This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinea... more This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinear wave equation with a random dynamical boundary condition. The white noises are taken into account not only in the model equation defined on a domain perforated with small holes, but also in the dynamical boundary condition on the boundaries of the small holes. An effective homogenized, macroscopic model is derived in the sense of probability distribution, which is a new stochastic wave equation on a unified domain, without small holes, with a usual static boundary condition.
This work concerns the effective approximation for a class of singularly perturbed stochastic par... more This work concerns the effective approximation for a class of singularly perturbed stochastic partial differential equations driven by a sufficiently small multiplicative noise with quadratic nonlinearities and random Neumann boundary conditions. By splitting the solution into two parts in the finite dimension kernel space and its complement space with some suitable multi-scale argument, it derives rigorously the dominant dynamics, which captures the essential dynamics of the original system as a singular parameter is enough small.
This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from th... more This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from the invariant random cone and employing the backward solvability argument, this wave system is approximated by a finite dimensional wave equation with a white noise. Especially, the finite dimension is explicit, accurate and determined by the coefficient of this wave system; and further originating from an Ornstein-Uhlenbek process and applying Banach space norm estimation, this wave system is approximated by a finite dimensional wave equation with a smooth colored noise.
This work is concerned with a stochastic sine-Gordon equation with a fast oscillation governed by... more This work is concerned with a stochastic sine-Gordon equation with a fast oscillation governed by a stochastic reaction–diffusion equation. It is shown that the fast component is ergodic, while the slow component is tight. Furthermore, employing the skill of partitioning time interval and borrowing from the averaging principle, the system is reduced into an effective equation. More precisely, the fast oscillation component is averaged out, and there exists an effective process, converging to the original stochastic sine-Gordon equation.
This work provides a finite dimensional reducing and a smooth approximating for a class of stocha... more This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.
This work is concerned with the invariant behavior of a stochastic atmosphere-ocean model with th... more This work is concerned with the invariant behavior of a stochastic atmosphere-ocean model with the degenerate noise, which describes the climate and geophysical phenomena. Through introducing a transform to deal with the complicated boundary conditions, first, the existence of the invariant measure is established. Since the degenerate noise only drives the latitudinal atmosphere surface temperature, it then proposes to combine the strong Feller property with the asymptotically strong Feller and the irreducibility to verify the uniqueness of the invariant measure. It further implies the ergodicity of the system.
This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a ... more This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.
This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinea... more This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinear wave equation with a random dynamical boundary condition. The white noises are taken into account not only in the model equation defined on a domain perforated with small holes, but also in the dynamical boundary condition on the boundaries of the small holes. An effective homogenized, macroscopic model is derived in the sense of probability distribution, which is a new stochastic wave equation on a unified domain, without small holes, with a usual static boundary condition.
In this paper, the long-time behaviour of solutions of the Cahn-Hilliard equations with fast grow... more In this paper, the long-time behaviour of solutions of the Cahn-Hilliard equations with fast growing nonlinearityu t+ΓΔ 2u-ΔG(u)=0, G(u)= uΦ(u), xun x∈Ω= x(Δu)n x∈Ω=0, u(0,x)=u 0(x),is considered. A new system is constructed. Existence and uniqueness of the solution of the new system and a smooth m-dimensional manifold (i.e. approximate inertial manifold) are given based on the contraction principle. It is proved that arbitrary trajectory of Cahn-Hilliard equation goes into a small neighbourhood of such manifold after large time.
In this paper, the long time behavor of the dissipative nonlinear long-short equations was studie... more In this paper, the long time behavor of the dissipative nonlinear long-short equations was studied in dynamics. By applying projecting operator and the operator eigenvalue methods, several linear and nonlinear approximate inertial manifolds of the system were constructed. And it is proved that arbitary trajectory of the long-short equations goes into a small neighbourhood of the approximate inertia manifolds after long time.
This article studies the asymptotic attractor of resction-diffusion equation by constructiong a s... more This article studies the asymptotic attractor of resction-diffusion equation by constructiong a solution sequence.At first,we prove that which the solution sequence doesn't go away from the global atrractor in terms of mathematical induction;the second ,it is proved that the solution sequence approaches to the global lattractor of the equation in long time,and the dimensional estimate of the asymptotic attractor is given.
This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a rand... more This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting skill is used to derive the approximating equation of the system in the sense of probability distribution, when the singular perturbation parameter is sufficiently small. The approximating equation is a stochastic parabolic equation when the power exponent of singular perturbation parameter is in [1/2, 1), but a deterministic hyperbolic (wave) equation when the power exponent is in (1, +∞).
Proceedings of the Edinburgh Mathematical Society, 2007
This paper is concerned with the critical nonlinear Gross–Pitaevskii equation, which describes th... more This paper is concerned with the critical nonlinear Gross–Pitaevskii equation, which describes the attractive Bose–Einstein condensate under a magnetic trap. We derive a sharp threshold between the global existence and the blowing-up of the system. Furthermore, we answer the question: how small are the initial data, such that the system has global solutions for the nonlinear critical power $p=1+(4/N)$?
