Cklund Transformation Research Articles

Integrability and Inverse Scattering Transform of the Modified Benjamin-Ono Equation

Abstract This paper presents a B{"a}cklund transformation, the Lax representation, and conserved quantities for the modified Benjamin--Ono equation. The initial problem of the modified Benjamin--Ono equation on the line was studied by the inverse scattering transform method, presenting a nonlocal Riemann--Hilbert problem in the inverse problem to reconstruct the explicit potential function. Further, the exact $N$-soliton solutions and long--time asymptotic behavior are provided, respectively. We also graphically show that the propagation of soliton solutions is consistent with the result of large-time asymptotic forms. As a revised version, the mBO equation incorporates the solution features of the BO equation, and its solutions are presented in logarithmic form, which seeks to describe related physical phenomena more accurately. Our results will contribute to further exploration of physical and mathematical problems related to the mBO equation.

B\"{a}cklund transformations as integrable discretization. The geometric approach

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that B\"{a}cklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.

The asymptotic stability of solitons for the focusing mKdV equation with weak weighted Sobolev initial data

In this paper, we consider the Cauchy problem for the focusing modified Korteweg–de Vries (mKdV) equation in line with the weak weighted Sobolev initial data and without the small-norm assumption. We use the inverse scattering transform, the auto-Bäcklund transformation, and the ∂̄-steepest descent method to obtain the asymptotic stability of the solitons of the mKdV equation.

B<i>Ä</i>CKLUND TRANSFORMATIONS AND ROUGE WAVES IN THE FRAME OF A FRACTIONAL ORDER MODEL IN MAGNETIZED DUSTY PLASMA

<abstract> Dusty plasma has become a hot topic in physics in recent years because of its wide application in the space environment, industrial processing and fusion reaction. The (3+1)-dimensional modified Zakharov-Kuznetsov (mZK) equation describing waves propagation in dusty plasma is derived via the reduced perturbation method, based on the governing equation. Furthermore, integer-order equation is derived as the fractional modified Zakharov-Kuznetsov (TF-mZK) equation. The exact solution and B$ \ddot{a} $cklund transformation are obtained by the fractional transformation and Bell polynomials. Finally, the rouge wave phenomenon in magnetized dusty plasma is described, and the effects of fractional order, phase velocity and dust-cyclotron frequency on the propagation characteristics of dust acoustic rogue waves are analyzed. </abstract>

Q-DISCRETE PAINLEVÉ EQUATIONS: THEIR HIERARCHIES AND PROPERTIES

The main objective of this thesis is to derive hierarchies of q-discrete PainelevA© equations. Some of the important properties of these hierarchies will also be given, namely Lax pairs, BA¤cklund transformations, solutions of their asso- ciated linear problems for special values of parameters and their symmetry groups. To construct these hierarchies, we apply a geometric reduction and a stair- case method on a multi-parameteric generalized lattice modified Korteweg-de Vries equation. In addition, the property of consistency around the cube is used in order to find BA¤cklund transformations. Starting with the base case of q-discrete second, third and fourth PainlevA© equations on A5 initial-values surface, new hierarchies of q-discrete third and fourth PainlevA© equations are discovered, and we also rediscover the hierarchy of q-discrete second PainlevA© equation. In this thesis, we provide the Lax pairs for each member in these hierarchies. Using the consistency around the cube, we also provide the BA¤cklund transformation for the entire hierarchy of q-discrete second and third PainlevA© hierarchies. We generate a hierarchy of special solutions starting with seed solutions for q-discrete second and third PainlevA© hierarchies. An assumption made is that particular parameter values would enable the ability to diagonalize the Lax pair. As a consequence, we found that the as- sociated linear problem for the three hierarchies can be solved in terms of q-Gamma function. Furthermore, the hierarchy of q-discrete fourth PainlevA© hierarchy can be reduced to one equation that can be linearlized to become Riccati equation which has hypergeometric special solutions. Finally, we investigated the affine Weyl group structure of the symmetry group for each hierarchy. In this thesis, we construct the explicit representation of the symmetry group for the first and second member of these hierarchies. The collection of new hierarchies, their Lax pairs, BA¤cklund transforma- tions, the resultant symmetry groups and special solutions comprise the new results of this thesis. This thesis contains material published in [10] in collabo- ration with N. Joshi and D. Tran and myself. The material of this paper is pre- sented in Chapter 3, and is related to qPII and qPIII hierarchies, their Lax pair and examples. In Chapter 4, BA¤cklund transformation of qPII and qPIII hierarchies, includes material from the above-mentioned paper. Similarly, Chapter 5 reports on results about solutions of the linear problem from the above paper. However, we emphasize that all the results about qPIV through out the thesis are completely new and unpublished. Chapter 6 includes unpublished material even for qPII and qPIII hierarchies.

