General Soliton Research Articles

Accessible spatiotemporal parabolic-cylinder solitons

We study analytically and numerically ‘accessible’ spatiotemporal solitons in a three-dimensional strongly nonlocal nonlinear medium. A general localized soliton solution of the ‘acceptable’ type is obtained in the Cartesian coordinates, using even and odd parabolic-cylinder functions. Characteristics of these accessible spatiotemporal solitons are discussed. The validity of the analytical solutions and their stability is verified by means of direct numerical simulations.

Resonant solitons from the 3 × 3 operator

The features of the resonant soliton of the 3×3 operator is studied and detailed. The scattering data of this operator contains four transmission coefficients, two in each half complex ζ-plane, where ζ is the spectral parameter. When the potential matrix has anti-hermitian symmetry (Qα,β=−Qβ,α*), each of the two transmission coefficients in the lower half plane become equal to the complex conjugates of one of the two in the upper half plane, leaving only two independent transmission coefficients, which is taken to be those in the upper half plane. The bound state scattering data for this operator consists of the zeros of these two transmission coefficients (bound state eigenvalues) and a normalization coefficient associated with each eigenvalue. With two transmission coefficients, there is a wider variety of possible soliton solutions than in the Zakharov–Shabat (ZS) 2×2 case. First, the eigenvalues could belong to only one transmission coefficient, with the other transmission coefficient having none. In this case the soliton solution will only exist in one of the two conjugate pairs of components of Q lying next to the diagonal, with all other components of Q vanishing. These soliton solutions will be equivalent to those in the ZS case, up to scaling factors. If one would distribute the eigenvalues among both transmission coefficients so that each transmission coefficient would have at least one eigenvalue, the soliton structure becomes more complex. In this case, any one eigenvalue will be found to generally make contributions to all non-diagonal components of Q. Of particular interest, inside this class is a special class of solitons which exist only when these two transmission coefficients have exactly equal eigenvalues. Equality of eigenvalues from different transmission coefficients give rise to “resonant solitons”. This latter class is the case which is treated here. It is found that there are two different states for the resonant soliton solution, both of which arise from a bifurcation. The bifurcation is shown to arise from the algebraic structure of the 3×3 scattering matrix. The linear dispersion relations (the solution of the inverse scattering problem) are also extended to the case when resonant solitons are present. It is further shown that the same result could be obtained by taking the limit of the eigenvalues in each transmission coefficient approaching the other. The general soliton solution under anti-hermitian symmetry is studied when there is only one eigenvalue in each transmission coefficient, with each unequal to the other. The asymptotics of this solution is detailed and then the same is done when the eigenvalues are exactly equal, showing that the latter contains the well known parametric interactions of “up-conversion” and “down-conversion”. Lastly, it is pointed out how this equality of eigenvalues in different transmission coefficients can be interpreted as a “nonlinear resonance condition”.

New exact solutions to the generalized KdV equation with generalized evolution

In this paper, by using a transformation and an application of Fan subequation, we study a class of generalized Korteweg–de Vries (KdV) equation with generalized evolution. As a result, more types of exact solutions to the generalized KdV equation with generalized evolution are obtained, which include more general single-hump solitons, multihump solitons, kink solutions and Jacobian elliptic function solutions with double periods.

The dynamics of nonstationary solutions in one-dimensional two-component Bose—Einstein condensates

This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose—Einstein condensates. It gives three kinds of stationary solutions to this model and develops a general method of constructing nonstationary solutions. It obtains the unique features about general evolution and soliton evolution of nonstationary solutions in this model.

KP line solitons and Tamari lattices

The Kadomtsev–Petviashvili (KP) II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at ‘critical’ events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow us to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).

Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations

We used what we called extended Fan's sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (DS) equations. The basic idea of the proposed extended Fan's subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.

The improved Fan sub-equation method and its application to the SK equation

In this paper, many types of exact solutions of a first-order nonlinear ordinary differential equation called Fan sub-equation, is further investigated by using bifurcation method of dynamical systems. As a result, more types of exact solutions to Sawada–Kotera (SK) equation are obtained, which include more general soliton solutions, kink solutions, triangular function solutions, Jacobian elliptic function solutions with double periods and so on.

Soliton Solutions of the Complex Ginzburg-Landau Equation

The function transformed method is applied to an n-dimensional complex Ginzburg-Landau equation (CGLE), which is being transformed to physically signif- icant well known nonlinear waves equations depending only on the function �, and these equations can be exactly solved. The solution of these equations inis shown to lead to a general soliton solution of the CGLE.

Application of singular perturbation method in analyzing traffic density waves

Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Korteweg-de-Vries (KdV) equation and the modified Korteweg-de-Vries (mKdV) equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently.

Exact discrete Exp-function solutions of the relativistic Toda lattice system

By means of the modified Exp-function method, the relativistic Toda lattice system is researched. And some new discrete Exp-function solutions, including discrete kink-type soliton solutions and general soliton solutions are obtained via the method and symbolic computation system Maple.

