This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest.On the one hand,by proving a compactness lemma and solvin...This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest.On the one hand,by proving a compactness lemma and solving a variational problem,the existence of the standing wave with ground state for the aforementioned equation is proved.On the other hand,the authors derive the instability of the standing wave by applying the potential well argument,the concavity method and an invariant region under the solution flow of the Cauchy problem for the equation under study,and the invariance of the region aforementioned can be shown by introducing an auxiliary functional and a supplementary constrained variational problem.展开更多
The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singu...The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system.On the one hand,the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrdinger equation.On the other hand,in the limit system,it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.展开更多
In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr¨odinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generat...In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr¨odinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.展开更多
基金supported by the National Natural Science Foundation of China (Nos.10801102, 0771151)the Sichuan Youth Sciences and Technology Foundation (No.07ZQ026-009) the China Postdoctoral Science Foundation
文摘This paper deals with the standing wave for a Hamiltonian nonlinear wave equation which can be viewed as a representative of the class of equations of interest.On the one hand,by proving a compactness lemma and solving a variational problem,the existence of the standing wave with ground state for the aforementioned equation is proved.On the other hand,the authors derive the instability of the standing wave by applying the potential well argument,the concavity method and an invariant region under the solution flow of the Cauchy problem for the equation under study,and the invariance of the region aforementioned can be shown by introducing an auxiliary functional and a supplementary constrained variational problem.
基金Project supported by the National Natural Science Foundation of China (Nos.10801102,10771151)the Sichuan Youth Sciences and Technology Foundation (No.07ZQ026-009)the China Postdoctoral Science Foundation
文摘The authors study the compressible limit of the nonlinear Schrdinger equation with different-degree small parameter nonlinearities in small time for initial data with Sobolev regularity before the formation of singularities in the limit system.On the one hand,the existence and uniqueness of the classical solution are proved for the dispersive perturbation of the quasi-linear symmetric system corresponding to the initial value problem of the above nonlinear Schrdinger equation.On the other hand,in the limit system,it is shown that the density converges to the solution of the compressible Euler equation and the validity of the WKB expansion is justified.
基金supported by the National Natural Science Foundation of China(No.11571254)
文摘In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr¨odinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.
