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 $p$-generalized KdV equation 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 the higher dimensional discrete convolution operation for several functions:$\underset{p}{\underbrace{c\times \cdots \times c}}\left(\text{total distance}\right):=\sum _{\begin{array}{c}{♣}_1,\cdots ,{♣}_p\in Z^{\nu }\\ {♣}_1+\cdots +{♣}_p=\text{total distance}\end{array}}\prod _{j=1}^{p}c\left({♣}_j\right).$ 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 (i.e., a Cauchy sequence). The result has been known for $p=2$ [11], and the combinatorics become harder for larger values of $p$. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, $p=3$. Next, we prove existence and uniqueness in the general case $p\ge 2$, which then covers the remaining cases $p\ge 4$. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general $p$.