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 transformations, we derive new vector Yang-Baxter map and integrable discrete vector sineGordon equation on a sphere.Mathematics Subject Classification. 35Q51, 37K10, 37K35, 39A14.

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Dressing method for the vector sine-Gordon equation and its soliton interactionsAlexander V. Mikhailov ⋆ , Georgios Papamikos † and Jing Ping Wang † * † School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, UK ⋆ School of Mathematics, University of Leeds, Leeds, UK AbstractIn this paper, we develop the dressing method to study the exact solutions for the vector sineGordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multi-soliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices.

We construct the Darboux transformation with Dihedral reduction group for the 2-dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund 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.

We construct birational maps that satisfy the parametric set-theoretical Yang-Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable Nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense. [1,5,13,24,25]. Moreover, there exist solutions which can be associated to matrix refactorisation problems and can be further related to Darboux and Bäcklund transformations [15] of soliton equations.Due to the significance of equation (1), its connection with various fields of mathematical and physics and its numerous applications in the theory of Integrable Systems, it is important to construct new solutions and study their properties. The motivation to study solutions of the Yang-Baxter equation related to Darboux transformations arises naturally from some recently obtained results [15,23] where Yang-Baxter maps were obtained using Darboux transformations related to several integrable models. In particular, in [15], Darboux transformations associated with the nonlinear Schrödinger equation (NLS), derivative NLS (DNLS) and deformation of the DNLS (dDNLS) equation were employed in order to construct birational integrable Yang-Baxter maps, while in [23] a Yang-Baxter map on the sphere constructed in relation to a vectorial generalisation of the sine-Gordon equation. The Lax operators of the DNL and the dDNLS equation are invariant under the action of reduction groups isomorphic to Z 2 and D 2 , with rational dependence in the spectral parameter and having poles in C = C ∪ {∞} at generic and degenerate (nongeneric) orbits; in [15], only the operators corresponding to degenerate orbits were used. In the first part of this paper, we study the refactorisation problem of the product of two Darboux matrices associated to the DNLS Lax operator, invariant under the Z 2 reduction group with poles in C that form a single generic orbit under the action of the reduction group. In the second part, we employ different Darboux transformations related to the NLS operator and the DNLS operator, invariant under Z 2 having poles at degenerate points and we construct solutions to the entwining Yang-Baxter equation.The paper is organised as follows. In the next section, we provide preliminary material, definitions and we fix the notation. In particular, we give the definition of the entwining Yang-Baxter equation and we explain the relation between entwining Yang-Baxter maps and matrix refactorisation problems. For entwining maps defined by matrix refactorisation problems, we discuss their birationality, we prove a statement regarding and their invariants and we give the definition of their integrability in the Liouville sense. Finally, we list all the Darboux matrices we are using throughout the text. In...

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We design a consistent Galerkin scheme for the approximation of the vectorial modified Kortewegde Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.