Positon Solutions Research Articles

Positon and negaton solutions of the mKdV equation with self-consistent sources

By using the generalized binary Darboux transformation with arbitrary functions at time t for the negative modified KdV equation with self-consistent sources (mKdV−ESCSs) which offers a non-auto-Bäcklund transformation between two mKdV−ESCSs with different degrees of sources, some new solutions for the mKdV−ESCSs such as singular multisoliton, multipositon, multinegaton, multisoliton–positon, multisoliton–negaton and multipositon–negaton solutions are found by taking the special initial solution for auxiliary linear problems and the special functions of t-time. At the same time, the properties of these solutions are analyzed in detail.

Restricted Flows and the Soliton Equation with Self-Consistent Sources

The KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Backlund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions.

Nonsingular positon and complexiton solutions for the coupled KdV system

Taking the coupled KdV system as a simple example, analytical and nonsingular complexiton solutions are firstly discovered in this Letter for integrable systems. Additionally, the analytical and nonsingular positon–negaton interaction solutions are also firstly found for S-integrable model. The new analytical positon, negaton and complexiton solutions of the coupled KdV system are given out both analytically and graphically by means of the iterative Darboux transformations.

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Abstract Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.

The solutions of the NLS equations with self-consistent sources

We construct the generalized Darboux transformation with arbitrary functions in time $t$ for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-B\"{a}cklund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is proposed. By reduction the generalized Darboux transformation with arbitrary functions in time $t$ for the Nonlinear Schr\"{o}dinger equation with self-consistent sources (NLSESCS) is obtained and enables us to find the dark soliton, bright soliton and positon solutions for NLS$^{+}$ESCS and NLS$^{-}$ESCS. The properties of these solution are analyzed.

Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation

A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.

Soliton, Positon and Negaton Solutions to a Schrödinger Self-consistent Source Equation

A Darboux transformation is furnished for a Schrodinger self-consistent source equation, along with a solution formula. Soliton, positon and negaton solutions to the Schrodinger self-consistent source equation are further constructed. The resulting solutions resemble those of the KdV equation with self-consistent sources, but the space and time variables are involved in a slightly different manner.

Negaton and positon solutions of the soliton equation with self-consistent sources

The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for $N$-times repeated GBDT are presented. This GBDT provides non-auto-B\"{a}cklund transformation between two KdV equations with different degrees of sources and enable us to construct more general solutions with $N$ arbitrary $t$-dependent functions. By taking the special $t$-function, we obtain multisoliton, multipositon, multinegaton, multisoliton-positon, multinegaton-positon and multisoliton-negaton solutions of KdVES. Some properties of these solutions are discussed.

Positon Solutions of the KdV Equation with Self-Consistent Sources

We construct a binary Darboux transformation with an arbitrary time function for the KdV equation with self-consistent sources. With this transformation, we obtain positon solutions of the KdV equation with self-consistent sources. We also discuss the properties of these solutions.

Darboux transformation for classical acoustic spectral problem

We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows steps similar to those for the Schrödinger operator. However, there is no one‐to‐one correspondence between the two problems. The technique developed enables one to construct new families of integrable potentials for the acoustic problem, in addition to those already known. The acoustic problem produces a nonlinear Harry Dym PDE. Using the technique, we reproduce a pair of simple soliton solutions of this equation. These solutions are further used to construct a new positon solution for this PDE. Furthermore, using the dressing‐chain approach, we build a modified Harry Dym equation together with its LA pair. As an application, we construct some singular and nonsingular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.

Positons: Slowly Decreasing Analogues of Solitons

We present an introduction to positon theory, almost never covered in the Russian scientific literature. Positons are long-range analogues of solitons and are slowly decreasing, oscillating solutions of nonlinear integrable equations of the KdV type. Positon and soliton–positon solutions of the KdV equation, first constructed and analyzed about a decade ago, were then constructed for several other models: for the mKdV equation, the Toda chain, the NS equation, as well as the sinh-Gordon equation and its lattice analogue. Under a proper choice of the scattering data, the one-positon and multipositon potentials have a remarkable property: the corresponding reflection coefficient is zero, but the transmission coefficient is unity (as is known, the latter does not hold for the standard short-range reflectionless potentials).

