Connections defining representations of zero curvature and the solitons of sine-Gordon and Korteweg-de Vries equations

It is claimed that solutions of travelling-wave type (and, in particular, soliton solutions) of partial differential equations can be created by using connections defining representations of zero curvature. In this paper, we construct solitons of the sine-Gordon and Korteweg-de Vries equations. By previous results of the author, the connections defining representations of zero curvature for a given differential equation generate Bäcklund transformations for this equation. It can be shown that the well-known Lax system (the so-called Lax pair) for the Korteweg-de Vries equation is a special case of a Bäcklund system (i.e., the system of partial differential equations defining a Bäcklund transformation). Note that the creation of solitons by means of the inverse scattering method is in fact a creation of solitons by means of the Lax system (without using connections defining the representations of zero curvature from the very beginning). Moreover, the inverse scattering method is essentially more labor-consuming than the method suggested in the present paper. Further, it is not required to involve any physical notions when using the suggested method. In the final section of the paper, we consider the so-called 2-soliton solutions of sine-Gordon and Korteweg — de Vries equations. Here we systematically use the invariant analytic method developed by G. F. Laptev, which is well-known in differential geometry under the title of Cartan-Laptev method.     

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parameterized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel–Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.     

LAX PAIR,BINARY DARBOUX TRANSFORMATION AND NEW GRAMMIAN SOLUTIONS OF NONISOSPECTRAL KADOMTSEV–PETVIASHVILI EQUATION WITH THE TWO-SINGULAR-MANIFOLD METHOD

In this letter, the two-singular-manifold method is applied to the (2+1)-dimensional nonisospectral Kadomtsev–Petviashvili equation with two Painlevé expansion branches to determine auto-Bäcklund transformation, Lax pairs and Darboux transformation. Based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the Nth iterated transformation formula in the form of Grammian is also presented. By using these Darboux transformations, we obtain some new grammian solutions.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

Ableitung einer kontinuierlichen Mannigfaltigkeit von Erhaltungssätzen der modifizierten Korteweg-de-Vries-Gleichung nach dem Noetherschen Satz

Derivation of a Continuous Set of Conservation Laws for the Modified Korteweg-de Vries Equation by Noether's Theorem A method developed recently to derive a continuous set of conservation laws from extended Bäcklund transformations by means of Noether's theorem is applied to the modified Korteweg-de Vries (m. KdV) equation that describes Alfvén waves in a plasma. The corresponding conserved currents are equivalent to those found by WADATI , SANUKI and KONNO . It is shown that the extended Bäcklund transformation B?_α for the m. KdV equation, which coincides with that for the sine-Gordon equation, by MIURA'S transformation becomes the extended Bäcklund transformation β_x for the Korteweg-de Vries equation where x = 1/2α.     

Connections defining representations of zero curvature and their Lax and Bäcklund mappings

Systems of differential forms are considered which define representations of zero curvature on specified fiber bundles through a set of structure equations. The curvature term which appears in the structure equations generates the partial differential equation. Bäcklund and Lax connections on associated bundles can also be established. This allows a unified treatment of Bäcklund transformations and Lax systems for the cases in which the differential equation appears as the curvature term in the structure equations.     

Bäcklund transformations for chiral field equations

We give a general procedure to obtain Bäcklund transformations for nonlinear partial differential equations derived as compatibility conditions between some given generalized Lax pair of operators. We apply this technique to obtain a set of Bäcklund transformations for three-dimensional and axially symmetric two-dimensional chiral field equations.     

Painlevé analysis of the Calogero-Degasperis-Fokas equation

It is found that the Calogero-Degasperis-Fokas equation has the Painlevé property, Bäcklund transformations as well as Miura transformations and the Lax pair for this equation are derived by the Weiss-Painlevé analysis.     

