Quasi-periodic Solutions for the Derivative Nonlinear SchrÖdinger Equation with Finitely Differentiable Nonlinearities

Citation:

Meina GAO,Kangkang ZHANG.Quasi-periodic Solutions for the Derivative Nonlinear SchrÖdinger Equation with Finitely Differentiable Nonlinearities[J].Chinese Annals of Mathematics B,2017,38(3):759~786
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Authors:

Meina GAO; Kangkang ZHANG

Foundation:

This work was supported by the National Natural Science Foundation of China (No.11201292), Shanghai Natural Science Foundation (No.12ZR1444300) and the Key Discipline "Applied Mathematics" of Shanghai Second Polytechnic University (No.XXKZD1304).
Abstract: The authors are concerned with a class of derivative nonlinear Schr\"{o}dinger equation $$\mathbf{i} u_t+u_{xx}+\mathbf{i} \epsilon f(u,\ov{u}, \omega t)u_x=0, \quad (t,x)\in\mathbb{R}\times [0, \pi],$$ subject to Dirichlet boundary condition, where the nonlinearity $f(z_1,z_2,\phi)$ is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.

Keywords:

Derivative NLS, KAM theory, Newton iterative scheme, Reduction theory, Quasi-periodic solutions, Smoothing techniques

Classification:

37K55, 35B15, 35J10, 35Q40, 35Q55
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