness result in the case $\mathfrak p=3$. Next, we prove existence and uniqueness in the general case $\mathfrak p\geq 2$, which then covers the remaining cases $\mathfrak p\geq 4$. As a byproduct, we recover the local result from \cite{damanik16}. We exhibit the most important combinatorial index $\sigma$ and obtain a relationship with other indices, which is essential to our proofs in the case of general $\mathfrak p$.