Non-conservative variational approximation for nonlinear Schrödinger equations
Nonlinear Dynamical System Group, Computational Science Research Center, Department of Mathematics and Statistics, San Diego State University, 92182-7720, San Diego, CA, USA
2 Department of Mathematics and Statistics, University of Massachusetts, 01003-4515, Amherst, MA, USA
Accepted: 11 August 2020
Published online: 21 October 2020
In the work of Galley (Phys Rev Lett 110:174301, 2013) an initial value problem formulation of Hamilton’s principle was proposed and applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrödinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems, namely the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein. We showcase the relevance of the NCVA method by exploring test case examples within the NLS setting including combinations of linear and density-dependent loss and gain. We also present an example applied to exciton–polariton condensates that intrinsically feature loss and a spatially dependent gain term.
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