Related papers: Matrix solutions of a noncommutative KP equation a…
It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure $n$-soliton solutions satisfy an eigenvalue equation for ordinary differential operators in the spectral…
Quasipolynomial (or QP) mappings constitute a wide generalization of the well-known Lotka-Volterra mappings, of importance in different fields such as population dynamics, Physics, Chemistry or Economy. In addition, QP mappings are a…
We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…
We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair, conserved…
We demonstrate that commuting quasilinear systems of Jordan block type are parametrised by solutions of the modified KP hierarchy. Systems of this form naturally occur as hydrodynamic reductions of multi-dimensional linearly degenerate…
We present exact soliton solutions of anti-self-dual Yang-Mills equations for G=GL(N) on noncommutative Euclidean spaces in four-dimension by using the Darboux transformations. Generated solutions are represented by quasideterminants of…
This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of this solutions with respect to two parameters $\kappa$ and $\lambda$ that appears in…
Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations,…
We consider certain degenerations of trigonal curves and hyperelliptic curves, which we call one step degeneration. We compute the limits of corresponding quasi-periodic solutions using the Sato Grassmannian. The mixing of solitons and…
We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some rank one matrix T. A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such…
We investigate the conditions under which systems of two differential eigenvalue equations are quasi exactly solvable. These systems reveal a rich set of algebraic structures. Some of them are explicitely described. An exemple of quasi…
The correspondence between a high-order non symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving…
We construct solutions of an infinite Toda system and an analogue of the Painlev'e II equation over noncommutative differential division rings in terms of quasideterminants of Hankel matrices.
We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and…
Using an algebraic orbifold method, we present non-commutative aspects of $G_2$ structure of seven dimensional real manifolds. We first develop and solve the non commutativity parameter constraint equations defining $G_2$ manifold algebras.…
We present formulas for the homogenous multivariate resultant as a quotient of two determinants. They extend classical Macaulay formulas, and involve matrices of considerably smaller size, whose non zero entries include coefficients of the…
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone…
We present three complementary methods to study stationary nonlinear solutions in one-dimensional nonlinear metal-dielectric structures. Two of them use an approximate treatment of the Kerr type nonlinear term taking into account only the…
Nonisospectral integrable systems can describe solitary waves in nonuniform media. In this paper, we apply the Cauchy matrix approach to construct three types of nonisospectral matrix modified Korteweg-de Vries (mKdV) eqautions and present…
We survey several results connecting combinatorics and Wronskian solutions of the KP equation, contextualizing the successes of a recent approach introduced by Kodama, et. al. We include the necessary combinatorial and analytical background…