Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations

Abstract

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrödinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm }$ to the nearest forward and backward singularities; large $X^{+}$ or $X^{-}$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.

arXiv comment28 pages, 14 figures; accepted for Physica D

References in corpus (40)

• Optical frequency comb generation from a monolithic microresonator
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model
• Bosons in anisotropic traps: ground state and vortices
• Spatiotemporal Model for Kerr Comb Generation in Whispering Gallery Mode Resonators
• Stability Analysis of the Lugiato-Lefever Model for Kerr Optical Frequency Combs. Part I: Case of Normal Dispersion
• Dark-Bright Solitons in Inhomogeneous Bose-Einstein Condensates
• Landau and Dynamical Instabilities of the Superflow of Bose-Einstein Condensates in Optical Lattices
• Modulational instability in cigar shaped Bose-Einstein condensates in optical lattices
• Binary Bose-Einstein Condensate Mixtures in Weakly and Strongly Segregated Phases
• Stability analysis for solitons in PT-symmetric optical lattices
• Bose-Einstein Condensates in Optical Lattices: Band-Gap Structure and Solitons
• Model of a PT symmetric Bose-Einstein condensate in a delta-functions double well
• Structure of binary Bose-Einstein condensates
• Frequency combs and platicons in optical microresonators with normal GVD
• Dark-dark solitons and modulational instability in miscible, two-component Bose-Einstein condensates
• Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions
• Bifurcations and stability of gap solitons in periodic potentials
• Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs
• Dark solitons in the Lugiato-Lefever equation with normal dispersion
• Families of Matter-Waves for Two-Component Bose-Einstein Condensates
• Bright solitary matter waves: formation, stability and interactions
• Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials
• Nonlinear modes in the harmonic PT-symmetric potential
• Origin and stability of dark pulse Kerr combs in normal dispersion resonators
• Beating dark-dark solitons in Bose-Einstein condensates
• Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations. Application to two-component Bose-Einstein condensates
• Dissipative periodic waves, solitons and breathers of the nonlinear Schrodinger equation with complex potentials
• Controlling phase separation of a two-component Bose-Einstein condensate by confinement
• States without linear counterpart in Bose-Einstein condensates
• Nonlinear modes for the Gross-Pitaevskii equation -- demonstrative computation approach
• Phase Separation and Dynamics of Two-component Bose-Einstein Condensates
• Truncated-Bloch-wave solitons in optical lattices
• Existence threshold for the ac-driven damped nonlinear Schrödinger solitons
• Dark-bright soliton interactions beyond the integrable limit
• Coding of nonlinear states for the Gross-Pitaevskii equation with periodic potential
• Stationary states of Bose-Einstein condensates in single- and multi-well trapping potentials
• Stable dipole solitons and soliton complexes in the nonlinear Schrodinger equation with periodically modulated nonlinearity
• Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: coding and method for computation
• Spontaneous Parity--Time Symmetry Breaking and Stability of Solitons in Bose-Einstein Condensates