# On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

## Abstract

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an solution, a first-order error bound in the -regular norm is shown and used to derive a second-order error bound in the norm. For the cubic Schrödinger equation with an solution, first-order convergence in the -regular norm is used to obtain second-order convergence in the norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and stability for error propagation, where -conditional for the Schrödinger-Poisson system and for the cubic Schrödinger equation.

## 1. Introduction

In this paper we give an error analysis of the Strang splitting time integration method applied to nonlinear Schrödinger equations

where

in the case of the cubic nonlinear Schrödinger equation, and

in the case of the Schrödinger-Poisson equations. The equations are considered with asymptotic boundary conditions and The Poisson equation in ( .Equation 1.3) is thus to be interpreted as giving by the convolution with the fundamental solution of the negative Laplacian,

In both cases, the initial data is given as for .

The cubic nonlinear Schrödinger equation arises as a model equation from several areas of physics; see, e.g., Sulem and Sulem Reference 20. The one-dimensional problem ( is important in fiber optics; see Agrawal )Reference 1. Schrödinger-Poisson equations (Equation 1.1), (Equation 1.3) (also known as the Hartree equation), and generalizations are basic equations in quantum transport; see, e.g., Brezzi, and Markowich Reference 6 and Illner, Zweifel, and Lange Reference 13. The more elaborate Schrödinger-Poisson system considered there has the same mathematical difficulties as (Equation 1.1) with (Equation 1.3), so we restrict our attention to this simpler set of equations.

In this paper we study the approximation properties of a semi-discretization in time. The numerical integrator we consider is a Strang-type splitting method, yielding approximations to with for a step size via

Here, is the solution operator of the free Schrödinger equation, expressed in terms of Fourier transforms as and approximately computed by FFT in a Fourier spectral method, whereas the exponential of acts as a pointwise multiplication operator. Note that and hence Method ( .Equation 1.4) is therefore explicit and time-reversible. The method is the composition of the exact flows of the differential equations

Such splitting methods are widely used; see, e.g., the early references Strang Reference 19 and Hardin and Tappert Reference 11, the study of the split-step Fourier method for the cubic nonlinear Schrödinger equation by Weideman and Herbst Reference 21 and its use in fiber optics as in Agrawal Reference 1, Section 2.4, the use of splitting methods for the time-dependent Kohn-Sham equations (closely related to the above Schrödinger-Poisson equations) in time-dependent density functional theory by Appel and Gross Reference 2, and the papers by Bao, Mauser, and Stimming Reference 4 on the use in the Schrödinger-Poisson- model and by Bao, Jaksch, and Markowich Reference 3 on the numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, which is closely related to the cubic nonlinear Schrödinger equation. We further refer to the review of splitting methods by McLachlan and Quispel Reference 18.

To our knowledge, there is as yet no rigorous convergence result in the literature for the splitting method for the cubic nonlinear Schrödinger equation. We mention, however, the work by Besse, Bidégaray, and Descombes Reference 5, where an error analysis is given for globally Lipschitz-continuous nonlinearities, which is not the case with the cubic nonlinearity considered here. For the Schrödinger-Poisson equation, a first-order error bound over a time interval with suitably small for initial data in the Sobolev space has been shown by Fröhlich Reference 8.

Here, we derive error bounds for the Strang splitting over any given finite time interval that are second-order accurate in the norm under the condition of spatial regularity. This is more stringent than the regularity needed for linear Schrödinger equations with a smooth bounded potential Reference 14. The higher regularity requirement for the nonlinear equations considered here is caused by a term in the double Lie commutator of with the nonlinearity, whereas in the linear case there is a cancellation of higher derivatives that leaves only second-order derivatives. It is also interesting to compare with finite-difference time-stepping methods such as the Crank-Nicholson method or the implicit midpoint rule, for which second-order error bounds involve bounds on the third time derivative of the solution, which would require regularity. -spatial

We remark that Weideman and Herbst Reference 21 report an instability phenomenon in the Strang splitting for the cubic Schrödinger equation for certain step sizes, caused by resonances between the linear part, which has its spectrum on the imaginary axis, and the nonlinearity. This instability can lead to an exponential error growth that is stronger than in the error propagation by the equation itself, and can thus impair the long-time behaviour of the method. It should be noted, however, that this potential long-time instability is not at odds with the finite-time stability and convergence results given here.