Noether's Theorem and the Conservation Laws of the KORTEWEG-DE VRIES Equation

A method developed recently by the author to derive a continuum of conservation laws by Noether's theorem from the so-called extended Bäcklund transformations is applied to the KORTEWEG -DE VRIES equation that describes various nonlinear dispersive wave phenomena in hydrodynamics, plasma physics and solid state physics. Further applications of Noether's theorem concerning this equation are given. It is shown that the Galilean transformation in the present case has an analogous function as Lie's transformation has with respect to the sine-Gordon equation.     

The gauge field copies through the Bäcklund transformation for the Liouville equation

It is shown that the gauge field copies for the field strength tensor associated with the Lax pair of the Liouville equation are generated through the non-auto Bäcklund transformations.     

Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries

The complete lists of vector hyperbolic equations on the sphere that have integrable third-order vector isotropic and anisotropic symmetries are presented. Several new integrable hyperbolic vector models are found. By their integrability, we mean the existence of vector Bäcklund transformations depending on a parameter. For all new equations, such transformations are constructed.     

Bäcklund transformations,nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the $n$th Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type $N$-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.     

A short proof of the Gaillard–Matveev theorem based on shape invariance arguments

We propose a simple alternative proof of the Wronskian representation formula obtained by Gaillard and Matveev for the trigonometric Darboux–Pöschl–Teller (TDPT) potentials. It rests on the use of singular Darboux–Bäcklund transformations applied to the free particle system combined to the shape invariance properties of the TDPT.     

On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations

Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric Bäcklund transformations and generalized conservation laws are constructed.     

On Decomposition of the ABS Lattice Equations and Related Bäcklund Transformations

The Adler-Bobenko-Suris (ABS) list contains scalar quadrilateral equations which are consistent around the cube, and have D₄ symmetry and tetrahedron property. Each equation in the ABS list admits a beautiful decomposition. We revisit these decomposition formulas and by means of them we construct Bäcklund transformations (BTs). BTs are used to construct lattice equations, their new solutions and weak Lax pairs.     

The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

Abstract

We present in this paper the singular manifold method (SMM) derived from Painlevé analysis, as a helpful tool to obtain much of the characteristic features of nonlinear partial differential equations. As is well known, it provides in an algorithmic way the Lax pair and the Bäcklund transformation for the PDE under scrutiny.

Moreover, the use of singular manifold equations under homographic invariance consideration leads us to point out the connection between the SMM and so–called nonclassical symmetries as well as those obtained from direct methods. It is illustrated here by means of some examples.

We introduce at the same time a new procedure that is able to determine the Darboux transformations. In this way, we obtain as a bonus the one and two soliton solutions at the same step of the iterative process to evaluate solutions.     

The integrability of an extended fifth-order KdV equation with Riccati-type pseudopotential

The extended fifth-order KdV equation in fluids is investigated in this paper. Based on the concept of pseudopotential, a direct and unifying Riccati-type pseudopotential approach is employed to achieve Lax pair and singularity manifold equation of this equation. Moreover, this equation is classified into three categories: extended Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, extended Lax equation and extended Kaup–Kuperschmidt (KK) equation. The corresponding singularity manifold equations and auto-Bäcklund transformations of these three equations are also obtained. Furthermore, the infinitely many conservation laws of the extended Lax equation are found using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas.     

Discrete bäcklund transformation

A discretized Bäcklund transformation is given for the discrete sine-Gordon equation, and the general soliton solution is obtained.     

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.     

Analytical Bethe Ansatz,Canonical Bäcklund Transformation and Q-Operator For A New Discrete Integrable Heirarchy

A new discrete heiarchy of integrable equations is generated from a new Lax Operator and a canonical Bäcklund transformation of the system is derived using Sklyanin’s formalism, based on the classical r-matrix. By quantising the system a quantum analogue of the corresponding canonical Bäcklund transformation is obtained and certain properties of the associated Q-operator are examined. Finally the analytical Bethe Ansatz is used to solve for the spectrum.