We give new Backlund transformations (BTs) for some known integrable (in the sense of being multidimensionally consistent) quadrilateral lattice equations. As opposed to the natural auto-BT inherent in every such equation, these BTs are of two other kinds. Specifically, it is found that some equations admit additional auto-BTs (with Backlund parameter), whilst some pairs of apparently distinct equations admit a BT which connects them.
We study an integrable system related to the relativistic Toda lattice. The bilinear representation of this lattice is given and the Bäckulund transformation obtained. A fully discrete version is also introduced with its bilinear Bäcklund transformation and Lax pair. One-soliton solution of the discrete system is presented by use of Bäcklund transformation.
We construct a two-parameter family of Bäcklund transformations for the trigonometric classical Gaudin magnet. The approach follows closely the one introduced by E.Sklyanin and V.Kuznetsov (1998,1999) in a number of seminal papers, and takes advantage of the intimate relation between the trigonometric and the rational case. As in the paper by A.Hone, V.Kuznetsov and one of the authors (O.R.) (2001) the Bäcklund transformations are presented as explicit symplectic maps, starting from their Lax representation. The (expected) connection with the XXZ Heisenberg chain is established and the rationa…
It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the Bäcklund transformations. Several new examples of such transformations are found. In particular we obtained the Bäcklund transformations for a pair of three-component analogs of the dispersive water wave system, and auto-Bäcklund transformations for coupled three-component KdV-type systems.
In the paper we discuss the Bäcklund transformation of the KdV equation between solitons and solitons, between negatons and negatons, between positons and positons, between rational solution and rational solution, and between complexitons and complexitons. We investigate the conditions that Wronskian entries satisfy for the bilinear Bäcklund transformation of the KdV equation. By choosing suitable Wronskian entries and the parameter in the bilinear Bäcklund transformation, we obtain transformations between many kinds of solutions.
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.
An auto-Bäcklund transformation for the quad equation $\mathrm{Q1}_1$ is considered as a discrete equation, called $\mathrm{H2}^a$, which is a so called torqued version of $\mathrm{H2}$. The equations $\mathrm{H2}^a$ and $\mathrm{Q1}_1$ compose a consistent cube, from which a auto-Bäcklund transformation and a Lax pair for $\mathrm{H2}^a$ are obtained. More generally it is shown that auto-Bäcklund transformations admit auto-Bäcklund transformations. Using the auto-Bäcklund transformation for $\mathrm{H2}^a$ we derive a seed solution and a one-soliton solution. From this solution it is seen tha…
Addition formulae of trigonometric and elliptic functions are used to generate Bäcklund transformations together with their connecting quadrilateral equations. As a result we obtain periodic solutions for a number of multidimensionally consistent affine linear and multiquadratic quadrilateral equations.
We provide a method which takes an auto-Bäcklund transformation (auto-BT) and produces another auto-BT for a different equation. We apply the method to the natural auto-BTs for the ABS quad equations, which gives rise to torqued versions of ABS equations and explains the origin of each auto-BT listed in [J. Atkinson, J. Phys. A: Math. Theor. 41 (2008) 135202]. The method is also applied to non-natural auto-BTs for ABS equations, which yields 3D consistent cubes which have not been found in [R. Boll, J. Nonl. Math. Phys. 18 (2011) 337--365], and to a multi-quadratic ABS* equation giving rise to…
The Bäcklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the Bäcklund transformations can be described in the terms of nonrelativistic lattices of the Toda type. Several multifield generalizations are presented.
Ultradiscrete soliton equations and Bäcklund transformation for a generalized soliton solution are presented. The equations include the ultradiscrete KdV equation or the ultradiscrete Toda equation in a special case. We also express the solution by the ultradiscrete permanent, which is defined by ultradiscretizing the signature-free determinant, that is, the permanent. Moreover, we discuss a relation between Bäcklund transformations for discrete and ultradiscrete KdV equations.
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.
The permutability of two Backlund transformations is employed to construct a non linear superposition formula and to generate a class of solutions for the N=2 super sine-Gordon model.
Hirota's bilinear approach is a very effective method to construct solutions for soliton systems. In terms of this method, the nonlinear equations can be transformed into linear equations, and can be solved by using perturbation method. In this paper, we study the bilinear Boussinesq equation and obtain its bilinear Bäcklund transformation. Starting from this bilinear Bäcklund transformation, we also derive its Lax pair and test its integrability.
The Backlund transformations for the Nizhnik-Novikov-Veselov equation are presented. It is shown that these transformations can be iterated and that the resulting sequence can be described by the Volterra equations. The relationships between the Nizhnik-Novikov-Veselov equation and the Volterra hierarchy are discussed.
Bäcklund transformation for the Bullough-Dodd-Jiber-Shabat equation $u_{xx}-u_{tt}=e^u-e^{-2u}$ is found. The construction of integrable boundary condition for this equation together with the algebro-geometric solutions satisfying it are suggested.
Treating an integrable quad-equation along with its two generalised symmetries as a compatible system allows one to construct an auto-Bäcklund transformation for solutions of the related NLS-type system. A fixed periodic reduction of the quad-equation yields a quasi-periodic reduction of its generalised symmetries that turn them into differential constraints compatible with the NLS-type system.
