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 the 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 non-linear 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 non-singular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.
A new Darboux transformation is presented for the Hirota-Satsuma coupled KdV system. It is shown that this Darboux transformation can be constructed by means of two methods: Painlevé analysis and reduction of a binary Darboux transformation. By iteration of the Darboux transformation, the Grammian type solutions are found for the coupled KdV system.
Elementary Darboux--Laplace transformations for semidiscrete and discrete second order hyperbolic operators are classified. It is proved that in the (semi)-discrete case there are two types of elementary Darboux--Laplace transformations as well: Darboux transformations that are defined by the choice of particular element in the kernel of the initial hyperbolic operator and classical Laplace transformations that are defined by the operator itself. It is proved that in the discrete case on the level of equivalence classes any Darboux--Laplace transformation is a product of elementary ones.
In this paper, we consider the two component derivative nonlinear Schrödinger equation and present a simple Darboux transformation for it. By iterating this Darboux transformation, we construct a compact representation for the $N-$soliton solutions.
Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non auto-Bäcklund transformation between two n-th KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
The Lax representation for the nonstationary Schrödinger equation with rather arbitrary potential is proposed. Some examples of the construction of exact solutions are given by means of Darboux Transformation method.
We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Backlund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix we obtain local conservation laws of the system.
Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the variable separable solutions which can be obtained from the truncated Painlevé analysis and the twice Darboux transformation leads to some new variable separable solutions which are the generalization of the known results obtained by using a guess ansatz to solve the generalized trilinear equation.
We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on $SL(2,R)$ within the AKNS framework, this model is based on $SL(2,R)\otimes U(1)$. The connection between the scalar and the matrix descriptions in this case implies that the generic Darboux matrix for the TB hierarchy has a different structure from that in the models based on $SL(2,R)$ studied thus far. The conventional Darboux transformation is shown to be q…
We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schrödinger equation are given as explicit examples.
With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrodinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.
We construct the Darboux-Backlund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix depending on the background solution. In order to obtain the transformation we use a new, more general, spectral problem.
In this paper, we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB) equations which is governed by femtosecond pulse propagation through inhomogeneous doped fibre. The determinant representation of Darboux transformation is used to derive soliton solutions, positon solutions of the IH-MB equations.
We construct a Darboux transformation of a general $su(3)$-valued spin system called the $Γ$-spin system. Using this Darboux transformation we derive a recursive formula for the soliton solutions of this spin system. Then using these results we present explicit expressions for the 1-soliton solution of the coupled $su(2)$-valued spin systems, namely, of the coupled M-LIII equation.
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.
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transform…
A nonlocal derivative nonlinear Schrodinger equation is introduced. By constructing its basic Darboux transformations of degrees one and two, the explicit expressions of new solutions are derived from seed solutions by Darboux transformation of degree 2n. Usually the derived solutions of this nonlocal equation may have singularities. However, by suitable choice of eigenvalues and the parameters describing the ratio of the two entries of the solutions of the Lax pair, global bounded solutions of the nonlocal derivative nonlinear Schrodinger equation are obtained from zero seed solution by a Dar…
The Hirota equation is an integrable higher order nonlinear Schrödinger type equation which describes the propagation of ultrashort light pulses in optical fibers. We present a standard Darboux transformation for the Hirota equation and then construct its quasideterminant solutions. The multisoliton and breather solutions of the Hirota equation are given explicitly.
In this paper, we obtain a uniform Darboux transformation for multi-component coupled NLS equations, which can be reduced to all previous presented Darboux transformation. As a direct application, we derive the single dark soliton and multi-dark soliton solutions for multi-component coupled NLS with defocusing case and mixed focusing and defocusing case. Some exact single and two-dark solitons of three-component NLS equation are shown by plotting the picture.
The Darboux transformations for the two dimensional elliptic affine Toda equations corresponding to all seven infinite series of affine Kac-Moody algebras, including $A_l^{(1)}$, $A_{2l}^{(2)}$, $A_{2l-1}^{(2)}$, $B_l^{(1)}$, $C_l^{(1)}$, $D_l^{(1)}$ and $D_{l+1}^{(2)}$, are presented. The Darboux transformation is constructed uniformly for the latter six series of equations with suitable choice of spectral parameters and the solutions of the Lax pairs so that all the reality symmetry, cyclic symmetry and complex orthogonal symmetry of the corresponding Lax pairs are kept invariant. The exact …
The $N=2 \;a=-2$ supersymmetric KdV equation is studied. A Darboux transformation and the corresponding Bäcklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation. The Bäcklund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems. The continuum limits of these discrete systems are also considered.
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies.
The Sasa-Satsuma equation is an integrable higher-order nonlinear Schrödinger equation. Higher-order and multicomponent generalisations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations is the Sasa-Satsuma equation. We present the binary Darboux transformations for the Sasa-Satsuma equation and then construct its quasigrammians solutions by iterating its binary Darboux transformations. Periodic, one-soliton, two-solitons and breather solutions are given as explicit examples.
