Related papers: Evolution equations: Frobenius integrability, cons…
We derive the Noether identities and the conservation laws for general gravitational models with arbitrarily interacting matter and gravitational fields. These conservation laws are used for the construction of the covariant equations of…
This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces.
Using an interpolant form for the gradient of a function of position, we write an integral version of the conservation equations for a fluid. In the appropriate limit, these become the usual conservation laws of mass, momentum and energy.…
This article is concerned with the mathematical analysis of a class of a nonlinear fractional Schrodinger equations with a general Hartree-type integrand. We prove existence and uniqueness of global-in-time solutions to the associated…
There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs. The same result holds for…
We proved the existence and uniqueness of a traveling wave solution to the thin film equation with a Navier slip condition at the liquid-solid interface. We obtain explicit lower and upper bounds for the solution and an absolute error…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…
In this work, we study convection-diffusion equations in the cases of bounded drifts and drifts induced by the gradient of a potential. We define a new notion of solution and prove its existence and uniqueness. Furthermore, we show the…
We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection…
The Novikov-Veselov (NV) equation is a dispersive (2+1)-dimensional nonlinear evolution equation that generalizes the (1+1)-dimensional Korteweg-deVries (KdV) equation. This paper considers the stability of plane wave soliton solutions of…
Electromagnetic waves and fluids have locally conserved mechanical properties associated with them and we may expect these to exist for matter waves. We present a semiclassical description of the continuity equations relating to these…
The Whitham equation is a model for the evolution of small-amplitude, unidirectional waves of all wavelengths on shallow water. It has been shown to accurately model the evolution of waves in laboratory experiments. We compute…
A new method for the solution of initial-boundary value problems for evolution PDEs recently introduced by Fokas is generalised to multidimensions. Also the relation of this method with the method of images and with the classical integral…
A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found…
The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity $f(u) = \tfrac{1}{2}u^2$ of the Korteweg-de Vries equation and the full linear dispersion relation $\Omega(k) = \sqrt{k\tanh…
A method for constructing evolution equations admitting a master symmetry is proposed. Several examples illustrating the method are presented. It is also noted that for certain evolution equations master symmetries can be useful for…
We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order…
We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate…
We discuss the time evolution of the wave function which is solution of a stochastic Schroedinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and…
We consider different measure-valued solvability concepts from the literature and show that they could be simplified by using the energy-variational structure of the underlying system of partial differential equations. In the considered…