Related papers: Nonabelian Integrable Systems, Quasideterminants, …
In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial…
In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference…
We develop an algebraic quantisation approach, based on quantisation ideals, and apply it to integrable non-Abelian differential--difference equations. We show that the Toda hierarchy admits a bi-quantum structure whose classical…
A new parameterisation of the solutions of Toda field theory is introduced. In this parameterisation, the solutions of the field equations are real, well-defined functions on space-time, which is taken to be two-dimensional Minkowski space…
This paper aims to find new explicit solutions including multi-soliton, multi-positon, multi-negaton, and multi-periodic for a coupled Volterra lattice system which is an integrable discrete version of the coupled KdV equation. The…
The general solutions of many three-dimensional Lotka-Volterra systems, previously known to be at least partially integrable, are constructed with the aid of special functions. Examples include certain ABC and May-Leonard systems. The…
In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems…
We study a class of integrable nonhomogeneous Lotka-Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials,…
In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher ${\bf Z}$--grading subspaces of a simple Lie algebra, and construct their general solutions. One…
We argue that one of the basic ingredients for the appearance of soliton solutions in integrable hierarchies, is the existence of ``vacuum solutions'' corresponding to Lax operators lying in some abelian subalgebra of the associated affine…
The Volterra lattice equations are completely integrable and possess bi-Hamiltonian structure. They are integrated using partitioned Lobatto IIIA-B methods which preserve the Poisson structure. Modified equations are derived for the…
The new class of integrable mappings and chains is introduced. Corresponding (1+2) integrable systems invariant with respect to such discrete transformations are represented in explicit form. Soliton like solutions of them are represented…
Nonlinear integrable models with two spatial and one temporal variables: Kadomtsev-Petviashvili equation and two-dimensional Toda lattice are investigated on the subject of correct formulation for boundary problem that can be solved within…
Brief review of the methods for solving the multicomponent nonlinear Schrodinger (MNLS) equations and analysis of their Hamiltonian structures is given. Main attention is paid to the MNLS related to the C.II- and D.III-types symmetric…
A direct method is developed for constructing the bright $N$-soliton solution of a multi-component modified nonlinear Schr\"odinger equation. Specifically, the two different expressions of the solution are obtained both of which are…
We make a generalization of the type C monomial space of a single variable, which was introduced in the construction of type C N-fold supersymmetry, to several variables. Then, we construct the most general quasi-solvable second-order…
We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schr\"odinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous…
In this paper we construct explicit smooth solutions to the Strominger system on generalized Calabi-Gray manifolds, which are compact non-K\"ahler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers.
Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for…
We show how to derive noncommutative versions of integrable partial difference equations using Darboux transformations. As an illustrative example, we use the nonlinear Schr\"odinger (NLS) system. We derive a noncommutative nonlinear…