This work is concerned with the dynamics of a slow-fast stochastic evolutionary system quantified with a scale parameter. An invariant foliation decomposes the state space into geometric regions of different dynamical regimes, and thus helps understand dynamics. A slow invariant foliation is established for this system. It is shown that the slow foliation converges to a critical foliation (i.e., the scale parameter is zero) in probability distribution, as the scale parameter tends to zero. The approximation of slow foliation is also constructed with error estimate in distribution. Furthermore, the geometric structure of the slow foliation is investigated: every fiber of the slow foliation parallels each other, with the slow manifold as a special fiber. In fact, when an arbitrarily chosen point of a fiber falls in the slow manifold, the fiber must be the slow manifold itself.
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.
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, +∞).
This paper is concerned with the nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under the magnetic trap. By combining the best constant of Gagliardo-Nirenberg's inequality with the characteristic of this equation, we derive out a global existence condition for the supercritical equation which coincides with the critical case.
This paper is concerned with the damped nonlinear Schrödinger 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 Schrödinger equation with any positive damped parameter.
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