Algebraic Solitons Research Articles

On Recurrent Solutions as Imbricate Series of Rational Growing-and-Decaying Modes: Solutions to the Davey-Stewartson Equation

where p = ±1, r is a constant. Equation (1) with p = 1 and p = −1 are called the DS I and DS II equation, respectively. Although it is known that the spatially periodic solitons can be constructed by the imbricate series of the algebraic solitons, we feel it is to be pointed out that the recurrent solutions can be constructed by the imbricate series of solutions. So in this note, it is shown that the growing-and-decaying mode and the breather solution to the DS equation are constructed as imbricate series of rational growing-and-decaying modes. The growing-and-decaying mode solution is given by

Quasi-two-dimensional solitons in ferroelectrics

The production and dynamics of nonlinear localized acoustic formations in ferroelectrics with the relaxation absorption are investigated. These formations are interpreted as algebraic solitons (lumps). It is shown that they are stable for transverse perturbations.

Moving near-gap solitons in the nonlinear optical medium

For a one-dimensional nonlinear optical medium with a periodic refraction index, new two-parameter soliton solutions of electrodynamics equations have been found. These solutions represent two interacting waves that propagate in two opposite directions. The oscillation frequency of each wave may fall either into the forbidden gap in the linear spectrum or outside it, and the group velocity may vary from zero to a maximal value that is determined by the parameters of the medium. Algebraic soliton solutions have been found as the limit of the nonlinear solutions, when the nonlinear wave frequency tends to the frequency of one of the linear-spectrum branches.

Stochastic Kadomtsev-Petviashvili equation

The example of Kadomtsev-Petviashvili equations with a random time-dependent force (stochastic Kadomtsev-Petviashvili equations) is used to show that the theory of Brownian particle motion can be applied to the theory of the stochastic behavior of solitons of model hydrodynamic equations which are completely integrable in the absence of forces and interrelated by the generalized Galilean transformation. The Brownian motion of two-dimensional algebraic solitons of the Kadomtsev-Petviashvili equations with positive dispersion leads to their diffusion broadening similar to the broadening of one-dimensional solitons of other fully integrable hydrodynamic equations. However, for longer times the rate of decay of algebraic solitons is higher because of the degeneracy of the momentum integral for these solitons. The behavior of a periodic chain of algebraic solitons is established under the action of a random force. Tilted plane solitons of the Kadomtsev-Petviashvili equations with negative dispersion vary under the action of a random force similar to the solitons of the Korteweg-de Vries equation. Several of these solitons interact via “virtual solitons” and generate new solitons provided that resonance conditions are satisfied whose dimensions increase as a result of the influence of the random force.

Structural instability of a soliton for the Davey-Stewartson II equation

An algebraic soliton of the Davey-Stewartson II equation looses structural stability under a small perturbation of the initial data.

Resonant Interactions of Y-Periodic Soliton with Line Soliton and Algebraic Soliton: Solutions to the Davey-Stewartson I Equation

The exact solutions to the Davey-Stewartson I equation are analyzed to study the nature of the interactions between y -periodic soliton and line soliton and between y -periodic soliton and algebraic soliton. The interactions are classified into several types according to the phase shifts due to collisions. There are two types of singular interactions: one is the resonant interaction where two solitons interact so as to make one soliton and the other is the extremely long-range interaction where two solitons interchange each other infinitely apart. Detail behaviors of interactions are illustrated graphically.

Stationary waves in a third-order nonlinear Schrödinger equation

Frequency modulated nonlocalized stationary wave solutions in a general type equation of the third-order approximation of the nonlinear dispersion wave theory (third-order nonlinear Schrödinger equation) are studied. The case of a polynomial type dependence of additional frequency on wave amplitude Ω ∼ A is considered. Sech-like and algebraic soliton solutions are obtained.

On the fission of algebraic solitons

The fission of an algebraic internal soliton climbing onto a shelf in a two–fluid system of infinite depth is investigated. It is shown that, by using conservation laws of the Benjamin—Ono equation with variable coefficients, we can predict the number of solitons emerging from an incident solitary wave and their amplitudes. These predictions are also verified with numerical solutions.

