Conservation laws and solitons for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

Abstract

Under investigation in this paper are the coupled cubic–quintic nonlinear Schrödinger equations describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained via the Ablowitz–Kaup–Newell–Segur scheme and the corresponding Darboux transformation is constructed. One-, two- and three-soliton solutions are presented and an infinite number of conservation laws are also derived. The features of solitons are graphically discussed: (i) head-on and overtaking elastic collisions of the two solitons; (ii) periodic attraction and repulsion of the bounded states of two solitons; (iii) energy-exchanging collisions of the three solitons.

Dark–bright soliton interactions for the coupled cubic-quintic nonlinear Schrödinger equations in fiber optics

“…When ι 1 = ρ 1 and ι 2 = ρ 2 , the bright soliton solutions for equations (2) have been derived via the Darboux transformation [23]. Conservation laws have been derived via the Ablowitz-Kaup-Newell-Segur scheme [24].…”

“…Vector solitons governed by coupled NLS-type equations can be generated via the two orthogonally polarized components of a single optical field, the two cores of an optical waveguide, or the two fields of different frequencies but with the same polarization [20][21][22]. With multi-mode effects considered, the following coupled NLS equations with cubic-quintic nonlinearity have been proposed [23,24]:…”

The Darboux Transformation and N-Soliton Solutions of Coupled Cubic-Quintic Nonlinear Schrödinger Equation on a Time-Space Scale

“…where ρ 1 is a real parameter [21,22]. This equation has been one of the universal mathematical models in the field of nonlinear science, which is applied widely into optics [23,24], Bose-Einstein condensation [25,26], and other fields [27].…”

Nonautonomous Solitons for the Coupled Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equations with External Potentials in the Non-Kerr Fibre

“…Based on what is aforementioned and extending (1), researchers have investigated the coupled variablecoefficient cubic-quintic NLS equations with external potentials as follows [28,[38][39][40] : 2 2 1 1 1 2 1 2 22 2 2 1 1 2 2 1 1 1 2 2 1 1 1 1 2 2 2 1 1 2 ) 2 2 2 2 1 2 2 2 22 2 2 1 1 2 2 2 1 1 2 2 2 1 1 1 2 2 2 2 1 2 …”

“…(2) turns into (1) [28,39,40], of which the bilinear forms and the corresponding soliton solutions [28], conservation laws, Lax pair and Darboux transformation [39], general Darboux transformation and the corresponding Nth-order rogue wave solitons [40,41] have been investigated. However, to our knowledge, the nonautonomous solitons for (2), which are nonautonomous systems, have not been studied yet.…”