The improved Fan sub-equation method and its application to the Boussinseq wave equation

The improved Fan sub-equation method is used to construct the exact travelling wave solutions to the Boussinseq wave equation in mathematical physics. The key idea of this method is to take full advantage of the Fan sub-equation method and bifurcation method of dynamical systems which can help us obtain the phase portraits and dynamical behavior of the general sub-equations and find more new exact solutions by solving the sub-equations. As a result, more exact travelling wave solutions to the Boussinesq wave equation are obtained which include more general soliton solutions, kink solutions, solitary pattern solutions, triangular solutions, singular periodic solutions and double periodic solutions.

Obtaining general soliton solutions of nonlinear Boussinesq equations by trigonometric function method

A suitable transformation (trigonometric function method) is found to change nonlinear Boussinesq differential equations into nonlinear algebra equations, which are solved by Wu elimination method and therewith the general soliton solutions of Boussinesq differential equations are obtained.

The extended Fan sub-equation method and its applications for a class of nonlinear evolution equations

The extended Fan sub-equation method with a computerized symbolic is used to construct the exact travelling wave solutions for a class of nonlinear evolution equations in mathematical physics. The key idea of this method is to take full advantage of general elliptic equations involving five parameters which have more new solutions and whose degeneracies can lead to special sub equation involving three parameters like Riccati equation, first kind elliptic equation, auxiliary ordinary equation, and generalized Riccati equation. As a result, many exact travelling wave solutions are obtained which include a more general soliton solutions, triangular solutions and double periodic solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

General soliton solutions for nonlinear dispersive waves in convective type instabilities

The two-dimensional Ginzburg–Landau equation (GLE) is obtained from basic equations by a linear stability analysis. This equation governs the evolution of slowly varying envelopes of periodic spatio-temporal patterns related to Rayleigh–Bénard convective instabilities. In addition, the phase instabilities of the complex GLE (CGLE) with quintic and space-dependent cubic terms modelling the Eckhaus and zigzag convective instabilities are reported. We find soliton solution classes to the elliptic and hyperbolic CGLE, by applying the function transformation method. The two-dimensional CGLE is transformed to a sine-Gordon equation, a sinh-Gordon equation and other equations, which depend only on one function χ. The general solution of the equation in χ leads to a general soliton solution of the two-dimensional CGLE. The obtained solutions contain some interesting specific solutions such as plane solitons, N multiple solitons and propagating breathers. We also discuss the soliton stability of the CGLE.

Kelvin-Helmholtz instability in magnetohydrodynamic flows

The Rayleigh-Taylor instability (RTI) of a continuously stratified fluid has implica- tions on the stability of solar and planetary interiors. A nonlinear stage of the two-dimensional RTI is studied by including various effects. By using the multiple scale method, we derived a non- linear Schrodinger equation (NLSE) in 2+1 dimensions. We show the general soliton solutions of the NLSE and this allows to discuss their stability.

Soliton collisions with shape change by intensity redistribution in mixed coupled nonlinear Schrödinger equations

A different kind of shape changing (intensity redistribution) collision with potential application to signal amplification is identified in the integrable -coupled nonlinear Schrödinger (CNLS) equations with mixed signs of focusing- and defocusing-type nonlinearity coefficients. The corresponding soliton solutions for the N=2 case are obtained by using Hirota's bilinearization method. The distinguishing feature of the mixed sign CNLS equations is that the soliton solutions can both be singular and regular. Although the general soliton solution admits singularities we present parametric conditions for which nonsingular soliton propagation can occur. The multisoliton solutions and a generalization of the results to the multicomponent case with arbitrary N are also presented. An appealing feature of soliton collision in the present case is that all the components of a soliton can simultaneously enhance their amplitudes, which can lead to a different kind of amplification process without induced noise.

Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space

In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.

Gap solitons in waveguide arrays.

Bright and dark spatial gap solitons are demonstrated in waveguide arrays. These gap solitons travel across the array at zero transverse velocity, in complete analogy with stationary (immobile) temporal gap solitons. Furthermore, the launching configuration for observing these stationary gap solitons is shown to be the analog of an "ideal experiment" for observing stationary temporal gap solitons, never observed so far. A clear distinction is established between the family of Floquet-Bloch solitons in general and discrete solitons in particular, and the limiting case of gap solitons.

General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations

We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann–Hilbert problem. We have applied these general results to the three-wave interaction system, and derived new classes of higher-order soliton and two-soliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave (u3) into the other two components (u1 and u2) and merger of u1 and u2 waves into the pumping u3 wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumping u3 wave into the u1 and u2 components, and the reverse process. In the nongeneric cases, these two-soliton solutions could describe the elastic interaction of the u1 and u2 waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Éksp. Teor. Fiz. 69, 1654 (1975)] and Kaup [Stud. Appl. Math. 55, 9 (1976)].

Exact soliton solution and inelastic two-soliton collision in a spin chain driven by a time-dependent magnetic field.

We investigate dynamics of exact N-soliton trains in a spin chain driven by a time-dependent magnetic field by means of an inverse scattering transformation. The one-soliton solution indicates obviously the spin precession around the magnetic field and periodic shape variation induced by the time-varying field as well. In terms of the general soliton solutions, N-soliton interaction and particularly various two-soliton collisions are analyzed. The inelastic collision by which we mean the soliton shape change before and after collision appears is generally due to the time-varying field. We, moreover, show that complete inelastic collisions can be achieved by adjusting spectrum and field parameters. This may lead to a potential technique of shape control of soliton.