Bäcklund transformations and divisor doubling

In classical mechanics well-known cryptographic algorithms and protocols can be very useful for construction canonical transformations preserving form of Hamiltonians. We consider application of a standard generic divisor doubling for construction of new auto B\"{a}cklund transformations for the Lagrange top and H\'{e}non-Heiles system separable in parabolic coordinates.

Traveling Wave Solutions to the Davey-Stewartson Equation With the Riccati-Bernoulli Sub-ODE Method

The Riccati-Bernoulli subsidiary ordinary differential equation (sub-ODE) method was proposed to construct the exact traveling wave solutions to the nonlinear partial differential equations (NLPDEs). Through traveling wave transformation, the NLPDE was reduced to a nonlinear ODE. With the aid of the Riccati-Bernoulli sub-ODE, the nonlinear ODE was converted into a set of nonlinear algebraic equations. The exact traveling wave solutions to the NLPDE were obtained as soon as this set of nonlinear algebraic equations were solved. Application of this method to the Davey-Stewartson equation directly gave the exact traveling wave solutions. The Bcklund transformation of the Davey-Stewartson equation was also given. The results were compared with those of the first-integral method. The proposed method is effective and easy to be generalized to deal with other types of nonlinear partial differential equations.

Myers-Perry black hole in an external gravitational field

We obtain a new exact solution of the 5D Einstein equations in vacuum describing a distorted Myers-Perry black hole with a single angular momentum. Locally, the solution is interpreted as a black hole distorted by a stationary $U(1)\times U(1)$ symmetric distribution of external matter. Technically, the solution is constructed by applying a 2-fold B\"{a}cklund transformation on a 5D distorted Minkowski spacetime as a seed. The physical quantities of the solution are calculated, and a local Smarr-like relation on the black hole horizon is derived. It possesses the same form as the Smarr-like relation for the asymptotically flat Myers-Perry black hole. It is demonstrated that in contrast to the asymptotically flat Myers-Perry black hole, the ratio of the horizon angular momentum and the mass $J^2/ M^3$ is unbounded, and can grow arbitrarily large. We study the properties of the ergoregion and the horizon surface. The external field does not influence the horizon topology. The horizon geometry however is distorted, and any regular axisymmetric geometry is possible.

Auto-B&amp;#228;cklund Transformation and Extended Tanh-Function Methods to Solve the Time-Dependent Coefficients Calogero-Degasperis Equation

In this paper, the Auto-B?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Calogero-Degasperis (VCCD) equation. New soliton and periodic solutions of many types are obtained. Furthermore, the soliton propagation is discussed under the effect of the variable coefficients.

Bäcklund transformation, auto-Bäcklund transformation and exact solutions for KdV-type equations

In plasma physics, fluid dynamics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena (ion acoustic wave in plasma, quantum hydrodynamic model, wave motion on the surface of shallow water and the unidirectional propagation of long wave of small amplitude and exists in many physical branches). In this paper, KdV-type equations are investigated. We are used Bäcklund Transformation to obtain new exact solutions for the (KdV)-type equations. The method of characteristics is used and the Bäcklund transformation are employed to generate new solutions from the old ones. By the homogenous balance method, we derive an auto–Bäcklund Transformation (ABT) for the KdV equation. Thus, families of solution for KdV-type equations are obtained. Keywords : KdV-type equations B ä cklund transformation, auto- B ä cklund transformation and exact solutions.

