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非线性Schrödinger方程；基态解；集中紧致原理<br>Nonlinear Schrödinger Equation "
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Abstract:
本文考虑如下形式的非线性Schr？dinger方程 (P)。利用有界区域逼近和集中紧致原理，当位势函数不恒等于常数，非线性项 不恒等于 ，本文证明了方程(P)存在最低能量解。
In this paper, we are concerned with the following nonlinear Schr？dinger equation
(P). By using the bounded domain approximate scheme and concen-tration compactness principle, we prove the existence of a ground state solution of (P) on the Nehari manifold when constant and .

Abstract:
The global Cauchy problem of nonlinear Schrodinger equations is considered by using generalized Strichartz inequalities and the contraction mapping principle.Under some restrictions on parameter $\alpha$, if the initial value is sufficiently small in some weak $L^p$ space, then there exists a global solution. Moreover, the global unique existence of self-similar solutions is obtained in weak $L^p$ space for the small initial value with self-similar structure.

Abstract:
Search for exact solutions to the generalized nonlinear Schr dinger equation is one of the essential directions in studies of the nonlinear dynamics in optical soliton and Bose-Einstein condensates. Stable soliton modes are of great significance for the experimental realization and potential application. In this paper, based on the introduction of a similarity transformation, the variable-coefficient nonlinear Schr dinger equation is transformed into the nonlinear Schr dinger equation, and then the single soliton solution, two-soliton solution and soliton solution in continuous-wave background for the variable coefficient nonlinear Schr dinger equation are obtained by using the known solutions. Meanwhile, their image analysis and relative discussion are given by selecting the different parameters in detail.

Abstract:
本文研究了如下Schr？dinger-Maxwell方程基态解的存在性问题 {-△u+V（x）u+K（x）？（x）u=b（x）|u|p-1u+λg（x，u） in R3， -△？=K（x）u2 in R3， 其中λ > 0；V（x）∈ C1（R3，R），且V（x）> 0.在K，g，b 满足一定的假设条件下，且0 < p < 1时，利用变分法和临界点理论，获得了基态解的存在性.该结论推广了文献[7]的结果. In this paper, we study the existence of ground state solutions for Schr？dingerMaxwell equations {-△u +V(x)u+K(x)？(x)u=b(x)|u|p-1u+λg(x,u) in R3, -△？=K(x)u2 in R3, where λ > 0, V (x) ∈ C1(R3, R) and V (x) > 0.Under certain assumptions on K, g and 0 < p < 1, we obtain the ground state solutions by using variational methods and critical point theory, which promotes the results of literature[7]

Abstract:
In this paper, we consider existence and concentration phenomena of least energy solutions of coupled nonlinear Schr dinger systems with Neumann boundary conditions. The focus is on the locations of peaks (maximum points) of the least energy solutions. Following Tai-Chia Lin and Juncheng Wei's procedure for Dirichlet problem, least energy solutions for Neumann problem are obtained. As the small perturbed parameter goes to zero, we prove that the peaks of the least energy solutions approach to the boundary of domain and the energy concentrates around these peaks. On the other hand, peaks of the two states attract or repulse each other depending on the interaction between them to be attractive or repulsive.

Abstract:
By applying the direct perturbation method to the 3-dimensional nonlinear Schr dinger equation with perturbation, we obtain its asymptotical solutions, which contain not only the zero-order solutions, but also the first-order corrections. Through these solutions, the effect of perturbation on the soliton is analyzed as well.

Abstract:
An extended Jacobian elliptic function expansion method is presented and successfully applied to the nonlinear Schr?dinger (NLS) equation and Zakharov equation. We obtain some new solutions besides Fu et al's results. The results show that our method is more powerful to construct Jacobian elliptic function and can be applied to other nonlinear physics systems.

Abstract:
By using the solutions of an auxiliary Lam\'e equation, a direct algebraic method is proposed to construct the exact solutions of $N$-coupled nonlinear Schr\"{o}dinger equations. The abundant higher-order exact periodic solutions of a family of $N$-coupled nonlinear Schr\"{o}dinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.

Abstract:
Using the wave packet theory, we obtain all the solutions of the weakly damped nonlinear Schr?dinger equation. These solutions are the static solution, and solutions of planar wave, solitary wave, shock wave and elliptic function wave and chaos. The bifurcation phenomenon exists in both steady and non-steady solutions. The chaotic and periodic motions can coexist in a certain parametric space region.