Elementary, one- and two-point, Backlund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time-discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2x2 Lax matrix.
The Adler-Bobenko-Suris (ABS) list contains all scalar quadrilateral equations which are consistent around the cube. Each equation in the ABS list admits a beautiful decomposition. In this paper, we first revisit these decomposition formulas, by which we construct Bäcklund transformations (BTs) and consistent triplets. Some BTs are used to construct new solutions, lattice equations and weak Lax pairs.
New infinite number of one- and two-point Bäcklund transformations (BTs) with explicit expressions are constructed for the high-order constrained flows of the AKNS hierarchy. It is shown that these BTs are canonical transformations including Bäcklund parameter $η$ and a spectrality property holds with respect to $η$ and the 'conjugated' variable $μ$ for which the point $(η, μ)$ belongs to the spectral curve. Also the formulas of m-times repeated Darboux transformations for the high-order constrained flows of the AKNS hierarchy are presented.
It is shown that explicit Bäcklund transformations (BTs) for the high-order constrained flows of soliton hierarchy can be constructed via their Darboux transformations and Lax representation, and these BTs are canonical transformations including Bäcklund parameter $η$ and possess a spectrality property with respect to $η$ and the 'conjugated' variable $μ$ for which the pair $(η, μ)$ lies on the spectral curve. As model we present the canonical explicit BTs with the spectrality for high-order constrained flows of the Kaup-Newell hierarchy and the KdV hierarchy.
We show that the Drinfeld-Sokolov system of equations has a nontrivial prolongation structure. The closure process for prolongation algebra gives rise to the sl(4,c) algebra which is used to derive the scattering problem for the system of equations under consideration. The nontrivial new Backlund transformations and some explicit solutions are given.
In this paper, we define the modified formal variable separation approach and show how it determines, in a remarkably simple manner, the decomposition solutions, the Bäcklund transformations, the Lax pair, and the linear superposition solution of the B-type Kadomtsev-Petviashvili equation. Also, the decomposition solutions, the Bäcklund transformation and the Lax pair relating to the C-type Kadomtsev-Petviashvili equation is obtain by the same technique. This indicates that the decomposition may provide a description of integrable behavior in nonlinear systems, while, at the same time, establi…
We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic map $\cal B$ on the 4-dimensional variety ${\cal V}(2,3)$. The map admits two different reductions, namely, to the Lie group SO(3) and to the coalgebra $so^*(3)$. The first reduction provides a discretization of the motion of the classical Euler top in space and has a transparent…
We construct a Backlund transformation for the trigonometric classical Gaudin magnet starting from the Lax representation of the model. The Darboux dressing matrix obtained depends just on one set of variables because of the so-called spectrality property introduced by E. Sklyanin and V. Kuznetsov. In the end we mention some possibly interesting open problems.
We study additional non-isospectral symmetries of constrained (reduced) N=2 supersymmetric KP hierarchies of integrable ``soliton''-like evolution equations. These symmetries are shown to form an infinite-dimensional non-Abelian superloop superalgebra. Furthermore we study the general Darboux-Backlund (DB) transformations (including adjoint-DB and binary DB) of N=2 super- KP hierarchies preserving (most of) the additional symmetries. Also we derive the explicit form of the general DB (N=2 ``super- soliton''-like) solutions in the form of generalized Wronskian-like super-determinants.
In this paper, from the algebraic reductions from the Lie algebra $gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many multi-component Sine-Gordon type and Sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon equations which can generate new solutions from seed solutions. To see the $Z_n$-systems clearly, we consider the $Z_2$-Sine-Gordon and $Z_3$-Sine-Gordon equations explicitly including their Bäcklund transformations, the nonlin…
Binary symmetry constraints are applied to constructing Bäcklund transformations of soliton systems, both continuous and discrete. Construction of solutions to soliton systems is split into finding solutions to lower-dimensional Liouville integrable systems, which also paves a way for separation of variables and exhibits integrability by quadratures for soliton systems. Illustrative examples are provided for the KdV equation, the AKNS system of nonlinear Schrödinger equations, the Toda lattice, and the Langmuir lattice.
We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlevé equations. The $A^{(1)}_{N-1}$ Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.
We demonstrate the way to derive the second Painlevé equation $P_2$ and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of $P_2$ while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra $J_{_{{\rm Mat}(N,N)}}$ with the Jordan product in the form…
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.
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…
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.
A two-parameters family of Backlund transformations for the classical elliptic Gaudin model is constructed. The maps are explicit, symplectic, preserve the same integrals as for the continuous flows and are a time discretization of each of these flows. The transformations can map real variables into real variables, sending physical solutions of the equations of motion into physical solutions. The starting point of the analysis is the integrability structure of the model. It is shown how the analogue transformations for the rational and trigonometric Gaudin model are a limiting case of this one…
We give Bäcklund transformations for first and second Painlevé hierarchies. These Bäcklund transformations are generalization of known Bäcklund transformations of the first and second Painlevé equations and they relate the considered hierarchies to new hierarchies of Painlevé-type equations.
