Nonlinear Superposition Formula Research Articles

Bäcklund transformation, nonlinear superposition formulae and infinite conserved laws of Benjamin equation

Backlund transformation, exact solitary wave solutions, nonlinear supperposition formulae and infinite conserved laws are presented by using TU-pattern. The algorithm involves wide applications for nonlinear evolution equations.

New integrable differential-difference systems: Lax pairs, bilinearforms and soliton solutions

Two new integrable differential-difference systems with their Lax pairs areproposed. By the dependent variable transformations, these integrable lattices can be transformed into bilinear equations.With the assistance of Mathematica, three-soliton solutions are explicitly obtained. We have also shown that these lattices can be obtained from a special case of the coupled bilinear equationsunder reduction. Furthermore a bilinear Bäcklundtransformation and the corresponding nonlinearsuperposition formula concerning the coupled bilinear equations are presented. Besides, it is also illustrated that the y-flow of these coupled bilinear equationscan be transformed into a latticepreviously derived by the authors.Starting from the corresponding bilinear Bäcklundtransformation, its corresponding Lax pair is obtained.

Application of Hirota's bilinear formalism to a two-dimensional lattice by Leznov

We consider the Leznov lattice in this paper. By a dependent variable transformation, the Leznov lattice is transformed into a quadri-linear form. This form is further transformed into a bilinear form by the introduction of an auxiliary variable. We present a Bäcklund transformation and a nonlinear superposition formula for the Leznov lattice. As an application of the results, soliton solutions and lump solutions are derived. Besides, starting from the bilinear BT, a Lax pair for the Leznov lattice is obtained.

Nonlinear superposition formula for the Kaup–Kupershmidt partial differential equation

The fifth order Kaup–Kupershmidt (KK) equation is one of the solitonic equations related to the integrable cases of the Hénon–Heiles system. As opposed to the Sawada–Kotera equation which is its dual equation, the construction of the N-soliton solutions of KK is not an easy task in using a perturbation scheme or the Hirota bilinear formalism. From the Bäcklund transformation obtained by singularity analysis considerations, we here establish the permutability theorem for the KK equation. This allows us to explain the “anomalous” structure of the N-soliton solution, empirically obtained by Parker [Physica D 137 (2000) 25–33, 34–48].

Symmetries, exact solutions, and nonlinear superposition formulas for two integrable partial differential equations

We recently introduced two new sixth-order partial differential equations (PDEs) associated with third-order scattering problems. Here we extend our study of these PDEs by considering the construction of exact solutions both by using the method of symmetry reduction due to Lie, and by using their Darboux transformations (DTs). Amongst the ordinary differential equations (ODEs) obtained by symmetry reduction is an ODE due to Cosgrove that is believed to define a new Painlevé transcendent. This ODE provides soliton solutions for our integrable PDEs that include arbitrary functions of time. The DTs for our PDEs allow the recovery of these solutions and in addition provide other solutions which are not associated with Lie symmetries (either classical or nonclassical). We also consider the iteration of the corresponding Bäcklund transformations (BTs) for these PDEs. The theorem of permutability allows us to reduce this process of iterating the DT from one of solving a third-order linear equation (the spatial part of the Lax pair) to that of solving either a second-order linear equation (for one PDE), or quite remarkably to that of solving a first-order linear equation (for the other PDE). These linear differential equations have coefficients involving three previous solutions of the PDE, and are a natural extension of the linear algebraic equation found by applying the theorem of permutability to the Korteweg–de Vries equation.

A Differential-Difference Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

A differential-difference Caudrey-Dodd-Gibbon-Kotera-Sawada equation is proposed. The corresponding Backlund transformation is presented. A Lax pair for it is found. A nonlinear superposition formula is also derived based on the permutability of the Backlund transformation.

The Hirota-Satsuma Coupled KdV Equation and a Coupled Ito System Revisited

The well-known Hirota-Satsuma coupled KdV equation and a coupled Ito system are reviewed. A new type of soliton solutions to these two systems under constant boundary condition at infinity is found. The so-called generalized Hirota-Satsuma coupled KdV system is also considered. Starting from its bilinear forms, we obtain a Bäcklund transformation and the corresponding nonlinear superposition formulae. As a result, soliton solutions first obtained by Satsuma and Hirota can be rederived. Moreover, rational solutions are also given.

Bäcklund transformation and its superposition principle of a Blaszak–Marciniak four-field lattice

A four-field lattice furnished by Blaszak and Marciniak is transformed into bilinear form upon introducing two auxiliary independent variables. A Bäcklund transformation in bilinear form is found for the lattice and the corresponding nonlinear superposition formula is rigorously established. As a consequence, soliton solutions to the lattice are derived.

Nonlinear superposition formulas based on imprimitive group action

Systems of nonlinear ordinary differential equations are constructed for which the general solution is expressed algebraically in terms of a finite number of particular solutions. The equations and the corresponding nonlinear superposition formula are based on a nonlinear action of the Lie group SL(N,C) on a homogeneous space M. The isotropy group of the origin of this space is a nonmaximal parabolic subgroup of SL(N,C). Such equations can occur as Bäcklund transformations for soliton equations on flag manifolds.

