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| Quasi-periodic Solutions for the Derivative Nonlinear SchrÖdinger Equation with Finitely Differentiable Nonlinearities |
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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 |
| Page view： 4390
Net amount： 3572 |
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). |
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| 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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