Related papers: Analytical and numerical solution of coupled KdV-M…
A new integrable class of Davey--Stewartson type systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev--Petviashvili equation by means of an asymptotically exact nonlinear…
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…
This paper discusses the construction of a new $(3+1)$-dimensional Korteweg-de Vries (KdV) equation. By employing the KdV's recursion operator, we extract two equations, and with elemental computation steps, the obtained result is $…
In this paper, we define the modified formal variable separation approach and show how it determines, in a remarkably simple manner, the decomposition solutions, the B\"acklund transformations, the Lax pair, and the linear superposition…
Symmetry reductions of systems of two nonlinear partial differential equations are studied. We find ansatzes reducing system of partial differential equations to system of ordinary differential equations. The method is applied to system…
We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under M\"obius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the…
We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two, by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows the steps, similar to…
A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The class differs from the class of the nonlinear nonlocal unidirectional wave equations previously studied by the addition…
It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a…
A method of looking for boundary conditions consistent with the integrability property of multidimensional Kadomtsev-Petviashvili (KP) type equations is discussed. The method is based on involutions of the Lax pair taken at the border…
In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit $\lambda_{j}$ $\rightarrow$ $\lambda_{1}$ of the Lax pair…
The numerical discretization of the Zakharov-Shabat Scattering problem using integrators based on the implicit Euler method, trapezoidal rule and the split-Magnus method yield discrete systems that qualify as Ablowitz-Ladik systems. These…
In [1], a generalized type of Darboux transformations defined in terms of a twisted derivation was constructed in a unified form. Such twisted derivations include regular derivations, difference operators, superderivatives and…
The KdV equation with self-consistent sources (KdVES) is used as a model to illustrate the method. A generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for…
Under the Flaschka-Newell Lax pair, the Darboux transformation for the Painlev\'{e}-II equation is constructed by the limiting technique. With the aid of the Darboux transformation, the rational solutions are represented by the Gram…
We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference…
Recently proposed nonholonomic deformation of the KdV equation is solved through inverse scattering method by constructing AKNS-type Lax pair. Exact and explicit N-soliton solutions are found for the basic field and the deforming function…
We consider the Cauchy problem for second order differential operators with two independent variables $P=D_t^2-D_x(b(t)a(x))D_x$. Assume that $b(t)$ is a nonnegative $C^{n,alpha}$ function and $a(x)$ is a nonnegative Gevrey function of…
A new approach to double-sub equation method is introduced to construct novel solutions for the nonlinear partial differential equations. It is applied to the Korteweg-de Vries (KdV) equation and yields new complexiton solutions of both the…
In this paper we consider the twice-renormalized, complex-valued modified KdV (mKdV) on the one-dimensional torus introduced by Chapouto. Our main result is the construction of an invariant measure supported at low-regularity. This work…