Related papers: B\"acklund Transformations for the Boussinesq Equa…
The B\"acklund transformation (BT) for the Camassa-Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent…
B\"acklund transformations (BTs) are traditionally regarded as a tool for integrating nonlinear partial differential equations (PDEs). Their use has been recently extended, however, to problems such as the construction of recursion…
Binary symmetry constraints are applied to constructing B\"acklund transformations of soliton systems, both continuous and discrete. Construction of solutions to soliton systems is split into finding solutions to lower-dimensional Liouville…
We summarize the results of our recent work on B\"acklund transformations (BTs), particularly focusing on the relationship of BTs and infinitesimal symmetries. We present a BT for an associated Degasperis-Procesi (aDP) equation and its…
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…
The B\"{a}cklund transformation for the three-level Maxwell-Bloch equation is presented in the matrix potential formalism. By applying the B\"{a}cklund transformation to a constant electric field background, we obtain a general solution for…
We study soliton solutions of matrix "good" Boussinesq equations, generated via a binary Darboux transformation. Essential features of these solutions are revealed via their "tropical limit", as exploited in previous work about the KP…
The elementary and systematic binary Bell polynomial approach is applied to the good Boussinesq equation. The bilinear representation, $n$-soliton solutions, bilinear B\"acklund transformation, Lax pair and infinite conservation laws of the…
The derivation of the Backlund transformations (BTs) is a standard problem of the theory of the integrable systems. Here, I discuss the equations describing the BTs for the Ablowitz-Ladik hierarchy (ALH), which have been already obtained by…
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…
We treat the lattice sine-Gordon equation and two of its generalised symmetries as a compatible system. Elimination of shifts from the two symmetries of the lattice sine-Gordon equation yields an integrable NLS-type system. An…
The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward constructed…
We construct B\"acklund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the $sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an exact…
The Backlund transformation related symmetry is nonlocal, which is hardly to apply in constructing solutions for nonlinear equations. In this paper, we first localize nonlocal residual symmetry to Lie point symmetry by introducing multiple…
A class of rational solutions of Toda lattice satisfying certain Backlund transformations and a class of mixed rational-soliton solutions (quasisolitons) in wronskian formare obtained using the method of Ablowitz and Satsuma. Also an…
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…
By introducing generalized Backlund Transformations depending on arbitrary functions, wave and localized soliton solutions of the Davey- Stewartson equations are generated. Moreover explicit soliton solutions of the Hamiltonian DSI and…
We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a…
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how B\"acklund transformations lead to equations that can be…
We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary…