Short-lived two-soliton bound states in weakly perturbed nonlinear Schrodinger equation

Phase controlling of collisions between solitons in the two-dimensional complex Ginzburg-Landau equation without viscosity

We present a systematic analysis of the outcome of soliton collisions upon variation of the relative phase φ of the solitons, in the two-dimensional cubic-quintic complex Ginzburg-Landau equation in the absence of viscosity. Three generic outcomes are identified: merger of the solitons into a single one, creation of an extra soliton, and quasielastic interaction. The velocities of the merger soliton and the extra soliton can be effectively controlled by φ. In addition, the range of the outcome of creating an extra soliton decreases to zero with the reduction of gain or the increasing of loss. The above features have potential applications in optical switching and logic gates based on interaction of optical solitons.

Generation of solitons and breathers in the extended Korteweg-de Vries equation with positive cubic nonlinearity

The initial-value problem for box-like initial disturbances is studied within the framework of an extended Korteweg-de Vries equation with both quadratic and cubic nonlinear terms, also known as the Gardner equation, for the case when the cubic nonlinear coefficient has the same sign as the linear dispersion coefficient. The discrete spectrum of the associated scattering problem is found, which is used to describe the asymptotic solution of the initial-value problem. It is found that while initial disturbances of the same sign as the quadratic nonlinear coefficient result in generation of only solitons, the case of the opposite polarity of the initial disturbance has a variety of possible outcomes. In this case solitons of different polarities as well as breathers may occur. The bifurcation point when two eigenvalues corresponding to solitons merge to the eigenvalues associated with breathers is considered in more detail. Direct numerical simulations show that breathers and soliton pairs of different polarities can appear from a simple box-like initial disturbance.

Radiationless energy exchange in three-soliton collisions

We revisit the problem of the three-soliton collisions in the weakly perturbed sine-Gordon equation and develop an effective three-particle model allowing us to explain many interesting features observed in numerical simulations of the soliton collisions. In particular, we explain why collisions between two kinks and one antikink are observed to be practically elastic or strongly inelastic depending on relative initial positions of the kinks. The fact that the three-soliton collisions become more elastic with an increase in the collision velocity also becomes clear in the framework of the three-particle model. The three-particle model does not involve internal modes of the kinks, but it gives a qualitative description to all the effects observed in the three-soliton collisions, including the fractal scattering and the existence of short-lived three-soliton bound states. The radiationless energy exchange between the colliding solitons in weakly perturbed integrable systems takes place in the vicinity of the separatrix multi-soliton solutions of the corresponding integrable equations, where even small perturbations can result in a considerable change in the collision outcome. This conclusion is illustrated through the use of the reduced three-particle model.

Universal map for fractal structures in weak interactions of solitary waves

Fractal scatterings in weak solitary-wave interactions are analyzed for generalized nonlinear Schrödiger equations (GNLS). Using asymptotic methods, these weak interactions are reduced to a universal second-order map. This map gives the same fractal-scattering patterns as those in the GNLS equations both qualitatively and quantitatively. Scaling laws of these fractals are also derived.

Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations

Weak interactions of solitary waves in the generalized nonlinear Schrödinger equations are studied. It is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities. Then by using the Karpman-Solov'ev method, a universal system of dynamical equations is derived for the velocities, amplitudes, positions, and phases of interacting solitary waves. These dynamical equations contain a single parameter, which accounts for the different forms of nonlinearity. When this parameter is zero, these dynamical equations are integrable, and the exact analytical solutions are derived. When this parameter is nonzero, the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively. Thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established. The origin of these fractal structures is also explored. It is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite-time singularities. Based on this observation, an analytical criterion for the existence and locations of fractal structures is obtained. Lastly, these analytical results are applied to the generalized nonlinear Schrödinger equations with various nonlinearities such as the saturable nonlinearity, and predictions on their weak interactions of solitary waves are made.

Lattice solitons in quasicondensates

We analyze finite temperature effects in the generation of bright solitons in condensates in optical lattices. We show that even in the presence of strong phase fluctuations solitonic structures with a well defined phase profile can be created. We propose a novel family of variational functions which describe well the properties of these solitons and account for the nonlinear effects in the band structure. We discuss also the mobility and collisions of these localized wave packets.

