Related papers: On completely integrable polynomial PDEs arising f…
We show some classes of higher order partial difference equations admitting a zero-curvature representation and generalizing lattice potential KdV equation. We construct integrable hierarchies which, as we suppose, yield generalized…
We systematically construct a distinct class of complex potentials including parity-time ($\cal PT$) symmetric potentials for the stationary Schr\"odinger equation by using the soliton and periodic solutions of the four integrable real…
We study nonexistence results and gradient estimates for solutions of \[ \Delta_p v + a v^{q}=0 \] defined on complete Riemannian manifolds satisfying a \emph{$\chi$-type Sobolev inequality}. We establish a Liouville theorem under the…
In this study, we give the variation of parameters method from a different viewpoint for the Nth order inhomogeneous linear ordinary difference equations with constant coefficient by means of delta exponential function . Advantage of this…
We propose a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations. This allows us to give the general integrable evolution corresponding to an arbitrary time and space dependence of the…
Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. In this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized…
The question of complete integrability of evolution equations associated to $n\times n$ first order isospectral operators is investigated using the inverse scattering method. It is shown that for $n>2$, e.g. for the three-wave interaction,…
In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent…
In this paper we introduce a randomized version of the backward Euler method, that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we…
Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a…
We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which…
This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and…
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of integrable…
We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(\beta,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $\beta$-Grothendieck polynomials…
We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method…
A formula for certain exact solutions to the modified Korteweg-de Vries (mKdV) equation is obtained via the inverse scattering transform method. The kernel of the relevant Marchenko integral equation is written with the help of matrix…
The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it…
It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff…
We present a fundamental solution to an initial value problem for the KdV-Whitham system in an explicit integral form. Monotonically decreasing initial data with finite number of breaking points are considered. Generating function for the…
This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type.…