Related papers: A multidimensionally consistent version of Hirota'…
An "exact discretization" of the Schroedinger operator is considered and its direct and inverse scattering problems are solved. It is shown that a differential-difference nonlinear evolution equation depending on two arbitrary constants can…
A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz-KdV equation and generalize to the vector case.…
We define the non-commutative multiple bi-orthogonal polynomial systems, which simultaneously generalize the concepts of multiple orthogonality, matrix orthogonal polynomials and of the bi-orthogonality. We present quasideterminantal…
The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of…
We study from a Hamiltonian point of view the generalized dispersionless KdV hierarchy of equations. From the so called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and…
Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota…
Soliton Solutions of Korteweg-de Vries (KdV) were constructed for given degenerate curves $y^2 = (x-c)P(x)^2$ in terms of hyperelliptic sigma functions and explicit Abelian integrals. Connection between sigma functions and tau function were…
In these lectures we discuss how the Painleve equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and…
The quasi-integrable KdV equation has been obtained from the corresponding deformation of the Hamiltonian for the usual KdV system. Following suitable gauge-fixing, it has been found that the quasi-conservation condition is satisfied and an…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM…
We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported…
Using new generalized Landen transformations, we prove that the solutions of the KdV and other nonlinear equations obtained recently by using a kind of superposition principle for periodic solutions are in fact novel re-expressions of well…
Under the effect of common perturbations, the multiple-soliton solution of the KdV equation is transformed into a sum of an elastic and a first-order inelastic component. The elastic component is a perturbation series, identical in…
Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, and the nonlinear Schr\"{o}dinger equations, the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the…
This paper proposes a new class of arbitrarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is…
Generalization of the modified KdV equation to a multi-component system, that is expressed by $(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0, 1,…
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
In three dimensions, the construction of bi-Hamiltonian structure can be reduced to the solutions of a Riccati equation with the arclength coordinate of a Frenet-Serret frame being the independent variable. Explicit integration of conserved…
The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In…