Structural transformation of eigenvalues for a perturbed algebraic soliton potential

We present a modified perturbation technique for the AKNS spectral system to evaluate eigenvalues supported by a perturbed algebraic soliton potential. The results of this technique are applied to the problem of the structural instability of algebraic solitons in the modified Korteweg-de Vries equation. It is shown analytically and numerically that the algebraic soliton is destroyed under the action of small initial perturbations and transforms either to a steady-state soliton with exponentially decaying tails or to a pulsating “breather”-type wave packet.

Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves.

The stochastic Benjamin-Ono equation is introduced, which models the propagation of nonlinear random waves in a two-layer fluid system with and without uneven bottom topography. In the case of the flat bottom, the effect of the external random flow field on the evolution of both soliton and periodic wave is investigated. In particular, the mean value and the correlation function of these nonlinear wave fields are calculated exactly under the assumption that the flow obeys the Gaussian stochastic process with a white noise. It is found that in the limit of large time, the mean value of an algebraic soliton approaches a Gaussian wave packet whereas that of a periodic wave is represented by Jacobi's theta function. In the case of the uneven bottom, a perturbation analysis is performed to evaluate the mean value of an algebraic soliton under the influence of random change of bottom topography. The large time asymptotic of the soliton is shown to exhibit a Gaussian wave packet with a small amount of the phase shift caused by the interaction between the soliton and the random bottom topography.

Optical solitons with power-law asymptotics.

It is shown that self-guided optical beams with power-law asymptotics, i.e., algebraic optical solitons, can be regarded as a special case of sech-type solitons (i.e. solitons with exponentially decaying asymptotics) in the limit where the beam propagation constant coincides with the threshold for linear wave propagation. This leads to the conjecture that algebraic optical solitons should be inherently unstable due to interactions with linear waves, even in cases when the corresponding family of sech-type solitons is stable. This conjecture is verified numerically for a wide class of optical solitons described by the generalized nonlinear Schr\"odinger equation with two competing nonlinearities. \textcopyright{} 1996 The American Physical Society.

Algebraic geometry and soliton dynamics

Algebro-geometric sectors of solutions of the KP hierarchy are described in terms of τ-functions and vertex operators. Some useful identities involving theta functions and prime forms on Riemann surfaces are provided which are applied to obtain explicit solutions in the bilinear formalism. By using a dressing method for τ-functions the soliton dynamics against the background of quasiperiodic solutions is characterized. Furthermore, a formula for the soliton shifts in terms of prime forms on Riemann surfaces is obtained.

Forced Benjamin-Ono equation and its application to soliton dynamics.

The propagation of long interfacial waves of finite amplitude is investigated in a two-layer fluid system where a layer of a light fluid overlies a layer of heavier fluid resting on an uneven bottom. The upstream condition is imposed such that the constant flow is in the same direction in both layers. Under the appropriate balance among nonlinearity, disperison, and topographic effect, the long-term evolution of the interfacial elevation is shown to be governed by a forced Benjamin-Ono (fBO) equation. A direct soliton perturbation theory is applied to the fBO equation to study the interaction of an algebraic soliton with a bottom topography when the Froude number is nearly equal to unity. A system of ordinary differential equations describing the change of the amplitude and position of the soliton is derived for a simple bottom profile. The solutions of these equations exhibit a variety of phenomena, such as the capture and repulsion of the soliton by topography and the occurrence of solitonlike phase shifts due to the interaction of the soliton with topography. We also examine the effect of small dissipation on the dynamics of the soliton. Special emphasis is given to the appearance of a branch of stationary states of the soliton which has not previously been observed in the absence of dissipation. (c) 1995 The American Physical Society

DYNAMICS OF INTERACTING ALGEBRAIC SOLITONS

A survey is made which highlights recent topics on the dynamics of algebraic solitons, which are exact solutions to a certain class of nonlinear integrodifferential evolution equations. The model equations that we consider here are the Benjamin-Ono (BO) and its higher-order equations together with the BO-Burgers equation, a model equation for deep-water waves, the sine-Hilbert (sH) equation and a damped sH equation. While these equations have their origin either in physics or in mathematics, each equation exhibits a novel type of algebraic soliton solution and hence its characteristic is worth studying in its own right. After deriving these equations, we are concerned with each equation separately. We first present explicit N-soliton solutions and then summarize related mathematical properties of the equation. Subsequently, a detailed description is given to the interaction process of two algebraic solitons using the pole expansion of the solution. Particular attention is paid to investigating the effects of small perturbations on the overtaking collision of two BO solitons by employing a direct multisoliton perturbation theory. It is shown that the dynamics of interacting algebraic solitons reveal new aspects which have never been observed in the interaction process of usual solitons expressed in terms of exponential functions.

