Nonlinear Schrödinger equation: looking for new waves

The nonlinear Schrödinger equation is one of the most important and most studied equations in mathematical physics. It is a relatively simple and compact model which provides a paradigmatic description of nonlinear waves in a variety of physical systems: water waves in oceans, light waves in optical fibers, Bose Einstein condensates (peculiar gases of ultra cold atoms) and plasmas. Its universality is definitely a reason for its success and for the continuous research which scholars devote to it since more than 50 years.

Most importantly the nonlinear Schrödinger equation is integrable, it can be solved exactly by the inverse scattering transform and possess an infinite number of conserved quantities (integrals of motions) which makes its solutions extremely rich and complex.

Notably solitons are some of the most interesting solutions of the nonlinear Schrödinger equation. Solitons, or solitary waves are waves which are localized in space (or in time) and which propagate without changing their shape and without dispersing. Furthermore they exhibit a particle like behavior, they are unchanged upon collision with other solitons.

The richness and complexity of the nonlinear Schrödinger equation is even increased when its generalized forms are considered. Such generalized forms contain further terms to include the description of more phenomena (in addition to cubic nonlinearity and diffraction/dispersion).

Very interesting are terms describing net exchange of energy between the wave and the environment where it propagates, and also variations and non homogeneities of the environment itself, since these are very common scenarios occurring in nature.

On the right A bullet (pulse localized both in the direction of its motion and in the one orthogonal to it) of small waves (the Bogoliubov-de Gennes excitations) travels on top of a “see” ( the condensate) without spreading (unlike a normal pulse shown on the left)! (From: S. Kumar, A. M. Perego and K. Staliunas, Linear and Nonlinear Bullets of the Bogoliubov–de Gennes Excitations, Phys Rev. Lett. 118, 044103 (2017). Video credit: Shubham Kumar)

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