In this paper, we shall study the initial boundary value problem of Schrodinger equation. The second order gradient superconvergence estimates for the problem are obtained solving by linear finite elements.
We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1...We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.展开更多
This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear S...This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear Schrodinger equations and gives correctors for the homogenization of linear and semilinear Schrodinger equations.展开更多
The bilinear equation of the genera/nonlinear Schrodinger equation with derivative （GDNLSE） and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively....The bilinear equation of the genera/nonlinear Schrodinger equation with derivative （GDNLSE） and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrodinger equation with derivative （DNLSE） and its multisoliton solutions are given by reduction.展开更多
The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odin...The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.展开更多
In this paper,we prove an infinite dimensional KAM theorem and apply it to study 2-dimensional nonlinear Schrodinger equations with different large forcing terms and(2p+1)-nonlinearities iu_(t)-Δu+φ_(1)(ω_(1)+t)u+...In this paper,we prove an infinite dimensional KAM theorem and apply it to study 2-dimensional nonlinear Schrodinger equations with different large forcing terms and(2p+1)-nonlinearities iu_(t)-Δu+φ_(1)(ω_(1)+t)u+φ_(2)(ω_(2)+t)|u|^(2p)u=0,t∈R,x∈T^(2) under periodic boundary conditions. As a result, the existence of a Whitneysmooth family of small-amplitude reducible quasi-periodic solutions is obtained.展开更多
文摘In this paper, we shall study the initial boundary value problem of Schrodinger equation. The second order gradient superconvergence estimates for the problem are obtained solving by linear finite elements.
基金supported by National Natural Science Foundation of China(Grant No.11171351)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162110021)
文摘We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.
基金The research was supported in part by the grant of ZARCF and NSFC
文摘This paper considers the initial and boundary value problem of some linear and semilinear Schrodinger equation with real potential and H01 initial data. The author obtains the homogenization of linear and semilinear Schrodinger equations and gives correctors for the homogenization of linear and semilinear Schrodinger equations.
基金The project supported by National Natural Science Foundation of China under Grant No.10671121
文摘The bilinear equation of the genera/nonlinear Schrodinger equation with derivative （GDNLSE） and the N-soliton solutions are obtained through the dependent variable transformation and the Hirota method, respectively. The bilinear equation of the nonlinear Schrodinger equation with derivative （DNLSE） and its multisoliton solutions are given by reduction.
文摘The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schr¨odinger equation and its modified form are formally obtained.
文摘In this paper,we prove an infinite dimensional KAM theorem and apply it to study 2-dimensional nonlinear Schrodinger equations with different large forcing terms and(2p+1)-nonlinearities iu_(t)-Δu+φ_(1)(ω_(1)+t)u+φ_(2)(ω_(2)+t)|u|^(2p)u=0,t∈R,x∈T^(2) under periodic boundary conditions. As a result, the existence of a Whitneysmooth family of small-amplitude reducible quasi-periodic solutions is obtained.