Nonlinear Differential Equations and Applications NoDEA, 2008
This paper is concerned with the coupled nonlinear Schrödinger equations with harmonic potentials... more This paper is concerned with the coupled nonlinear Schrödinger equations with harmonic potentials which describe the Bose-Einstein condensate under the magnetic trap. Two types of the invariant evolution flows are obtained, then a sharp criterion of global existence and blowing up of solutions for the equations is given in terms of the Hamiltonian invariants. Furthermore, we would answer: How small are the initial data such that the global solutions of the system exist?
This paper is concerned with a nonlinear defocusing fourth-order dispersive Schrödinger equation ... more This paper is concerned with a nonlinear defocusing fourth-order dispersive Schrödinger equation with unbounded potentials which models propagation in fiber arrays. We analyze the global existence of the solution and also obtain the existence of the standing waves for the system. Furthermore, we prove that the standing waves are orbitally stable.
Journal of Physics A: Mathematical and Theoretical, 2009
This paper is concerned with the damped nonlinear Schrodinger equation. Through analyzing the cha... more This paper is concerned with the damped nonlinear Schrodinger equation. Through analyzing the characteristics of the equation and the effect of the damping on the global existence, we construct a variational problem. Then combining the variational problem, we establish a crucial invariant evolution flow to derive an explicit and computed criterion to answer: how small are the initial data such that the solutions of the system globally exist? Moreover, the small initial data criterion can be applied in the nonlinear Schrodinger equation with any positive damped parameter.
This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic pot... more This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under a magnetic trap. We establish two types of new invariant evolution flows. Then, in terms of the Hamiltonian invariants, we derive a new sharp energy criterion for global existence and blowing up of solutions of the equation, which can
Invariant manifolds play an important role in the study of the qualitative dynamical behaviors fo... more Invariant manifolds play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. However, the geometric shape of these manifolds is largely unclear. The purpose of the present paper is to try to describe the geometric shape of invariant manifolds for a class of stochastic partial differential equations with multiplicative white noises. The local geometric shape of invariant manifolds is approximated, which holds with significant likelihood. Furthermore, the result is compared with that for the corresponding deterministic partial differential equations.
Journal of Mathematical Analysis and Applications, 2008
This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations iϕ ... more This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations iϕ t + ϕ − |x| 2 ϕ + μ|ϕ| p−1 ϕ + γ |ϕ| q−1 ϕ = 0, which describes the attractive Bose-Einstein condensates under a magnetic trap. We establish the existence of the standing wave of the equation. Furthermore, we prove that the standing wave is nonlinearly unstable.
This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinea... more This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinear wave equation with a random dynamical boundary condition. The white noises are taken into account not only in the model equation defined on a domain perforated with small holes, but also in the dynamical boundary condition on the boundaries of the small holes. An effective homogenized, macroscopic model is derived in the sense of probability distribution, which is a new stochastic wave equation on a unified domain, without small holes, with a usual static boundary condition.
This work concerns the effective approximation for a class of singularly perturbed stochastic par... more This work concerns the effective approximation for a class of singularly perturbed stochastic partial differential equations driven by a sufficiently small multiplicative noise with quadratic nonlinearities and random Neumann boundary conditions. By splitting the solution into two parts in the finite dimension kernel space and its complement space with some suitable multi-scale argument, it derives rigorously the dominant dynamics, which captures the essential dynamics of the original system as a singular parameter is enough small.
This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from th... more This work is concerned with a stochastic wave equation driven by a white noise. Borrowing from the invariant random cone and employing the backward solvability argument, this wave system is approximated by a finite dimensional wave equation with a white noise. Especially, the finite dimension is explicit, accurate and determined by the coefficient of this wave system; and further originating from an Ornstein-Uhlenbek process and applying Banach space norm estimation, this wave system is approximated by a finite dimensional wave equation with a smooth colored noise.
This work is concerned with a stochastic sine-Gordon equation with a fast oscillation governed by... more This work is concerned with a stochastic sine-Gordon equation with a fast oscillation governed by a stochastic reaction–diffusion equation. It is shown that the fast component is ergodic, while the slow component is tight. Furthermore, employing the skill of partitioning time interval and borrowing from the averaging principle, the system is reduced into an effective equation. More precisely, the fast oscillation component is averaged out, and there exists an effective process, converging to the original stochastic sine-Gordon equation.
This work provides a finite dimensional reducing and a smooth approximating for a class of stocha... more This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.
This work is concerned with the invariant behavior of a stochastic atmosphere-ocean model with th... more This work is concerned with the invariant behavior of a stochastic atmosphere-ocean model with the degenerate noise, which describes the climate and geophysical phenomena. Through introducing a transform to deal with the complicated boundary conditions, first, the existence of the invariant measure is established. Since the degenerate noise only drives the latitudinal atmosphere surface temperature, it then proposes to combine the strong Feller property with the asymptotically strong Feller and the irreducibility to verify the uniqueness of the invariant measure. It further implies the ergodicity of the system.
This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a ... more This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.