The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time $t$ for the KPESCS which, in contrast with the binary Darboux transformation of the KP equation, provides a non-auto-Bäcklund transformation between two KPESCSs with different degrees. The formula for N-times repeated generalized binary Darboux transformation is proposed and enables u…
The (2+1)-dimensional Hirota-Maxwell-Bloch equation (HMBE) is integrable by the Inverse Scattering Method. In this paper, we construct a Darboux transformation (DT) of the (2+1)-dimensional HMBE. Also the one-soliton solution obtained by means of the one-fold DT. For the $n$-soliton solution the general form is presented.
In this paper, we construct a generalized Darboux transformation for nonlinear Schrödinger equation. The associated $N$-fold Darboux transformation is given both in terms of a summation formula and in terms of determinants. As applications, we obtain compact representations for the $N$-th order rogue wave solutions of the focusing nonlinear Schrödinger equation and Hirota equation. In particular, the dynamics of the general third order rogue wave is discussed and shown to exhibit interesting structure.
The application of the Darboux Transformation method to the integrable model of Cylindrically Symmetrical Chiral field has been considered. The associated linear system of matrix equations has been proposed and the properties of symmetrie for its solutions has been obtained. The necessary form of Darboux Transformation has been found and formal one- and N-soliton solutions have been constructed. With the use of Pohlmayer's Transformation the equation sin-Gordon type have been given and the hypothesis about its integrability has been deduced.
We present a new approach to the construction of the Darboux matrix. This is a generalization of the recently formulated method based on the assumption that the square of the Darboux matrix vanishes for some values of the spectral parameter. We consider the multisoliton case, the reduction problem and the discrete case. The relationship between our approach and the standard dressing method is discussed in detail.
In the recent comment quoted in the title (arXiv:1407.7852v1), a comment is presented on our recent work which derive a generalized nonlinear wave solution formula for mixed coupled nonlinear Schödinger equations by performing the unified Darboux transformation (arXiv:1407.5194). Here we would like to reply to the comment and clarify some facts in arXiv:1407.5194.
The Myrzakulov-Lakshmanan-II (ML-II) equation is one of a (2+1)-dimensional generalizations of the Heisenberg ferromagnetic equation. It is integrable and has a non-isospectral Lax representation. In this paper, the Darboux transformation (DT) for the ML-II equation is constructed. Using the DT, the 1-soliton and 2-soliton solutions of the ML-II equation are presented.
In this paper we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from an ${\mathfrak{sl}}_3({\mathbb{C}})$-based automorphic Lie algebra. The integrability of the found systems is demonstrated via Lax pairs and generalised symmetries.
We analyze a certain class of integral equations related to Marchenko equations and Gel'fand-Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to Darboux transformations associated with various systems of ordinary differential operators. We illustrate our theory by deriving the Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when …
The mKP equation with self-consistent sources (mKPESCS) is treated in the framework of the constrained mKP hierarchy. We introduce a new constrained mKP hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy with self-consistent sources. This offers a natural way to obtain the Lax representation for the mKPESCS. Based on the conjugate Lax pairs, we construct the generalized Darboux transformation with arbitrary functions in time $t$ for the mKPESCS which, in contrast with the Darboux transformation for the mKP equation, provides a non-auto-Bäcklund transformation betwee…
Integrable Heisenberg ferromagnetic equations are an important subclass of integrable systems. The M-XCIX equation is one of a generalizations of the Heisenberg ferromagnetic equation and are integrable. In this paper, the Darboux transformation of the M-XCIX equation is constructed. Using the DT, a 1-soliton solution of the M-XCIX equation is presented.
A version of the iterated Bäcklund-Darboux transformation, where Darboux matrix takes a form of the transfer matrix function from the system theory, is constructed for the discrete canonical system and Non-Abelian Toda lattice. Results on the transformations of the Weyl functions, insertion of the eigenvalues, and construction of the bound states are obtained. A wide class of the explicit solutions is given. An application to the semi-infinite block Jacobi matrices is treated.
In this paper we review two concepts directly related to the Lax representations: Darboux transformations and Recursion operators for integrable systems. We then present an extensive list of integrable differential-difference equations together with their Hamiltonian structures, recursion operators, nontrivial generalised symmetries and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra type equations, integrable discretization of derivative nonlinear Schrödinger equations such as the Kaup-Newell lattice, the Chen-L…
The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov-Witten total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities.
The technique of Darboux transformation is applied to nonlocal partner of two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with quantized spectral parameter. The generalization of the Darboux transformation technique on linear equations of such a kind is given. The connections between the solutions of overdetermined linear systems and their expansions in series at singular points neighborhood are presented. The solutions of the nonlocal Toda lattice and infinite hierarc…
In this paper, we propose a new approach to calculate multi-soliton solutions of Camassa-Holm (CH) equation and modified Camassa-Holm (MCH) equation with aid of Darboux transformation (DT). The new approach simplifies the approach presented in {\it Proc. R. Soc. Lond. A} {\bf 460} 2617-2627 (2004). We first map the CH and MCH equation to a negative order KdV (NKdV) equation by a reciprocal transformation. Then we proceed to apply the DT to solve the NKdV equation in the usual way. Finally we invert the reciprocal transformation to recover the solutions of the CH equation and MCH equation.
