Three-dimensional rogue waves in non-stationary parabolic potentials

TL;DR

This paper develops a symmetry-based transformation to connect 3D inhomogeneous NLS equations with variable coefficients to 1D constant coefficient NLS equations, enabling the construction of 3D rogue wave solutions with potential applications.

Contribution

It introduces a systematic similarity transformation that reduces the (3+1)D inhomogeneous NLS to a (1+1)D integrable NLS, facilitating the derivation of 3D rogue wave solutions.

Findings

Derived 3D rogue wave-like solutions with complex temporal evolution.

Demonstrated the transformation using rational solutions of the NLS.

Showed potential for experimental realization in optics and BECs.

Abstract

Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous nonlinear Schrodinger (NLS) equation with variable coefficients and parabolic potential to the (1+1)-dimensional NLS equation with constant coefficients. This transformation allows us to relate certain class of localized exact solutions of the (3+1)-dimensional case to the variety of solutions of integrable NLS equation of (1+1)-dimensional case. As an example, we illustrated our technique using two lowest order rational solutions of the NLS equation as seeding functions to obtain rogue wave-like solutions localized in three dimensions that have complicated evolution in time including interactions between two time-dependent rogue wave solutions. The obtained three-dimensional rogue wave-like solutions may raise the possibility of relative experiments and potential applications in nonlinear optics and BECs.