This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinea... more This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinear wave equation with a random dynamical boundary condition. The white noises are taken into account not only in the model equation defined on a domain perforated with small holes, but also in the dynamical boundary condition on the boundaries of the small holes. An effective homogenized, macroscopic model is derived in the sense of probability distribution, which is a new stochastic wave equation on a unified domain, without small holes, with a usual static boundary condition.
In this paper, the long-time behaviour of solutions of the Cahn-Hilliard equations with fast grow... more In this paper, the long-time behaviour of solutions of the Cahn-Hilliard equations with fast growing nonlinearityu t+ΓΔ 2u-ΔG(u)=0, G(u)= uΦ(u), xun x∈Ω= x(Δu)n x∈Ω=0, u(0,x)=u 0(x),is considered. A new system is constructed. Existence and uniqueness of the solution of the new system and a smooth m-dimensional manifold (i.e. approximate inertial manifold) are given based on the contraction principle. It is proved that arbitrary trajectory of Cahn-Hilliard equation goes into a small neighbourhood of such manifold after large time.
In this paper, the long time behavor of the dissipative nonlinear long-short equations was studie... more In this paper, the long time behavor of the dissipative nonlinear long-short equations was studied in dynamics. By applying projecting operator and the operator eigenvalue methods, several linear and nonlinear approximate inertial manifolds of the system were constructed. And it is proved that arbitary trajectory of the long-short equations goes into a small neighbourhood of the approximate inertia manifolds after long time.
This article studies the asymptotic attractor of resction-diffusion equation by constructiong a s... more This article studies the asymptotic attractor of resction-diffusion equation by constructiong a solution sequence.At first,we prove that which the solution sequence doesn't go away from the global atrractor in terms of mathematical induction;the second ,it is proved that the solution sequence approaches to the global lattractor of the equation in long time,and the dimensional estimate of the asymptotic attractor is given.
This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a rand... more This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting skill is used to derive the approximating equation of the system in the sense of probability distribution, when the singular perturbation parameter is sufficiently small. The approximating equation is a stochastic parabolic equation when the power exponent of singular perturbation parameter is in [1/2, 1), but a deterministic hyperbolic (wave) equation when the power exponent is in (1, +∞).
Proceedings of the Edinburgh Mathematical Society, 2007
This paper is concerned with the critical nonlinear Gross–Pitaevskii equation, which describes th... more This paper is concerned with the critical nonlinear Gross–Pitaevskii equation, which describes the attractive Bose–Einstein condensate under a magnetic trap. We derive a sharp threshold between the global existence and the blowing-up of the system. Furthermore, we answer the question: how small are the initial data, such that the system has global solutions for the nonlinear critical power $p=1+(4/N)$?
Nonlinear Differential Equations and Applications NoDEA, 2008
This paper is concerned with the coupled nonlinear Schrödinger equations with harmonic potentials... more This paper is concerned with the coupled nonlinear Schrödinger equations with harmonic potentials which describe the Bose-Einstein condensate under the magnetic trap. Two types of the invariant evolution flows are obtained, then a sharp criterion of global existence and blowing up of solutions for the equations is given in terms of the Hamiltonian invariants. Furthermore, we would answer: How small are the initial data such that the global solutions of the system exist?
This paper is concerned with a nonlinear defocusing fourth-order dispersive Schrödinger equation ... more This paper is concerned with a nonlinear defocusing fourth-order dispersive Schrödinger equation with unbounded potentials which models propagation in fiber arrays. We analyze the global existence of the solution and also obtain the existence of the standing waves for the system. Furthermore, we prove that the standing waves are orbitally stable.
Journal of Physics A: Mathematical and Theoretical, 2009
This paper is concerned with the damped nonlinear Schrodinger equation. Through analyzing the cha... more This paper is concerned with the damped nonlinear Schrodinger equation. Through analyzing the characteristics of the equation and the effect of the damping on the global existence, we construct a variational problem. Then combining the variational problem, we establish a crucial invariant evolution flow to derive an explicit and computed criterion to answer: how small are the initial data such that the solutions of the system globally exist? Moreover, the small initial data criterion can be applied in the nonlinear Schrodinger equation with any positive damped parameter.
This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic pot... more This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under a magnetic trap. We establish two types of new invariant evolution flows. Then, in terms of the Hamiltonian invariants, we derive a new sharp energy criterion for global existence and blowing up of solutions of the equation, which can
Invariant manifolds play an important role in the study of the qualitative dynamical behaviors fo... more Invariant manifolds play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. However, the geometric shape of these manifolds is largely unclear. The purpose of the present paper is to try to describe the geometric shape of invariant manifolds for a class of stochastic partial differential equations with multiplicative white noises. The local geometric shape of invariant manifolds is approximated, which holds with significant likelihood. Furthermore, the result is compared with that for the corresponding deterministic partial differential equations.
Journal of Mathematical Analysis and Applications, 2008
This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations iϕ ... more This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations iϕ t + ϕ − |x| 2 ϕ + μ|ϕ| p−1 ϕ + γ |ϕ| q−1 ϕ = 0, which describes the attractive Bose-Einstein condensates under a magnetic trap. We establish the existence of the standing wave of the equation. Furthermore, we prove that the standing wave is nonlinearly unstable.
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