Energy exchange in collisions of intrinsic localized modes

Energy transfer process is examined numerically for the binary collision of intrinsic localized modes (ILMs) in the Fermi-Pasta-Ulam beta lattice. Unlike "solitons" in the integrable systems, ILMs exchange their energy in collision due to the discreteness effect. The mechanism of this energy exchange is examined in detail, and it is shown that the phase difference is the most dominant factor in the energy exchange process and, generally speaking, the ILM with advanced phase absorbs energy from the other. Heuristic model equations which describe the energy transfer of ILMs are proposed by considering the ILMs as interacting "particles." The results due to these equations agree qualitatively very well with those of the numerical simulations. In some cases, the relation between the phase difference of the ILMs and the transferred energy becomes singular, which may be regarded as one of the major mechanisms responsible for the generation of "chaotic breathers."

Two-soliton collisions in a near-integrable lattice system

We examine collisions between identical solitons in a weakly perturbed Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which corresponds to the Salerno model, and may be realized as an array of strongly overlapping nonlinear optical waveguides) or a quintic perturbation, or both. Complex dependences of the outcomes of the collisions on the initial phase difference between the solitons and location of the collision point are observed. Large changes of amplitudes and velocities of the colliding solitons are generated by weak perturbations, showing that the elasticity of soliton collisions in the AL model is fragile (for instance, the Salerno's perturbation with the relative strength of 0.08 can give rise to a change of the solitons' amplitudes by a factor exceeding 2). Exact and approximate conservation laws in the perturbed system are examined, with a conclusion that the small perturbations very weakly affect the norm and energy conservation, but completely destroy the conservation of the lattice momentum, which is explained by the absence of the translational symmetry in generic nonintegrable lattice models. Data collected for a very large number of collisions correlate with this conclusion. Asymmetry of the collisions (which is explained by the dependence on the location of the central point of the collision relative to the lattice, and on the phase difference between the solitons) is investigated too, showing that the nonintegrability-induced effects grow almost linearly with the perturbation strength. Different perturbations (cubic and quintic ones) produce virtually identical collision-induced effects, which makes it possible to compensate them, thus finding a special perturbed system with almost elastic soliton collisions.

Soliton collisions in the discrete nonlinear Schrödinger equation

We report analytical and numerical results for on-site and intersite collisions between solitons in the discrete nonlinear Schrödinger model. A semianalytical variational approximation correctly predicts gross features of the collision, viz., merger or bounce. We systematically examine the dependence of the collision outcome on initial velocity and amplitude of the solitons, as well as on the phase shift between them, and location of the collision point relative to the lattice; in some cases, the dependences are very intricate. In particular, merger of the solitons into a single one, and bounce after multiple collisions are found. Situations with a complicated system of alternating transmission and merger windows are identified too. The merger is often followed by symmetry breaking (SB), when the single soliton moves to the left or to the right, which implies momentum nonconservation. Two different types of the SB are identified, deterministic and spontaneous. The former one is accounted for by the location of the collision point relative to the lattice, and/or the phase shift between the solitons; the momentum generated during the collision due to the phase shift is calculated in an analytical approximation, its dependence on the solitons' velocities comparing well with numerical results. The spontaneous SB is explained by the modulational instability of a quasiflat plateau temporarily formed in the course of the collision.

Chaotic character of two-soliton collisions in the weakly perturbed nonlinear Schrödinger equation

We analyze the exact two-soliton solution to the unperturbed nonlinear Schrödinger equation and predict that in a weakly perturbed system (i) soliton collisions can be strongly inelastic, (ii) inelastic collisions are of almost nonradiating type, (iii) results of a collision are extremely sensitive to the relative phase of solitons, and (iv) the effect is independent on the particular type of perturbation. In the numerical study we consider two different types of perturbation and confirm the predictions. We also show that this effect is a reason for chaotic soliton scattering. For applications, where the inelasticity of collision, induced by a weak perturbation, is undesirable, we propose a method of compensating it by perturbation of another type.