DeepAI AI Chat
Log In Sign Up

Time splitting method for nonlinear Schrödinger equation with rough initial data in L^2

by   Hyung Jun Choi, et al.

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in L^2, {[ i∂_t u +Δ u = λ |u|^p u, (x,t) ∈ℝ^d ×ℝ_+, u (x,0) =ϕ (x), x∈ℝ^d, ]. where λ∈{-1,1} and p >0. While the Lie approximation Z_L is known to converge to the solution u when the initial datum ϕ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data ϕ∈ L^2 (ℝ^d), we prove the convergence of the Lie approximation Z_L to the solution u in the mass-subcritical range, max{1,2/d}≤ p < 4/d. Furthermore, our argument can be extended to the case of initial data ϕ∈ H^s (ℝ^d) (0<s≤1), for which we obtain a convergence rate of order s/2-s that breaks the natural order barrier s/2.


page 1

page 2

page 3

page 4


For the splitting method of the nonlinear heat equation with initial datum in W^1,q

In this paper, we analyze an operator splitting scheme of the nonlinear ...

A modified splitting method for the cubic nonlinear Schrödinger equation

As a classical time-stepping method, it is well-known that the Strang sp...

Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System

This article analyses the convergence of the Lie-Trotter splitting schem...

Low regularity error estimates for the time integration of 2D NLS

A filtered Lie splitting scheme is proposed for the time integration of ...

Embedded exponential-type low-regularity integrators for KdV equation under rough data

In this paper, we introduce a novel class of embedded exponential-type l...

Temporal approximation of stochastic evolution equations with irregular nonlinearities

In this paper we prove convergence for contractive time discretisation s...

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Motivated by the challenge of incorporating data into misspecified and m...