Solitonic model of the condensate
Andrey Gelash, Dmitry Agafontsev, Pierre Suret, Stéphane Randoux
Abstract
We consider a spatially extended box-shaped wave field that consists of a plane wave (the condensate) in the middle and equals zero at the edges, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. Within the inverse scattering transform theory, the scattering data for this wave field is presented by the continuous spectrum of the nonlinear radiation and the soliton eigenvalues together with their norming constants; the number of solitons N is proportional to the box width. We remove the continuous spectrum from the scattering data and find analytically the specific corrections to the soliton norming constants that arise due to the removal procedure. The corrected soliton parameters correspond to symmetric in space N-soliton solution, as we demonstrate analytically in the paper. Generating this solution numerically for N up to 1024, we observe that, at large N, it converges asymptotically to the condensate, representing its solitonic model. Our methods can be generalized for other strongly nonlinear wave fields, as we demonstrate for the hyperbolic secant potential, building its solitonic model as well.