Multisoliton perturbation theory for the Benjamin-Ono equation and its application to real physical systems.

A direct perturbation theory is developed to study the effects of small perturbations on the interaction process of algebraic solitons of the Benjamin-Ono (BO) equation. Using the method of multiple scales, the modulation equations for the amplitude and the phase of each soliton are derived in the lowest approximation. As practical applications of the theory, the interaction of two solitons is investigated for the two different types of perturbations that appear in real physical systems. One is a dissipative perturbation (BO--Burgers equation) and the other is a dispersive perturbation (higher-order BO equation). In both cases, the changes of the soliton parameters due to small perturbation are calculated by numerical integrations and their characteristics are elucidated in detail. Among them, the phase shift caused by the dispersive perturbation is a remarkable feature that has never been observed in the collision process of algebraic solitons.

New type of gap soliton in a coupled Korteweg-de Vries wave system.

We show that, in a narrow gap in the spectrum of two linearly coupled Korteweg--de Vries equations with opposite signs of the dispersion coefficient, a two-parameter family of solitons of a novel type may exist. These are envelope solitons with decaying oscillating tails, which are radically different from the gap solitons previously known in nonlinear optics. In particular, they may become singular at some value of the velocity, and degenerate into algebraic solitons in another special case. It is demonstrated that gap solitons of the same type may also exist in a nonlinear optical system consisting of focusing and defocusing tunnel-coupled planar lightguides.

Resonant Interactions betweenY-Periodic Soliton and Algebraic Soliton: Solutions to the Kadomtsev-Petviashvili Equation with Positive Dispersion

There are two types of singular interactions between algebraic soliton and y -periodic soliton (i.e. the array of the localized structures in the y -direction, which propagates in the x -direction): one is the resonant interaction where the algebraic soliton joins in the line of y -periodic soliton after collision, while the other is the extremely repulsive interaction where the algebraic soliton affects the motion of the y -periodic soliton infinitely apart. Both interactions are associated with the boundary between attractive interaction and repulsive one in the parameter space.

K-P Burgers Equation for the Decayof Solitary Magnetosonic Waves Propagating Obliquelyin a Warm Collisional Plasma

A nonlinear evolution equation governing the two dimensional propagation of fast and slow magnetosonic modes in a warm collisional plasma has been derived. This equation is a combination of Kadomtsev-Petviashvili (K-P) equation and the Burgers equation. The two dimensional K-P equation has two types of solitary wave solutions depending on the sign of the coefficients. One type is the usual planar type, the other is the lump solution. Both types of solitary wave solutions decay with time in the weak collisional limit. The two dimensional features of the amplitude and width of the lump or algebraic soliton have been discussed and the decay rates computed numerically. The decay rates depend on plasma beta and on the angle of propagation and the rates are different for the fast and the slow waves. At a certain angle of propagation the decay rates of both the modes are equal in the case of low beta plasma. Because of 2-D effects, the slow mode, although it has a lower collisional decay rate than the fast mode,...

Two-Dimensional Multisoliton Solutions: Periodic Soliton Solutions to the Kadomtsev-Petviashvili Equation with Positive Dispersion

N -periodic soliton solutions that consist of N sequences of the algebraic localized solitons in arbitrary directions are presented to the Kadomtsev-Petviashvili equation with positive dispersion using the bilinear transformation method. The existence conditions and some properties of them are also given.

New Type of Algebraic Solitons Expressed in Terms of Pfaffians

The algebraic N -soliton solution of a model equation for deep water waves presented by the author (J. Math. Phys. 29 (1988) 49) is shown to be expressed in terms of Pfaffians. It exhibits a new type of solution which has quite different characteristics in comparison with those of the well-known algebraic N -soliton solution of the Benjamin-Ono equation. It is also verified that the N -soliton solution satisfies certain quadratic identity for Pfaffians and this provides a direct proof of the solution without recource to mathematical induction.