Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation

We study a nonlinear ground state of the Gross-Pitaevskii equation with a parabolic potential in the hydrodynamics limit often referred to as the Thomas-Fermi approximation. Existence of the energy minimizer has been known in literature for some time but it was only recently when the Thomas-Fermi approximation was rigorously justified. The spectrum of linearization of the Gross-Pitaevskii equation at the ground state consists of an unbounded sequence of positive eigenvalues. We analyze convergence of eigenvalues in the hydrodynamics limit. Convergence in norm of the resolvent operator is proved and the convergence rate is estimated. We also study asymptotic and numerical approximations of eigenfunctions and eigenvalues using Airy functions.     

Some properties of the spinor soliton

In this paper, the solitons of nonlinear Dirac equation are discussed in detail, and several functions which reflect their characteristics are computed. The numerical results show that, the nonlinear Dirac equation has only finite meaningful solitons, and these solitons have 1/2-spin and positive mass; the spinor soliton has two kinds of parity states, and each parity state has two kinds of energy states; the larger the self-coupling coefficientw, the more the excitation states, and ifw is less than a critical value, then the meaningful soliton does not exist. These properties may have relations with some fundamental particles.     

Two-dimensional solitons in irregular lattice systems

We compute and study localized nonlinear modes (solitons) in the semi-infinite gap of the focusing two-dimensional nonlinear Schrödinger (NLS) equation with various irregular lattice-type potentials. The potentials are characterized by large variations from periodicity, such as vacancy defects, edge dislocations, and a quasicrystal structure. We use a spectral fixed-point computational scheme to obtain the solitons. The eigenvalue dependence of the soliton power indicates parameter regions of self-focusing instability; we compare these results with direct numerical simulations of the NLS equation. We show that in the general case, solitons on local lattice maximums collapse. Furthermore, we show that the Nth-order quasicrystal solitons approach Bessel solitons in the large-N limit.     

Multiscale expansions avector solitons of a two-dimensional nonlocal nonlinear Schrödinger system

One- and two-dimensional solitons of a multicomponent nonlocal nonlinear Schrödinger (NLS) system are constructed. The model finds applications in nonlinear optics, where it may describe the interaction of optical beams of different frequencies. We asymptotically reduce the model, via multiscale analysis, to completely integrable ones in both Cartesian and cylindrical geometries; we thus derive a Kadomtsev-Petviashvili equation and its cylindrical counterpart, Johnson's equation. This way, we derive approximate soliton solutions of the nonlocal NLS system, which have the form of: (a) dark or antidark soliton stripes and (b) dark lumps in the Cartesian geometry, as well as (c) ring dark or antidark solitons in the cylindrical geometry. The type of the soliton, namely dark or antidark, is determined by the degree of nonlocality: dark (antidark) soliton states are formed for weaker (stronger) nonlocality. We perform numerical simulations and show that the derived soliton solutions do exist and propagate undistorted in the original nonlocal NLS system.     

Doubly localized two-dimensional rogue waves generated by resonant collision in Maccari system

In this paper, the Maccari system is investigated, which is viewed as a two-dimensional extension of nonlinear Schrödinger equation. We derive doubly localized two-dimensional rogue waves on the dark solitons of the Maccari system with Kadomtsev–Petviashvili hierarchy reduction method. The two-dimensional rogue waves include line segment rogue waves and rogue-lump waves, which are localized in two-dimensional space and time. These rogue waves are generated by the resonant collision of rational solitary waves and dark solitons, the whole process of transforming elastic collision into resonant collision is analytically studied. Furthermore, we also discuss the local characteristics and asymptotic properties of these rogue waves. Simultaneously, the generating conditions of the line segment rogue wave and rogue-lump wave are also given, which provides the possibility to predict rogue wave. Finally, a new way to obtain the high-order rogue waves of the nonlinear Schrödinger equation are given by proper reduction from the semi-rational solutions of the Maccari system.     

On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity

We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic–quintic nonlinear Schrödinger equation with a periodic potential.     

Spectral and dispersion properties of pair soliton states in a quasi-one-dimensional anisotropic antiferromagnet withS=1/2

By solution of the Schrödinger equation in the continuum approximation, it is shown analytically that there exist excited eigenstates of the quasi-one-dimensional Ising antiferromagnet with spinS=1/2 in the form of spatially localized quantum states. Computer modeling of a discrete model of interacting solitons with allowance for the symmetry of the solutions gives eigenvalues of the Sturm sequence that differ from the solutions of the continuum approximation. The spectral and dispersion properties of the nonlinear bound states of lowest energy and the selection rules in resonance transitions in an external magnetic field applied parallel to and perpendicular to the axis of magnetic anisotropy are calculated.L. V. Kirenski Physics Institute, Siberian Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 112–119, April, 1992.     

Limite semi-classique pour l'équation de Schrödinger non-linéaire avec potentiel harmonique

This Note is dedicated to the semiclassical limit of the nonlinear focusing Schrödinger equation with a harmonic potential. The method does not use a linearization argument as is usually done, but the conservation laws (quantum and classical) and the stability of the ground state. To cite this article: S. Keraani, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.      

