Solitons supported by intensity-dependent dispersion

Abstract

Soliton solutions are studied for paraxial wave propagation with intensity-dependent dispersion.Although the corresponding Lagrangian density has a singularity, analytical solutions, derived by the pseudo-potential method and the corresponding phase diagram, exhibit one-and two-humped solitons with almost perfect agreement to numerical solutions. The results obtained in this work reveal a hitherto unexplored area of soliton physics associated with nonlinear corrections to wave dispersion. 1 arXiv:2001.01631v1 [nlin.PS] 6 Jan 2020 Chromatic dispersion is the dependence of the phase velocity of a wave on its frequency [1] or, equivalently, frequency dependence of the refractive index. Nonlinear corrections to the chromatic dispersion as a function of the wave intensity arise for various waves, such as shallow water waves [2, 3], acoustic waves in micro-inhomogeneous media [4], or ultrafast coherent pulses in GaAs/AlGaAs quantum well waveguide structures [5]. In the context of photon-atom interactions, nonlinear dispersion effects may come about from the saturation of the atomic-level population [6], electromagnetically-induced transparency (EIT) in a chain-Λ configuration [7], or nonlocal nonlinearity mediated by dipole-dipole interactions [8].

Solitary waves with intensity-dependent dispersion: variational characterization

“…It is also interesting to investigate how the singularity locations can change in the time evolution of the solitary waves, our analytical and numerical results rely on the fixed location of the singularities. Finally, it is interesting to study Lyapunov stability stability of other (sign-changing) solitary waves and periodic solutions discussed both in [5] and [8]. It is also worth exploring generalizations of the NLS model in the settings of the discrete (waveguide) systems, as well as in higher-dimensional systems.…”

“…This interplay is responsible for the formation of smooth solitary waves in a wide class of dispersive nonlinear systems. Nevertheless, some physical systems feature intensity-dependent dispersion (IDD); relevant examples include the femtosecond pulse propagation in quantum well waveguides [3], the electromagnetically induced transparency in coherently prepared multistate atoms [4], and fiber-optics communication systems [5]. Such features have been discussed in the context of both photonic, and in phononic (acoustic) crystals [6] and have even been argued to arise at the quantum-mechanical level (between Fock states) in the work of [7].…”

“…Out focus here is in addressing the NLS model with the prototypical IDD introduced in the context of nonlinear optics in [5]:…”

“…where ψ = ψ(x, t) is the complex wave function. It was shown in [5] that the NLS equation ( 1) admits formally two conserved quantities:…”

Solitary waves in the resonant nonlinear Schrödinger equation: Stability and dynamical properties

“…(1) has been shown to reduce to a reaction-diffusion system (RDS) which in turn represents the simplest two-component integrable system contained in the AKNS hierarchy of integrable systems [8,10]. It is worth noting that similar to the RNLS models involving nonlinear modifications of the dispersion term are of continued interest also in other fields such as nonlinear optics; see, e.g., [11] for a recent (albeit somewhat different in flavor) example. The exact solutions of Eq.…”