New Rogue Wave Solutions of (1+2)-Dimensional Non-Isospectral KP-II Equation

The generalized binary Darboux transformation for the (1+2)-dimensional non-isospectral KP-II equation is presented. Moreover, as a direct application, the new rogue wave solutions for the (1+2)-dimensional non-isospectral KP-II equation are constructed by the generalized binary Darboux transformation.     

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.     

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey–Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima–Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.     

Binary Darboux Transformation for the Modified Kadomtsev--Petviashvili Equation

Via the elementary Darboux transformation (DT) of the modified Kadomtsev--Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.     

On Coupled KdV Equations with Self-consistent Sources

The coupled Korteweg-de Vries （CKdV） equation with self-consistent sources （CKdVESCS） and its Lax representation are derived. We present a generalized binary Darboux transformation （GBDT） with an arbitrary time- dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Biicklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more generM solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton- positon solutions of the CKdVESCS.     

On the extended KdV equation with self-consistent sources

The positive extended KdV equation with self-consistent sources (eKdV⁺ ESCSs) is firstly presented and its related linear auxiliary equation is derived. The generalized binary Darboux transformation (DT) is applied to construct some new solutions of the eKdV⁺ ESCSs such as N-soliton solution, N-double pole solution and nonsingular N-positon solution. The properties of these solutions are analyzed. Moreover, the interaction of two solitons is discussed in detail.     

A look at the generalized Darboux transformations for the quasinormal spectra in Schwarzschild black hole perturbation theory: Just how general should it be?

In this article we take a close look at three types of transformations usable in the Schwarzschild black hole perturbation theory: a standard (DT), a binary (BDT) and a generalized (GDT) Darboux transformations. In particular, we discuss the absolutely crucial property of isospectrality of the aforementioned transformations which guarantees that the quasinormal mode (QNM) spectra of potentials, related via the transformation, completely coincide. We demonstrate that, while the first two types of the Darboux transformations (DT and BDT) are indeed isospectral, the situation is wildly different for the GDT: it violates the isospectrality requirement and is therefore only valid for the solutions with just one fixed frequency. Furthermore, it is shown that although in this case the GDT does provide a relationship between two arbitrary potentials (a short-ranged and a long-ranged potentials relationship being just a trivial example), this relationship ends up being completely formal. Finally, we consider frequency-dependent potentials. A new generalization of the Darboux transformation is constructed for them and it is proven (on a concrete example) that such transformations are also not isospectral. In short, we demonstrate how a little, almost incorporeal flaw may become a major problem for an otherwise perfectly admirable goal of mathematical generalization.     

Darboux Transformation and Explicit Solutions for Drinfel＇d-Sokolov-Wilson Equation

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.     

Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation

In this paper,we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB)equations which are governed by femtosecond pulse propagation through inhomogeneous doped fibre.The determinant representation of Darboux transformation is used to derive soliton solutions,positon solutions to the IH-MB equations.     

A Unified Explicit Construction of 2N-Soliton Solutions for Evolution Equations Determined by 2×2 AKNS System

An explicit N-fold Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed. By using the Darboux transformation, the solutions of the evolution equations are reduced to solving alinear algebraic system, from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given. Furthermore, a reduction technique for MKdV equation is presented, and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique. A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.     

Exact solutions of the Gerdjikov-Ivanov equation using Darboux transformations

We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton and breather solutions of the GI equation are given as explicit examples.     

Broer-Kaup系统的达布变换和新的精确解

利用Broer-Kaup系统Lax对的两个基本解，构造了该系统一个新的达布变换，并利用该变换求出了Broer-Kaup系统一组新的双向孤子解. 关键词： Lax对 达布变换 孤子解     

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev-Petviashvili equation

For a variable coefficient Kadomtsev-Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painleve analysis.The N-fold binary Darboux transformation is presented in a compact form.As an application,the multi-lump,higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained.Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.     

Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation

A method of determining the exact solutions to the Burgers equation on the basis of the Darboux transformation is described. It is shown that a single application of the Darboux transformation to the homogeneous Burgers equation transforms the latter into the inhomogeneous equation describing acoustic wave propagation against transonic flow in the de Laval nozzle. In this case, the contraction ratio of the nozzle is fixed and determined by the viscosity coefficient of the medium. Based on the exact solution of the homogeneous Burgers equation, for the aforementioned problem of the flow in the nozzle, all the possible regular steady-state solutions are presented and the evolution of nonstationary solutions is investigated. The algorithm of a multiple Darboux transformation, which allows an increase in the strength of inhomogeneity, i.e., in the contraction ratio of the nozzle, is determined. This approach leads to a discrete set of possible contraction ratios at which exact solutions can be obtained. The Crum’s theorem is used to derive a formula that allows determination of the exact solutions to the inhomogeneous Burgers equation from the solutions to the homogeneous heat transfer equation. It is noted that, in fact, the proposed algorithm of the multiple Darboux transformation makes it possible to decrease the viscosity coefficient of the medium in a discrete way.     

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 (2007)]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 (2011)], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.``     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrödinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrödinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrödinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrödinger equation is given.     

Darboux Transformation and Multi-Solitons for Complex mKdV Equation

An explicR N-fold Darboux transformation with multi-parameters for coupled mKdV equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur （AKNS） system spectral problem. By using the Darboux transformation and the reduction technique, some multi-soliton solutions for the complex mKdV equation are obtained.     

Explicit N-Fold Darboux Transformations and Soliton Solutions for Nonlinear Derivative Schrödinger Equations

An explicit N-fold Darboux transformation for a coupled of derivative nonlinear Schrödinger equations is constructed with the help of a gauge transformation of spectral problems. As a reduction, the Darboux transformation for well-known Gerdjikov-Ivanov equation is further obtained, from which a general form of N-soliton solutions for Gerdjikov-Ivanov equation is given.     

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.