In this thesis,we mainly study the quasi-periodic Cauchy problem for some nonlinear evolution PDEs,in particular,nonlinear dispersive PDEs.The key ingredient is to solve the Cauchy problem for infinite-dimensional ODE with higher-dimensional(alternating)discrete convolution.By an explicit combinatorial analysis technique,we obtain(local)existence and uniqueness results.This thesis mainly consists of three chapters as follows.The first chapter is the introduction,in which we introduce the concept of almost-periodic spatial structure,the KdV equation and the Deift conjecture,the nonlinear Schrodinger equation,main works and contributions,proof methods,and outline of this thesis.The second chapter is concerned with the quasi-periodic Cauchy problem for the generalized KdV equation.Under the exponential decay condition in the Fourier space,we use an explicit combinatorial analysis technique to prove that this problem has a unique local in time spatially quasi-periodic solution with the same frequency vector as the initial data.From the point of technique,a main contribution is that we give the combinatorial description of real-valued power-law nonlinearity and Feynman diagram.The third chapter studies the power-law nonlinear Schrodinger equation and the derivative nonlinear Schrodinger equation with quasi-periodic initial data.Using an explicit combinatorial analysis method,we prove that the former has a "global" in time spatially quasi-periodic solution under the polynomial decay condition in the Fourier space,that the latter has a local in time solution spatially quasi-periodic solution under the exponential decay condition in the Fourier space.From the technique perspective,our main contributions are:(i)obtaining the combinatorial structure with Feynman diagram and power of*[ยท]for the complex-valued powerlaw nonlinearity;(ii)establishing the combinatorial analysis in the framework of polynomial decay assumption. |