On Darboux-Bäcklund Transformation and Positon Type Solution of Coupled Nonlinear Optical Waves

A Darboux-Backlund transformation is used to obtain a positon type solution of the nonlinear equations describing the propagation of coupled nonlinear optical pulses.This form of the positon solution is then compared with that obtained by the special limiting procedure applied to a two-soliton solution. It is observed that though the algebraic form of the two solutions is different yet both of these have singularities and the position of the singularities remains on the similar curve in the (x, t) plane. We also depict the form of these solutions graphically. Finally, it may be added that the method of Darboux-Backlund transformation is convenient for generating more than one-positon solution. PACS numbers: 02.30.+g, 52.35.Sb, 52.35.Mw

`Positon' and `Dromion' Solutions of the (2+1) Dimensional Long Wave-Short Wave Resonance Interaction Equations

`Positon' and `dromion' solutions are derived for the long wave-short wave interaction equations in a two-layer fluid. Positons are new, exact solutions of nonlinear evolution equations that exhibit algebraic decay in the far field. A positon solution can be generated by taking a special limit of multi-soliton expansion. Variation of the limiting process yields different solutions. Dromions are exact, localized solutions of (2+1) dimensional (2 spatial, 1 temporal) nonlinear evolution equations that decay exponentially in all directions. One and higher dromion solutions are investigated, and a particular case of higher dromion solutions is considered in details. By applying another limiting procedure a new solution is generated. This method of `coalescence of eigenvalues' or `wavenumbers' is thus quite universal, and can be applied to a wide variety of expansions, not just the multi-soliton type. Finally, the case of propagation solutions on a continuous wave background is also studied.

Inverse scattering transformation for positons

The scattering problem for the one-dimensional Schrodinger operator with potential equal to the positon solution of the Korteweg-de Vries (KdV) equation is investigated. It is shown that the transition coefficient is equal to zero and different positon potentials can have the same reflection coefficient, i.e. the inverse scattering problem cannot be solved uniquely. It is observed that the reflection coefficient calculated for the positon solutions does not change with time in accordance with the inverse scattering method for KdV.

Do nonsingular globally bounded positon solutions exist?

The positon solutions discovered so far for several nonlinear evolution equations are singular solutions. We show that for a discrete version of the well-known sinh-Gordon equation nonsingular positon solutions do exist. Under appropriate restrictions on the parameters of the construction they are globally bounded. In the continuum limit the corresponding (singular) solutions of the sinh-Gordon equation are recovered. Bibliography: 11 titles.

Special solutions from the variable separation approach: the Davey - Stewartson equation

A variable separation procedure for the Davey - Stewartson (DS) equation is proposed by using a prior ansatz to its bilinear form. The reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables. The trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a Boussinesq-type equation system. Whenever a pair of solutions of the reduced fields are obtained, a corresponding solution of the DS equation can be obtained algebraically. The single dromion solution and some kinds of positon solutions are obtained explicitly.

Negaton and positon solutions of the KdV and mKdV hierarchy

We give a systematic classification and a detailed discussion of the structure, motion and scattering of the recently discovered negaton and positon solutions of the Korteweg - de Vries hierarchy. There are two distinct types of negaton solutions which we label and , where (n + 1) is the order of the Wronskian used in the derivation. For negatons, the number of singularities and zeros is finite and they show very interesting time dependence. The general motion is in the positive x direction, except for certain negatons which exhibit one oscillation around the origin. In contrast, there is just one type of positon solution, which we label . For positons, one gets a finite number of singularities for n odd, but an infinite number for even values of n. The general motion of positons is in the negative x direction with periodic oscillations. Negatons and positons retain their identities in a scattering process and their phase shifts are discussed. We obtain a simple explanation of all phase shifts by generalizing the notions of `mass' and `centre of mass' to singular solutions. Finally, it is shown that negaton and positon solutions of the KdV hierarchy can be used to obtain corresponding new solutions of the modified KdV hierarchy.

Positon and positon-like solutions of the Korteweg-de Vries and Sine-Gordon equations

It is shown that the positon solution, reported recently for the Korteweg-de Vries and other completely integrable equations, can be regarded as a limiting case of the well-known 2-soliton formula. The existence of integrals of motion related to singular positon solutions is also discussed.

Dynamic properties of positons

The dynamic properties of positon solutions of integrable nonlinear evolution equations are studied. The intricate relation between the dynamics of the pole of a positon and the motion of the positon itself is demonstrated for the Korteweg-de Vries equation. The new features of positons of higher order are shown. The soliton-positon interaction is sketched, a comparison with some well known singular solutions of the Korteweg-de Vries equation is given and the connection between these singular solutions is shown. The extension of the analysis to positons of other nonlinear evolution equations is indicated.

Supertransparent potentials for the Dirac equation

An extension of the classical Darboux transformations is applied to the one-dimensional Dirac equation in order to construct von Neumann-Wigner potentials allowing embedded eigenvalues. These potentials lead to a novel type of scattering problem with a trivial S-matrix composed of vanishing reflection coefficients and a trivial transmission coefficient. Related topics like the underlying symmetry of the Dirac equation and the connection with positon solutions of nonlinear evolution equations are discussed.