Effect of a Localized Random Forcing Term on the Korteweg-De Vries Equation

In this work, we numerically investigate the influence of a white noise-type forcing on the phenomenon of forced generation of solitons by a localized moving disturbance. Our numerical method is based on finite elements and least-squares. We present numerical experiments for different values of noise amplitude and Froude number, which describe some damping effects on the emission of solitons.     

Dynamics of Embedded Solitons in the Extended Korteweg–de Vries Equations

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is based on the development of a soliton perturbation theory. By obtaining the analytical formula for the tail amplitudes of nonlocal solitary waves, we demonstrate the existence of single-hump embedded solitons for both Hamiltonian and non-Hamiltonian perturbations. These embedded solitons can be isolated (existing at a unique wave speed) or continuous (existing at all wave speeds). Under small wave speed limit, our results show that the tail amplitudes of nonlocal waves are exponentially small, and the product of the amplitude and cosine of the phase is a constant to leading order. This qualitatively reproduces the previous results on the fifth-order KdV equation obtained by exponential asymptotics techniques. We further study the dynamics of embedded solitons and prove that, under Hamiltonian perturbations, a localized wave initially moving faster than the embedded soliton will asymptotically approach this embedded soliton, whereas a localized wave moving slower than the embedded soliton will decay into radiation. Thus, the embedded soliton is semistable. Under non-Hamiltonian perturbations, stable embedded solitons are found for the first time.     

Optical Soliton Bullets in (2+1)D Nonlinear Bragg Resonant Periodic Geometries

We consider light propagation in a Kerr-nonlinear 2D waveguide with a Bragg grating in the propagation direction and homogeneous in the transverse direction. Using Newton's iteration method we construct both stationary and travelling solitary wave solutions of the corresponding mathematical model, the 2D nonlinear coupled mode equations (2D CME). We call these solutions 2D gap solitons due to their similarity with the gap solitons of 1D CME (fiber grating). Long-time stable evolution preserving the solitary fashion is demonstrated numerically despite the fact that, as we show, for the 2D CME no local constrained minima of the Hamiltonian functional exist. Building on the 1D study of [ 1 ], we demonstrate trapping of slow enough 2D gap solitons at localized defects. We explain the mechanism of trapping as resonant transfer of energy from the soliton to one or more nonlinear defect modes. For a special class of defects, we construct a family of nonlinear defect modes by numerically following a bifurcation curve starting at analytically or numerically known linear defect modes. Compared to 1D the dynamics of trapping are harder to fully analyze and the existence of many defect modes for a given defect potential causes that slow solitons store a part of their energy for virtually all of the studied attractive defects.     

Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms

We report exact bright and dark solitary wave solution of the nonlinear Schrodinger equation (NLSE) in cubic–quintic non-Kerr medium adopting phase–amplitude ansatz method. We have found the solitary wave parameters along with the constraints under which bright or dark solitons may exist in such a media. Furthermore, we have also studied the modulation instability analysis both in anomalous and normal dispersion regime. The role of fourth order dispersion, cubic–quintic nonlinear parameter and self-steeping parameter on modulation instability gain has been investigated.     

Instabilities of Multihump Vector Solitons in Coupled Nonlinear Schrödinger Equations

Spectral stability of multihump vector solitons in the Hamiltonian system of coupled nonlinear Schrödinger (NLS) equations is investigated both analytically and numerically. Using the closure theorem for the negative index of the linearized Hamiltonian, we classify all possible bifurcations of unstable eigenvalues in the systems of coupled NLS equations with cubic and saturable nonlinearities. We also determine the eigenvalue spectrum numerically by the shooting method. In case of cubic nonlinearities, all multihump vector solitons in the nonintegrable model are found to be linearly unstable. In case of saturable nonlinearities, stable multihump vector solitons are found in certain parameter regions, and some errors in the literature are corrected.     

Soliton Solutions for a Generalized Inhomogeneous Variable-Coefficient Hirota Equation with Symbolic Computation

Under investigation in this paper is a generalized inhomogeneous variable- coefficient Hirota equation. Through the Hirota bilinear method and symbolic computation, the bilinear form and analytic one-, two- and N-soliton solutions for such an equation are obtained, respectively. Properties of those solitons in the inhomogeneous media are discussed analytically. We get the soliton with the property that the larger the amplitude is, the narrower and slower the pulse is. Dynamics of that soliton can be regarded as a repulsion of the soliton by the external potential barrier. During the interaction of two solitons, we observe that the larger the value of the coefficient β in the equation is, the larger the distance of the two solitons is.     

Soliton Resonances for the MKP-II

Using the second flow (derivative reaction-diffusion system) and the third one of the dissipative SL(2, ℝ) Kaup-Newell hierarchy, we show that the product of two functions satisfying those systems is a solution of the modified Kadomtsev-Petviashvili equation in 2+1 dimensions with negative dispersion (MKP-II). We construct Hirota’s bilinear representations for both flows and combine them as the bilinear system for the MKP-II. Using this bilinear form, we find one- and two-soliton solutions for the MKP-II. For special values of the parameters, our solution shows resonance behavior with the creation of four virtual solitons. Our approach allows interpreting the resonance soliton as a composite object of two dissipative solitons in 1+1 dimensions.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 133–142, July, 2005.     