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.     

An Instability Criterion for Nonlinear Standing Waves on Nonzero Backgrounds

A nonlinear Schrödinger equation with repulsive (defocusing) nonlinearity is considered. As an example, a system with a spatially varying coefficient of the nonlinear term is studied. The nonlinearity is chosen to be repelling except on a finite interval. Localized standing wave solutions on a non-zero background, e.g., dark solitons trapped by the inhomogeneity, are identified and studied. A novel instability criterion for such states is established through a topological argument. This allows instability to be determined quickly in many cases by considering simple geometric properties of the standing waves as viewed in the composite phase plane. Numerical calculations accompany the analytical results.     

On Soccer Balls and Linearized Inverse Statistical Mechanics

The classical inverse statistical mechanics question involves inferring properties of pairwise interaction potentials from exhibited ground states. For patterns that concentrate near a sphere, the ground states can range from platonic solids for small numbers of particles to large systems of particles exhibiting very complex structures. In this setting, previous work (von Brecht et?al., Math. Models Methods Appl. Sci. 22, 2012) allows us to infer that the linear instabilities of the pairwise potential accurately characterize the resulting nonlinear ground states. Potentials with a small number of spherical harmonic instabilities may produce very complex patterns as a result. This leads naturally to the linearized inverse statistical mechanics question: given a finite set of unstable modes, can we construct a potential that possesses precisely these linear instabilities? If so, this would allow for the design of potentials with arbitrarily intricate spherical symmetries in the ground state. In this paper, we solve our linearized inverse problem in full, and present a wide variety of designed ground states.     

Solitary waves in a quartic nonlinear elastic rod

The bright and dark solitons described by the nonlinear Schrödinger equation (NLSE) are given for a quartic nonlinear elastic rod. It has also been found that the KdV soliton does not exist in this system.     

Analytical studies of soliton pulses along two-dimensional coupled nonlinear transmission lines

A nonlinear network with many coupled nonlinear LC dispersive transmission lines is considered, each line of the network containing a finite number of cells. In the semi-discrete limit, we apply the reductive perturbation method and show that the wave propagation along the network is governed by a two-dimensional nonlinear partial differential equation (2-D NPDE) of Schrödinger type. Two regimes of wave propagation, the high-frequency and the low-frequency are detected. By the means of exact soliton solution of the 2-D NPDE, we investigate analytically the soliton pulse propagation in the network. Our results show that the network under consideration supports the propagation of kink and dark solitons.     

A minimization method and applications to the study of solitons

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior. In this paper, we prove a general, abstract theorem ( Theorem 26) which allows to prove the existence of a class of solitons. Such solitons are suitable minimizers of a constrained functional and they are called hylomorphic solitons. Then we apply the abstract theory to problems related to the nonlinear Schrödinger equation (NSE) and to the nonlinear Klein–Gordon equation (NKG).     

Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system with symbolic computation

In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via symbolic computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides.     

Asymptotic dynamics of nonlinear Schrödinger equations with many bound states

We consider a nonlinear Schrödinger equation with a bounded localized potential in . The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H¹, and that its ground state component is larger than n^3−ε with ε>0 small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.     

Anisotropic subdiffractive solitons

We study solitons in the two-dimensional defocusing nonlinear Schrödinger equation with the spatio-temporal modulation of the external potential. The spatial modulation is due to a square lattice; the resulting macroscopic diffraction is rotationally symmetric in the long-wavelength limit but becomes anisotropic for shorter wavelengths. Anisotropic solitons-solitons with the square (x, y)-geometry - are obtained both in the original nonlinear Schrödinger model and in its averaged amplitude equation.     

Dark-soliton dynamics in a two-species Bose–Einstein condensate with the external potential and thermal cloud effects

Considering the time-dependent external potential and thermal cloud effects, this paper investigates via symbolic computation the dark-soliton dynamics in a two-species Bose–Einstein condensate (BEC), which can be described by the quasi-one-dimensional coupled Gross–Pitaevskii equations. Under the balance between the harmonic potential and thermal cloud effects, dark multi-soliton solutions are derived for the two-species BEC through the Hirota bilinear method. Regions of the s$s$-wave scattering lengths are ascertained for the existence of the dark solitons in two species. Influence of the scattering lengths and external potential on the background density, soliton width and velocity is examined. Graphical analysis demonstrates that the harmonic and linear potentials can change the propagation paths, collision positions and collision time of the dark solitons, and that the thermal cloud effects can affect the number of atoms in the two-species BEC.     

Bright and dark solitons in a periodically attractive and expulsive potential with nonlinearities modulated in space and time

This work deals with soliton solutions of the nonlinear Schrödinger equation with a diversity of nonlinearities. We solve the equation in a potential which oscillates in time between attractive and expulsive behavior, in the presence of nonlinearities which are modulated in space and time. Despite the presence of the periodically expulsive behavior of the potential, the results show that the nonlinear equation can support a diversity of localized excitations of the bright and dark types.