Abstract An algorithm is described which decides if a given polynomial differential expression ∆ of multivariate functions is exact, i.e. whether there exists a first integral P such that D x P = ∆ for any one x of a set of n variables and to provide the integral P . A generalization is given

This letter presents some special features of a class of integrable PDE’s admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have rst derivative

analysis in order to single out equations that are likely to be integrable. New PDE’s are thus obtained, along with others already well-known for their integrability and for which we obtain here the trilinear expression. 1.

PDEs

A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison

. The numerical integration of infinite-dimensional Hamiltonian systems or Hamiltonian PDEs is much less explored. In this paper, we suggest a new theoretical framework for generalizing symplectic numerical integrators for ODEs to Hamiltonian PDEs in R 2 : time plus one space dimension. The central idea

equation and its Lax pair. Thus problem of construction of new integrable pde’s reduces to construction of commutator identities on associative algebras. 1.

Abstract We review recent classification results on the theory of systems of nonlinear Hamiltonian partial differential equations with one spatial dimension, including a perturbative approach to the integrability theory of such systems, and discuss universality conjectures describing critical

ABSTRACT. The article presents and refines the results which were proven in [1]. We give a condition for obtaining the optimal constant of the integral inequality for the numerical analysis of a nonlinear system of PDEs.

Dedicated to Professor Piet Wesseling for his numerous contributions in the ¯eld of numerical mathematics and computational °uid dynamics An integration method is discussed which has been designed to treat parabolic and hy-perbolic terms explicitly and sti ® reaction terms implicitly. The method