Interaction of a breather with a magnetization wave in a ferromagnet with light-axis anisotropy

We use the dressing method to find exact solutions of the Landau-Lifshitz equation for a ferromagnet with light-axis anisotropy. These solutions describe the interaction of a nonlinear precession wave of arbitrary amplitude with solitons. We analyze the change of the internal structure and the physical parameters of the solitons as a result of their interaction with the magnetization wave. We find an infinite series of integrals of motion that stabilize the soliton on the background of the pumping wave.     

Exponential Asymptotics for Line Solitons in Two‐Dimensional Periodic Potentials

As a first step toward a fully two‐dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two‐dimensional periodic potentials. For this two‐dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one‐dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence‐relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch‐band edge; and for each rational slope, two line‐soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline‐soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.     

Forced motion of breathers and domain boundaries against the background of nonlinear magnetization wave

Exact solutions of Landau–Lifshitz equation for a ferromagnet with an easy-axis anisotropy, which describe interaction of nonlinear precession wave of large amplitude with soliton-like objects, such as breathers, solitary domains and domain boundaries, are found by the “dressing” method. The change of the internal structure and physical parameters of solitons due to interaction with magnetization wave is analyzed. It is shown, that both solitary domains and domain walls move toward the wave. The conditions for the destruction of solitons by the nonlinear magnetization wave are obtained. An infinite series of integrals of motion, that stabilize the solitons on the background of magnetization wave, is found.     

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.     

Degenerate Four-Virtual-Soliton Resonance for the KP-II

We propose a method for solving the (2+1)-dimensional Kadomtsev-Petviashvili equation with negative dispersion (KP-II) using the second and third members of the disipative version of the AKNS hierarchy. We show that dissipative solitons (dissipatons) of those members yield the planar solitons of the KP-II. From the Hirota bilinear form of the SL(2, ℝ) AKNS flows, we formulate a new bilinear representation for the KP-II, by which we construct one- and two-soliton solutions and study the resonance character of their mutual interactions. Using our bilinear form, for the first time, we create a four-virtual-soliton resonance solution of the KP-II, and we show that it can be obtained as a reduction of a four-soliton solution in the Hirota-Satsuma bilinear form for the KP-II.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 162–170, July, 2005.     

Spatial Vector Solitons in Nonlinear Photonic Crystal Fibers

We study spatial vector solitons in a photonic crystal fiber (PCF) made of a material with the focusing Kerr nonlinearity. We show that such two-component localized nonlinear waves consist of two mutually trapped components confined by the PCF linear and the self-induced nonlinear refractive indices, and they bifurcate from the corresponding scalar solitons. We demonstrate that, in a sharp contrast with an entirely homogeneous nonlinear Kerr medium where both scalar and vector spatial solitons are unstable and may collapse, the periodic structure of PCF can stabilize the otherwise unstable two-dimensional spatial optical solitons. We apply the matrix criterion for stability of these two-parameter solitons, and verify it by direct numerical simulations.     

On weakly nonlinear waves in media exhibiting mixed nonlinearity

We obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. The coefficients of the nonlinear terms in the transport equation are obtained in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics; the amplitude of the disturbance is shown to become unbounded in the neighborhood of the point where the ray tube collapses. We also obtain a formula, akin to the one obtained by R. Rosales (1991), for the energy dissipated across shocks.     

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.     

Partially Coherent Waves in Nonlinear Periodic Lattices

We study the propagation of partially coherent (random-phase) waves in nonlinear periodic lattices. The dynamics in these systems is governed by the threefold interplay between the nonlinearity, the lattice properties, and the statistical (coherence) properties of the waves. Such dynamic interplay is reflected in the characteristic properties of nonlinear wave phenomena (e.g., solitons) in these systems. While the propagation of partially coherent waves in nonlinear periodic systems is a universal problem, we analyze it in the context of nonlinear photonic lattices, where recent experiments have proven their existence.     

Dipole and Quadrupole Solitons in Optically Induced Two-Dimensional Photonic Lattices: Theory and Experiment

Dipole and quadrupole solitons in a two-dimensional photorefractive optical lattice are investigated both theoretically and experimentally. It is shown theoretically that out-of-phase dipole solitons and quadrupole solitons exist and are linearly stable in the intermediate-intensity regime. In-phase dipole and quadrupole solitons, however, are always linearly unstable, but their instabilities are rather weak in the low-intensity regime. Experimentally, both types of dipole solitons are observed, and the experimental results agree qualitatively with the theoretical predictions. In addition, we have observed the anisotropic effect of the photorefractive crystal in the dipole-soliton formation.     

On Gradient Solitons of the Ricci–Harmonic Flow

In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.     

Dressing the boundary: On soliton solutions of the nonlinear Schrödinger equation on the half‐line

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schrödinger equation on the half‐line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux‐dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self‐modulated solitons on a constant background. Half‐line solitons in both cases are explicitly computed. In particular, the boundary‐bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.