Hirota bilinear approach to a new integrable differential-difference system

A new integrable differential-difference system is proposed. By the dependent variable transformation, the system is transformed into multilinear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Bäcklund transformation for it is obtained. Furthermore a nonlinear superposition formula is presented. As an application of the obtained results, soliton solutions to the system are derived.

Two Integrable Coupled Nonlinear Systems

Motivated by Hirota and Satsuma's results on their coupled KdV equation, two integrable coupled nonlinear systems are considered. One of them is a coupled Ito system. It is shown that the coupled Ito system is a special case of the (6,2)-reduction of the two component BKP hierarchy while the other coupled system can be obtained from the (5,1)-reduction of the two component BKP hierarchy. By using MATHEMATICA, we obtain the 3- and 4-soliton solutions of the coupled Ito system. In addition, starting from bilinear equations of the other coupled system, a Bäcklund transformation is found and nonlinear superposition formulae are established. Soliton solutions and rational solutions are also derived from these results.

A new integrable differential-difference system and its explicit solutions

A new integrable differential-difference system is proposed. By the dependent variable transformation, the system is transformed into multilinear form. By introducing an auxiliary variable, we further transform it into the bilinear form. Its corresponding Bäcklund transformation is obtained. Furthermore, nonlinear superposition formulae are presented. As an application of the obtained results, soliton solutions and rational solutions to the system are derived.

Some new results on the triple-humped soliton equation: Bäcklund transformation, superposition principle and new explicit solutions

A triple-humped soliton equation is considered. A corresponding Backlund transformation for it is obtained and a nonlinear superposition formula is presented. As an application of the results obtained, soliton solutions first found by Narita to the system are rederived. A sequence of rational solutions are found and other types of solutions which are a mix of exponentials and rational expressions are deduced.

Some new results on the Blaszak–Marciniak lattice: Bäcklund transformation and nonlinear superposition formula

The so-called Blaszak–Marciniak lattice is considered. By the dependent variable transformation, the Blaszak–Marciniak lattice is transformed into a quart-linear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Bäcklund transformation is obtained. Furthermore, a nonlinear superposition formula is proved rigorously. As an application of the obtained results, soliton solutions are derived.

Application of the Hirota bilinear formalism to a new integrable differential-difference equation

A new integrable differential-difference equation is proposed. By the dependent variable transformation, the equation is transformed into a multi-linear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Bäcklund transformation for it is obtained. Furthermore, a nonlinear superposition formula is presented. As an application of the obtained results, soliton solutions to the equation are derived.

Inversion of the Bloch equations withT2relaxation: An application of the dressing method

The Bloch equations, with time-varying driving field, and ${T}_{2}$ relaxation, are expressed as a scattering problem, with ${\ensuremath{\Gamma}}_{2}{=1/T}_{2}$ as the scattering parameter, or eigenvalue. When the rf pulse, describing the driving field, is real, this system is equivalent to the $2\ifmmode\times\else\texttimes\fi{}2$ Zakharov-Shabat eigenvalue problem. In general, for complex rf pulses, the system is a third-order scattering problem. These systems can be inverted, to provide the rf pulse needed to obtain a given magnetization response as a function of ${\ensuremath{\Gamma}}_{2}$. In particular, the class of ``soliton pulses'' are described, which have utility as ${T}_{2}$-selective pulses. For the third-order case, the dressing method is used to calculate these pulses. Constraints on the dressing data used in this method are derived, as a consequence of the structure of the Bloch equations. Nonlinear superposition formulas are obtained, which enable soliton pulses to be calculated efficiently. Examples of one-soliton and three-soliton pulses are given. A closed-form expression for the effect of ${T}_{1}$ relaxation for the one-soliton pulse is obtained. The pulses are tested numerically and experimentally, and found to work as predicted.

A Bäcklund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice

The so-called Belov-Chaltikian lattice is considered. By the dependent variable transformation, the Belov-Chaltikian lattice is transformed into a trilinear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Backlund transformation for it is obtained. Furthermore, a nonlinear superposition formula is proved rigorously. As an application of the obtained results, soliton solutions are derived.

A Bäcklund Transformation and Nonlinear Superposition Formula of the Caudrey-Dodd-Gibbon-Kotera-Sawada Hierarchy

In this paper, the Caudrey-Dodd-Gibbon-Kotera-Sawada hierarchy in bilinear form is considered. A Backlund transformation for the CDGKS hierarchy is presented. Under certain conditions, the corresponding nonlinear superposition formula is proved.

Bäcklund transformations and nonlinear superposition formulae of a differential-difference KdV equation

In this paper, a differential-difference KdV equation is considered. We present two Bäcklund transformations and corresponding nonlinear superposition formulae for it. As an application of the obtained results, N-soliton solutions first obtained by Ohta and Hirota are rederived. Furthermore, a sequence of rational solutions are also obtained. Thus the integrability of the differential-difference KdV equation is further confirmed.

Rational solutions of integrable equations via nonlinear superposition formulae

In this paper, a method is developed to obtain rational solutions of integrable equations which include KdV, Boussinesq, KP, Ito, (1 + 1)-dimensional CDGKS, Ramani and (2 + 1)-dimensional CDGKS equations. The key step that we use to find rational solutions of these equations is the use of nonlinear superposition formulae. All these rational solutions obtained for each equation are connected by a Bäcklund transformation, which enables us to find other new solutions through the nonlinear superposition of rational solutions and other known solutions.