A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of generalized Toda lattice while semi-discrete systems in the set define the Backlund transform for this Toda lattice and the fully discrete representative of the set can be obtained as a superposition of such kind Backlund transforms. Four linear Zakharov-Shabat type systems taken pairwise realize Lax pairs for these six dynamical systems.
We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type …
We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion" on the defect point via the space-like and time-like description. We then exploit the structural similarity of these equations with the discrete and continuous Backlund transformations. And although these equations are similar they are not exactly the same to the Backlund transformations. We also consider specific examples of integrable models to demonstrate our construction, i.e. the Toda chain and the sine-Gordon model. The equations of the time (space) e…
In this note we present explicitly the construction of the mKdV hierarchy and show that it decomposes into positive and negative graded sub-hierarchies. We extend the construction of the Backlund transformation for the sinh-Gordon model to all other positive and negative odd graded equations of motion generated by the same affine algebraic structure. Some simple examples of solutions are explicitly verified to satisfy, in a universal manner, the Backlund transformations for the first few odd (positive and negative) sub-hierarchies.
In this paper we study a non-linear partial differential equation (PDE), proposed by N. Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the Bäcklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrabl…
Proper lattices for the discrete BKP and the discrete DKP equaitons are determined. Linear Bäcklund transformation equations for the discrete BKP and the DKP equations are constructed, which possesses the lattice symmetries and generate auto-Bäcklund transformations
We present a geometric construction of Backlund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Backlund transformations, which are naturally parametrized 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…
The BT of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Backlund parameters) as a 2-cycle of the BT. In this article we will show that such solutions are associated with a commuting one-parameter family of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit solution of the mappings within this family and hence give the solutions of Adler's equation that are 2-cycles of the BT.
For the integrable case of the discrete self-trapping (DST) model we construct a Backlund transformation. The dual Lax matrix and the corresponding dual Backlund transformation are also found and studied. The quantum analog of the Backlund transformation (Q-operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q-operator as an explicit integral operator as well as describe its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable …
Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé ($P_{IV}$) hierarchies recently found by the same authors [{\em Publ. Res. Inst. Math. Sci. (Kyoto)} {\bf 37} 327--347 (2001)]. We show how the known pattern of BTs for $P_{IV}$ can be extended to our $P_{IV}$ hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water w…
The construction of generalized Backlund transformation for the $A_n$ Affine Toda hierarchy is proposed in terms of gauge transformation acting on the zero curvature representation. Such construction is based upon the graded structure of the underlying affine algebra which induces a classification of generalized Backlund transformations. Moreover, explicit examples for $su(3)$ and $su(4)$ lead to uncover interesting composition properties of various types of Backlund transformations. The universality character of the gauge-Backlund transformation method is extended to all equations of the hier…
Solitons in nonlinear optics holds a special role both in theoretical and experimental studies. Several types of evolution equations are seen to govern different situation of physical relevance. One such is the existence of both resonant and nonresonant situation in optical fibre. The corresponding evolution equation was devised by Doktorov et. al., which consists of a forced NLS equation along with two other equations for population difference and polarization. Here, we have followed an earlier formulation of Neugebauer for Darboux-Bäcklund transformation for this coupled systems. This formal…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order partial derivative taken as the new dependent variable. Associated Backlund transformations are constructed for evolution equations of general form (special cases of which are Burgers, Korteweg-de Vries, and many other nonlinear equations of mathematical physics). The results obtained are used for order reduction and constructing exact solutions of hydrodynamic…
The Hirota bilinear difference equation is generalized to discrete space of arbitrary dimension. Solutions to the nonlinear difference equations can be obtained via Bäcklund transformation of the corresponding linear problems.
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the von Mises transformation, with the unknown function taken as the new independent variable and an appropriate partial derivative taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form (special cases of which are Burgers, Korteweg--de Vries, and Harry Dym type equations as well as many other nonlinear equations of mathematical physics). The results obtained are used for order reduction and c…
The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformat…
The Bäcklund transformation and its nonlinear superposition principle are presented for the Krichever-Novikov equation $u_t= u_{xxx} - {3/(2u_x)} (u^2_{xx} - r(u)) + cu_x, r^{(5)}=0$.
A new super-Backlund transformation for the N=1 supersymmetric sinh-Gordon equation is constructed. Based on this construction we propose a type-II integrable defect for the supersymmetric sinh-Gordon model consistent with this new transformation through the Lagrangian formalism. Explicit expressions for the modified conserved energy, momentum and supercharges are also computed. In addition, we show for the model that the type-II defect can also been regarded as a pair of fused defects of a previously introduced type. The explicit derivation of the associated defect matrices is also presented …
We show that the auto-Backlund transformations of the sine-Gordon, Korteweg-deVries, nonlinear Schrodinger, and Ernst equations are related to their respective CPT symmetries. This is shown by applying the CPT symmetries of these equations to the Riccati equations of the corresponding pseudopotential functions where the fields are allowed to transform into new solutions while the pseudopotential functions and the Backlund parameter are held fixed.
