On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

2012 ◽  
Vol 129 (3) ◽  
pp. 249-271 ◽  
Author(s):  
G. Biondini ◽  
A. Bui
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Half Line ◽  
Robin Boundary Conditions ◽  
Nonlinear Schrödinger ◽  
Robin Boundary

scholarly journals The numerical unified transform method for the nonlinear Schrödinger equation on the half-line

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Xin Yang ◽  
Bernard Deconinck ◽  
Thomas Trogdon
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Computational Cost ◽  
Nonlinear Schrodinger Equation ◽  
Half Line ◽  
Transform Method ◽  
Nonlinear Schrödinger ◽  
Unified Transform Method

We implement the numerical unified transform method to solve the nonlinear Schrödinger equation on the half-line. For the so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the numerical inverse scattering transform solves whole-line problems. In particular, the method computes the solution at any x and t without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method’s computational cost does not increase for large x , t and the method is more accurate as x , t increase. Our ideas also apply to some cases where the boundary conditions are not linearizable.

EXPECTED AND UNEXPECTED SOLUTIONS TO THE STATIONARY ONE-DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION

2001 ◽  
Vol 15 (10n11) ◽  
pp. 1663-1667
Author(s):  
LINCOLN D. CARR ◽  
CHARLES W. CLARK ◽  
WILLIAM P. REINHARDT
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Einstein Condensate ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Bose Einstein Condensate ◽  
Dilute Gas

We present all stationary solutions to the nonlinear Schrödinger equation in one dimension for box and periodic boundary conditions. For both repulsive and attractive nonlinearity we find expected and unexpected solutions. Expected solutions are those that are in direct analogy with those of the linear Schödinger equation under the same boundary conditions. Unexpected solutions are those that have no such analogy. We give a physical interpretation for the unexpected solutions. We discuss the properties of all solution types and briefly relate them to experiments on the dilute-gas Bose-Einstein condensate.

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