Exact kink solitons in the presence of diffusion, dispersion, and polynomial nonlinearity (original) (raw)

Special polynomials and soliton dynamics

Special polynomials play a role in several aspects of soliton dynamics. These are differential polynomials in u, the solution of a nonlinear evolution equation, which vanish identically when u represents a single soliton. Local special polynomials contain only powers of u and its spatial derivatives. Non-local special polynomials contain, in addition, non-local entities (e.g., \delta x-1u). When u is a multiple-solitons solution, local special polynomials are localized in the vicinity of the soliton-collision region and fall off exponentially in all directions away from this region. Non-local ones are localized along soliton trajectories. Examples are presented of how, with the aid of local special polynomials, one can modify equations that have only a single-soliton solution into ones, which have that solution as well as, at least, a two-solitons solutions. Given an integrable equation, with the aid of local special polynomials, it is possible to find all evolution equations in hig...

EJTP 3, No. 10 (2006) 39–88 Electronic Journal of Theoretical Physics Nonlinear Field Equations and Solitons as Particles

2006

Abstract: Profound advances have recently interested nonlinear field theories and their exact or approximate solutions. We review the last results and point out some important unresolved questions. It is well known that quantum field theories are based upon Fourier series and the identification of plane waves with free particles. On the contrary, nonlinear field theories admit the existence of coherent solutions (dromions, solitons and so on). Moreover, one can construct lower dimensional chaotic patterns, periodic-chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution. We discuss in some detail a nonlinear Dirac field and a spontaneous symmetry breaking model that are reduced by means of the asymptotic perturbation method to a system of nonlinear evolution equations integrable via an appropriate change of variables. Their coherent, chaotic and fractal solutions ...

Solitons in the chiral equation

Communications in Mathematical Physics, 1990

The purpose of this paper is to give a geometric description of the solitons in the principal chiral equation in 1 + 1 dimensions in terms of Grassmannians, and a qualitative description of their behaviour in terms of Morse functions. Additionally it shows how a soliton can be "added" to an arbitrary solution of the chiral equation.

Envelope solitons for generalized forms of the phi-four equation

Journal of King Saud University - Science, 2013

We consider two variants of the generalized phi-four equation with arbitrary constant coefficients and general values of the exponents in the dissipation and nonlinear terms. By using solitary wave ansatze in terms of sech p (x) and tanh p (x) functions respectively, we find the non-topological (bright) as well as topological (dark) soliton solutions for the considered models. The physical parameters in the soliton solutions are obtained as a function of the dependent model coefficients. The conditions of existence of solitons are presented. Further, we show that the obtained soliton solutions depend on the exponent of the wave function u(x, t), positive or negative, and on all the dependent model coefficients as well.

Nonlinear Field Equations and Solitons as Particles

Electronic Journal of Theoretical Physics

Profound advances have recently interested nonlinear field theories and their exact or approximate solutions. We review the last results and point out some important unresolved questions. It is well known that quantum field theories are based upon Fourier series and the identification of plane waves with free particles. On the contrary, nonlinear field theories admit the existence of coherent solutions (dromions, solitons and so on). Moreover, one can construct lower dimensional chaotic patterns, periodic-chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution. We discuss in some detail a nonlinear Dirac field and a spontaneous symmetry breaking model that are reduced by means of the asymptotic perturbation method to a system of nonlinear evolution equations integrable via an appropriate change of variables. Their coherent, chaotic and fractal solutions are examined in some detail. Finally, we consider the possible identification of some types of coherent solutions with extended particles along the de Broglie-Bohm theory. However, the last findings suggest an inadequacy of the particle concept that appears only as a particular case of nonlinear field theories excitations.

Construction of soliton equations using special polynomials

Communications in Nonlinear Science and Numerical Simulation, 2013

A simple, algorithmic approach is proposed for the construction of the most general family of equations of a given scaling weight, possessing, at least, the same single-soliton solution as a given, lower scaling weight equation. The construction exploits special polynomialsdifferential polynomials in the solution, u, of an evolution equation, which vanish identically when u is a single-soliton solution. Applying the approach to different types of evolution equations yields new results concerning the most general families of evolution equations in a given scaling weight, which possess solitary wave solutions. The same method can be applied in the identification of families of evolution equations of mixed scaling weight (and, in general, of any structure), which admit single-soliton solutions of a desired form.

Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations

Symmetry, 2022

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are...

Soliton and vortex type solutions in non-linear chiral theories

Nuclear Physics B, 1976

Both static and time-dependent exact solutions of the classical field equations are presented for a chiral SU(2) X SU(2) non-linear pion Lagrangian. Of particular interest are the finite energy vortex-type solutions and the appearance of a topologically conserved charge. In two space-time dimensions, there are also genuine soliton solutions. * There is now a rapidly growing literature on the subject. For a review see ref. [ 11.