We restrict our attention in this paper to nonlinear Schrödinger equations (Equation 1.1) on the whole space Our arguments would apply similarly to problems with periodic boundary conditions and in lower space dimension, and could be extended to nonlinear Schrödinger equations with other power nonlinearities. .

We only study semi-discretization in time but we expect that the results extend to various types of full discretization, uniformly in the spatial discretization parameter. What needs to be checked is the discrete version of the Lie commutator bounds established in this paper for the spatially continuous case. Once such bounds are available, the theory extends to the fully discrete case without further ado. The same remark apparently applies to splitting methods for other nonlinear evolution equations such as the KdV equation, where similarly the scheme of proof given here becomes applicable once the necessary Lie bracket bounds are established.

Throughout the paper, denotes the Hilbert space of Lebesgue square integrable functions, and is the Sobolev space of having all generalized derivatives up to order -functions in We denote the solution of ( .Equation 1.1) at time by The . norm is preserved along the solution, and we assume it to be of unit norm: .

The paper is organized as follows. In the first part (Sections 2 to 6) we consider the Schrödinger-Poisson equation (Equation 1.1), (Equation 1.3) and then, in Sections 7 and 8, we extend the results and techniques to the cubic Schrödinger equation. Sections 2 and 7 state the results of this paper. In Section 3 we give some inequalities related to the nonlinearity in the Schrödinger-Poisson equation. In Section 4 we prove the first-order error bound in the Sobolev norm for solutions in and in Section 5 this is used to show the second-order error bound in , for solutions. Section 6 proves an -regular result of the numerical solution. Finally, Section 8 outlines the modifications in the proofs needed for the cubic Schrödinger equation. -regularity

## PART A. SCHRÖDINGER-POISSON EQUATIONS

## 2. Error bounds for solutions in Statement of results :

In this section we formulate error bounds in the and norm and state some related results. According to a result by Illner, Zweifel, and Lange Reference 13, the Schrödinger-Poisson equation (Equation 1.1), (Equation 1.3) has a global strong solution: implies for all The result can be extended to yield . regularity of solutions to initial data for any globally in time. We suppose that the solution to the Schrödinger-Poisson equation (Equation 1.1), (Equation 1.3) is in for and set ,

Our main result concerning the error of the Strang-type splitting scheme (Equation 1.4) reads as follows.

The following auxiliary results are of independent interest. We write the step of the splitting scheme (Equation 1.4) briefly as

Note that also the estimate depends on bounds in -stability The proof of Theorem .2.1 therefore proceeds by first showing the error bound, which, in particular, establishes the required bound of the norm of numerical solutions. We then are in the position to prove the error bound using the -conditional -stability.

## 3. Some inequalities

Hardy’s inequality (e.g., Reference 15, p. 350)

implies some further inequalities that play an important role in the following.

With the product rule of derivatives, Lemma 3.1 immediately yields the following bounds.

For further inequalities concerning we refer to Castella Reference 7 and Illner, Zweifel, and Lange Reference 13.

## 4. Proof of the first-order error bound in

### 4.1. stability: Proof of Proposition -conditional2.2

(a) Since preserves both the and the norm, we only need to compare and which are the solutions at time , of the linear initial value problems

with norms of and bounded by We rewrite the difference of the right-hand sides as .

(b) By Lemma 3.1, we thus obtain

and hence, recalling unit

so that by the Gronwall inequality,

where

(c) We proceed in the same way for the

Next we estimate the

which yields, using the bound