The B�cklund Transformation and Exact Solutions for the Equivalent Dodd-Bullough-Mikhailov (E-DBM) Equation

In this paper, the E-DBM equation is studied. The Btransformation for this equation is obtained by using the homogeneous balance (HB) method. And some new solitary wave solutions, including double peakon solitary wave solutions and peakon singular solitary wave solutions, are given by the transformation.

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley &amp; Sons, Ltd.

New infinite sequence exact solutions to the general lattice

To seek infinite sequence exact solutions to nonlinear difference-differential equations, a kind of auxiliary equation of hyperbolic function type is intruduced, and the B&amp;#228,cklund transformation of the solutions to the equation and the formula of nonlinear supperposition of solutions are presented. Based on these and combing auxiliary equation with function transformation, the general lattice equation is chosen as example for which the infinite sequence exact solutionsto the equation are obtained.

Bilinear approach to N = 2 supersymmetric KdV equations

The N=2 supersymmetric KdV equations are studied within the framework of Hirota's bilinear method. For two such equations, namely $N=2, a=4$ and $N=2, a=1$ supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear B\"{a}cklund transformation is given for the $N=2, a=1$ supersymmetric KdV equation.

Magnetic soliton transport over topological spin texture in chiral helimagnet with strong easy-plane anisotropy

We show the existence of an isolated soliton excitation over the topological ground state configuration in chiral helimagnet with the Dzyaloshinskii-Moryia exchange and the strong easy-plane anisotropy. The magnetic field perpendicular to the helical axis stabilizes the kink crystal state which plays a role of "topological protectorate" for the traveling soliton with a definite handedness. To find new soliton solution, we use the B\"{a}cklund transformation technique. It is pointed out that the traveling soliton carries the magnon density and a magnetic solition transport may be realized.

Transplantation method of solutions and the solutions of a coupled VCmKdV equation with source

According to the comparability of the nonlinear terms and dispersion terms between the variable coefficient modified Korteweg-de Vries （VCmKdV） equation and constant coefficient KdV-mKdV equation, we transform properly the KdV-mKdV equation with known solutions, transplant its solutions to the VCmKdV equation with unknown solutions, and thus construct the transplantation relation between the solutions of two different equations. Utilizing this transplantation method of solutions, we obtain new exact solutions and solitary wave-like solutions of the coupled VCmKdV system, which is derived from a two-layer fluid model with source or sink. Then we compare the Bcklund transformation and this transplantation method, and analyse the influence of source and sink on the amplitude.

Singular solitons of generalized Camassa-Holm models

In this paper we introduce one type of generalized dispersive Camassa-Holm model and make its singularity analysis. We prove that the model is Painlevé integrable by an alternative WTC-Kruskal test and obtain the Painlevé-Bcklund systems and the Bcklund transformation. Many new types of regular soliton, singular soliton, kink soliton, compacton and anti-compacton are explored. Particularly, we have found singular structures of periodic cuspon waves in kink solitons, which occur in their central regions. Based on the regular solitonic system, we do Bcklund transformation and obtain three sorts of singular solitons, namely the periodic blow-up wave with hump structure, kink soliton for the blow-up wave structure and the compacton.

Dromion soliton waves and the their evolution in the background of Jacobi sine waves

By an improved mapping approach, Bcklund transformation approach and a variable separation approach, a series of excitations of the (2+1)-dimensional generalized Breor-Kaup (GBK) system is derived. Based on the derived excitations, we find some complex waves of GBK equation. Then we discussed the evolution of dromion solitions in the background of the Jacobi sine waves.

On Darboux transformation of the supersymmetric sine-Gordon equation

Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super B\"{a}cklund transformation and is further used to obtain $N$ super soliton solutions.

New foldons and their elastic interactions in the (2+1)-dimensional modified Veselov-Novikov system

By using Mathematica and multi-linear variable separation(MLVS) approach which is based on the Bcklund transformations, a new exact solution which include low dimensional arbitrary functions of the (2+1)-dimensional modified Veselov-Novikov system is obtained. Two new foldons are constructed and their entirely elastic interactions are considered. In additon, foldon and ghoston interactions are derived. MLVS approach is also extended to solve a new (1+1)-dimensional nonlinear system.