Related papers: A complete list of conservation laws for non-integ…
We show that the so-called hidden potential symmetries considered in a recent paper [Gandarias M., Physica A, 2008, V.387, 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and…
We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation,…
In this paper we study the generalized variable-coefficient Gardner equations of the form $u_t + A(t)u^n\,u_x+ C(t)\,u^{2n}u_x + B(t)\,u_{xxx} + Q(t)\,u =0$. This class broadens out many other equations previously considered: Johnpillai and…
The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional extention, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly…
Using advantages of nonstandard computational techniques based on the light-cone variables, we explicitly find the algebra of generalized symmetries of the (1+1)-dimensional Klein-Gordon equation. This allows us to describe this algebra in…
All possible linearly independent local conservation laws for $n$-dimensional diffusion--convection equations $u_t=(A(u))_{ii}+(B^i(u))_i$ were constructed using the direct method and the composite variational principle. Application of the…
Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ The infinite hierarchies of polynomial and…
A weak formulation is devised for the K(m,n) equation which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones…
In 2003, A.J. Majda and J.A. Biello derived and studied the so-called reduced equations for equatorial baroclinic-barotropic waves, to which we refer as to the Majda-Biello system. The equations in question describe the nonlinear…
We study the class of generalized Korteweg-DeVries equations derivable from the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - { {(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right) dx, $ where the…
In \cite{RH3} Rasin and Hydon suggested a way to construct an infinite number of conservation laws for the discrete KdV equation (dKdV), by repeated application of a certain symmetry to a known conservation law. It was not decided, however,…
Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable…
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include higher order effects. Although this equation has only one conservation law, exact…
We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein--Gordon equation, we…
The core focus of this research work is to obtain invariant solutions and conservation laws of the (3+1)-dimensional ZK equation, a higher-dimensional generalization of the Korteweg--de Vries (KdV) equation, which describes the phenomenon…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…
The paper compares computational aspects of four approaches to compute conservation laws of single differential equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer…
Using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations thereby establishing their…
We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u-\partial_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then…
The \nl \cls for the N=1 supersymmetric KdV equation are shown to be related in a simple way to powers of the fourth root of its Lax operator. This provides a direct link between the supersymmetry invariance and the existence of \nl…