The methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark “soliton islands” in a “sea of solitary waves” of the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain or absorption. It is shown that solitons exist only under certain conditions and the parameter functions describing dispersion, nonlinearity, and gain or absorption inhomogeneities cannot be chosen independently. Fundamental soliton management regimes are discovered.

Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies.

We show that the methodology based on the generalized inverse scattering transform (IST) concept provides a systematic way to discover the novel exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain or absorption. The fundamental innovation of the present approach is to notice that it is possible both to allow for a variable spectral parameter with new dependent variables and to apply of the famous "moving in time focuses" concept of the self-focusing theory to the IST formalism. We show that for nonlinear optics this algorithm is a useful tool to design novel dispersion managed fiber transmission lines and soliton lasers. Fundamental soliton management regimes are predicted.

The fundamental concept of colored nonautonomous solitons in nonlinear and dispersive nonautonomous physical systems is introduced. Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The parallels between nonlinear guided wave phenomena in optics and nonlinear guided wave phenomena in Bose condensates are clearly demonstrated by considering optical and matter wave soliton dynamics in the framework of nonautonomous evolution equations. The exact analytical solutions and numerical experiments reveal many specific features of nonautonomous solitons. Fundamental laws of the soliton adaptation to the external potentials are derived. Bound states of colored nonautonomous solitons are studied in detail and a comparison of the canonical Satsuma–Yajima breather dynamics with a nonautonomous ‘agitated’ breather is presented. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies

By means of analytical and numerical methods, we reveal the main features of nonautonomous matter-wave solitons near the Feshbach resonance in a one-dimensional Bose-Einstein condensate confined by a harmonic potential with a varying-in-time longitudinal trapping frequency. Based on the generalized nonautonomous Gross-Pitaevskii model, we show that solitons in nonautonomous physical systems exist only under certain conditions so that varying-in-time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability scenarios and complement each other. We focus on the most physically important examples where the applied magnetic field is either a linearly or a periodically varying-in-time function. In the case of periodically varying scattering length, variations of confining harmonic potential are found to be sign-reversible (periodic attractive and repulsive) to support the soliton-management regime. We investigate the losses of validity of one-dimensional (1D) approximation in the cases when, by the joint action of varying-in-time nonlinearity and confining potential, the atom cloud can be compressed from an initially elongated quasi-1D cigar-shaped geometry to a final ball-shaped three-dimensional geometry and the induced soliton collapse may occur.

The novel stable “soliton islands” in a “sea of solitary waves” of the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain or absorption are discovered. Different soli