We generalize to the supersymmetric case the representation of the KP hierarchy as a set of conservation laws for the generating series of the conserved densities. We show that the hierarchy so obtained is isomorphic to the JSKP of Mulase and Rabin. We identify its ``bosonic content'' with the so-called Darboux-KP hierarchy, which geometrically encompasses the theory of Darboux-Bäcklund transformations, and is an extension both of the KP theory and of the modified KP theory. Finally, we show how the hierarchy can be linearized and how the supersymmetric counterpart of a wide class of rational …
We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of $\mathbb{Z}_\mathcal{N}$ graded discrete Lax pairs very recently. In this paper, the $\mathbb{Z}_\mathcal{N}$ graded discrete equations in coprime case and their corresponding Lax pairs are derived from the discrete Gel'fand-Dikii hierarchy by applying a transformation of the independent variables. The construction of the Darboux tranformations is r…
This paper is concerned with a generalized type of Darboux transformations defined in terms of a twisted derivation $D$ satisfying $D(AB)=D(A)+σ(A)B$ where $σ$ is a homomorphism. Such twisted derivations include regular derivations, difference and $q$-difference operators and superderivatives as special cases. Remarkably, the formulae for the iteration of Darboux transformations are identical with those in the standard case of a regular derivation and are expressed in terms of quasideterminants. As an example, we revisit the Darboux transformations for the Manin-Radul super KdV equation, studi…
A new form of Darboux-Bäcklund transformation and its higher order form is derived for Derivative Nonlinear Schrodinger Equation(DNLS). The new form arises due to the different form of Lax pair. It is observed that by a special choice of the eigenvalue of DB transformation one can generate a mixed form of solution(containing both algebraic and exponential dependence on (x, t) can be generated. On the other hand by adopting a new methodology due to Neugebauer et. al. it is found that purely rational solution can be constructed. The two different approach yields different class of solution and a…
The use of effective Darboux transformations for general classes Lax pairs is discussed. The general construction of ``binary'' Darboux transformations preserving certain properties of the operator, such as self-adjointness, is given. The classes of Darboux transformations found include the multicomponent BKP and CKP reductions of the KP hierarchy.
We systematically study Darboux-type transformations for the KdV and AKNS hierarchies and provide a complete account of their effects on hyperelliptic curves associated with algebro-geometric solutions of these hierarchies.
In this paper, we derive n-fold Darboux transformation of the two-component Hirota and the Maxwell-Bloch(TH-MB) equations and its determinant representation. Using Darboux determinant representation, we provide soliton solutions, positon solutions of the TH-MB equations.
In this paper we construct Darboux transformations for the supersymmetric Two-boson equation. Two Darboux transformations and associated Bäcklund transformations are presented. For one of them, we also obtain the corresponding the nonlinear superposition formula.
The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.
We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6-point difference scheme. The scheme is appropriate to solving a hyperbolic type initial-boundary value problem. We discuss several reductions and specifications of the transformations as well as construction of other Darboux covariant schemes by means of existing ones. In particular we introduce a 10-point scheme which can be regarded as the discretization of self-adjoint hyperbolic equation.
The Hirota-Miwa equation can be written in `nonlinear' form in two ways: the discrete KP equation and, by using a compatible continuous variable, the discrete potential KP equation. For both systems, we consider the Darboux and binary Darboux transformations, expressed in terms of the continuous variable, and obtain exact solutions in Wronskian and Grammian form. We discuss reductions of both systems to the discrete KdV and discrete potential KdV equations, respectively, and exploit this connection to find the Darboux and binary Darboux transformations and exact solutions of these equations.
For the n-dimensional integrable system with a twisted so(p,q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding T^n\to R^{2n}.
We characterize in terms of Darboux transformations the spaces in the Segal-Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of e^x. The resulting subgrassmannian is parametrized in terms of trigonometric Calogero-Moser matrices.
In this paper the Singular Manifold Method has allowed us to obtain the Lax pair, Darboux transformations and tau functions for a non-linear Schrödiger equation in 2+1 dimensions. In this way we can iteratively build different kind of solutions with solitonic behavior.
For a generalized super KdV equation, three Darboux transformations and the corresponding Bäcklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the Bäcklund-Darboux transformations and the corresponding discrete system are considered for Kupershmidt's super KdV equation. When all the odd variables vanish, a nonlinear superposition formula is obtained for Levi's Bäcklund transformation for the KdV equation.
Several types of Darboux transformations for supersymmetric integrable systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations are considered. We also present solutions such as supersolitons and superkinks.
For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the Bäcklund transformation found early by Liu and Xie \cite{liu2}. The Darboux transformation and the related Bäcklund transformation are used to construct integrable super differential-difference and difference-difference systems. The continuum limits of these discrete systems and of their Lax pairs are also considered.
We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.
