A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and its Application to the Nonlinear Schr'odinger Equation

When utilizing numerical methods to approximate the solutions to time-dependent partial differential equations (PDEs), proper handling of boundary conditions can be quite challenging. Sometimes, an otherwise stable numerical scheme will become unstable depending on how the boundary conditions are computed [1]. In addition, high-order schemes can degrade in accuracy to lower-order when using boundary conditions which are not compatible with the high-order accuracy [2] Often, researchers will forgo a complicated boundary condition implementation and instead use tried-and-true boundary conditions techniques which are very simple yet provide acceptable results. One of the most common is the use of Dirichlet boundary conditions, where the function value at the boundary of the domain is set to a constant value (commonly this value is zero). A Dirichlet boundary condition of zero-value is often used when simulating solutions which decay towards zero at infinity, and where most of the dynamics (or 'action') is expected to remain in the central regions of the computational grid.

Problems involving PDEs whose function values are complex cannot, in general, make use of standard Dirichlet boundary conditions because of the constant oscillation of the real and imaginary parts of the function due to the intrinsic frequency of the system and the dynamics of the solutions (when the solution decays to zero at infinity, standard Dirichlet conditions of zero-value can be used). There are situations (for examples, see Sec. 4) where the modulus-squared of the solution converges to a constant value at the boundary. In such a case, a modulus-squared Dirichlet (MSD) boundary condition (which keeps the modulus-squared of the solution at the boundaries constant) would be desirable.

In this paper, we present a simple way to simulate a modulus-squared Dirichlet boundary condition in time-dependent complex-valued PDEs. The MSD boundary condition is very easy to implement and has an accuracy equal to the interior numerical scheme (as long as the assumption of a constant modulus-squared value at the boundary is valid). This new boundary condition eliminates the need for overly large grids or expensive and complicated boundary conditions for many problems.

A very common time-dependent complex-valued PDE used in a wide range of applications is the nonlinear Schrödinger equation (NLSE). The NLSE is a universal model describing the evolution and propagation of complex filed envelopes in nonlinear dispersive media. As such, it is used to describe many physical systems [3] including nonlinear optics [4] and the mean-field dynamics of Bose-Einstein condensates (BECs) (in which case the NLSE is typically modified to include an external potential term, in which case the NLSE is referred to as the Gross-Pitaevskii equation) [5]. In systems such as optics and BECs, the modulus-squared of the solution (referred to as the ‘wavefunction’) represents the observable (intensity of light and atomic density respectively). In this situation, often the dynamics of ‘dark’ structures (dark solitons, vortices, vortex lines, vortex rings, etc.) which reside inside medium are studied. These are coherent structures of very low (or zero) central density which exist inside the bulk of the medium. The most basic form of the structures can be examined by assuming an infinite-extent domain, in which case the solutions exist within an infinite constant-density background. Such a situation is very well-suited for the use of MSD boundary conditions. As such, in Sec. 4, we use simulations of dark coherent structures in the NLSE to test the MSD boundary condition.

Many sophisticated boundary conditions have been developed for both the linear and nonlinear Schrödinger equations which simulate transparent or artificial boundaries (see the review of Ref. [6] and references therein). Most of these boundary conditions focus on eliminating reflections off the boundaries (the biggest problem when using Dirichlet boundary conditions) when studying dynamics which trail off to zero at infinity. Since our main concern are boundary conditions in a constant-density scenario, we do not make use of the boundary conditions described there. Additionally, one of the main features of the MSD boundary condition is that its simplicity of implementation allows small research projects to make common use of them. The boundary conditions shown in Ref. [6] can be very complicated to implement, and are therefore best suited for large projects.

The paper is organized as follows: In Sec. 2 we formulate the MSD boundary condition in general for any time-dependent complex-valued PDE. Then, in Sec. 3, we apply the MSD boundary condition to the NLSE. In Sec. 4 we numerically test the MSD boundary condition for simulating the NLSE using a Runge-Kutta finite-difference scheme. Stability effects when using the MSD boundary condition are discussed in Sec. 5. We conclude in Sec. 6.

For this paper we use t

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