On the reflection of solitons of the cubic nonlinear Schrödinger equation

Abstract

Abstract. In this paper we perform a numerical study on the interesting phenomenon of soliton reflection of solid walls. We consider the 2D cubic nonlinear Schrödinger equation as the underlying mathematical model and we use an implicit-explicit type Crank-Nicolson finite element scheme for its numerical solution. After verifying the perfect reflection of the solitons on a vertical wall, we present the imperfect reflection of a dark soliton on a diagonal wall.

Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

“…5). It is worth to note that, in the bibliography, the Relaxation Scheme is formulated along with the initial choice Φ 1 2 := ( 0 ) (see, e.g., [4,5,10,11]), which is a first order in time approximation of ( ( 1 2 , •)) and results a first order in time convergence of Φ + 1 2 to ( ( + 1 2 , •)) (see Thm. 4.7 and Tab.…”

“…Two decades ago, for the discretization in time of the nonlinear Schrödinger (NLS) equation, C. Besse [4] introduced a new linear-implicit, conservative time-stepping method (called Relaxation Scheme (RS)) as an attempt to avoid the numerical solution of the nonlinear systems of algebraic equations that the application of the implicit Crank-Nicolson method yields. The (RS) combined with a finite element or a finite difference space discretization, is computationally efficient (see, e.g., [3,8,11]) and performs as a second order method (see, e.g., [5,11]). Later, C. Besse [5], analysing the (RS), as a semidiscrete in time method to approximate the solution to the Cauchy problem for the (NLS) equation with power non-linearity, provided its convergence under small final time , without concluding a convergent rate with respect to the time-step.…”

“…Katsaounis and Kyza [10] first proposed a finite element version of the (RS) over a non uniform partition of the time interval, and then constructed an posteriori bound for the error only at the time-nodes, under the assumption that the proposed method has a second order convergence at the intermediate time nodes. At this point, we would like to observe, that the finite element version of the (RS) proposed in [10,11] requires the solution of two linear systems of algebraic equations at every time-step, and thus its computational complexity is two times higher than that of the corresponding finite difference version of the (RS).…”

“…2). Also, we undertook the challenge to investigate the influence on the convergence rate of the initial choice Φ 1 2 = ( 0 ), which is a first order (with respect to ) approximation of ( ( 12 , •)) and it is used in actual computations by several authors (see, e.g., [4,5,10,11]). First, we show that the latter initial choice affects the order of convergence at the intermediate node (see Cor.…”

Capturing of solitons collisions and reflections in nonlinear Schrödinger type equations by a conservative scheme based on MOL

“…Various numerical approaches have been presented for solving nonlinear Schrödinger-type equations [15][16][17][18][19][20]. Recently, the reflection of solitons due to walls of the (2 + 1)-dimensional cubic NLS is studied numerically using an explicit-implicit scheme by Crank-Nicolson finite element technique [21]. The authors studied the reflection of a single soliton to wall for the NLS subjected to three boundary condition types, whereas they did not consider soliton collisions in their study.…”