On hyperbolic systems of equations with zero generalized Laplace invariants

A criterion for the interruption of the sequence of generalized Laplace invariants is found. A general solution of a system of linear hyperbolic equations with zero invariants is constructed.     

Exact solutions of the problem of adsorption-desorption dynamics with a nonlinear sorption isotherm

Analytic solutions of the problem of adsorption-desorption dynamics with a nonlinear Sorption isotherm are obtained for cases of practical importance in which the Danckwerts condition is satisfied at the inlet to the porous medium and the adsorbate concentration in the mobile phase at the surface of the porous medium is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 107–112, September–October, 1989.     

New example of a nonlinear hyperbolic equation possessing integrals

We discover an important new case in the classical problem of the classification of nonlinear hyperbolic equations possessing integrals. In the general (least degenerate) case, in addition, we obtain a formula describing the splitting of the right-hand side of such equations with respect to the first derivatives. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 20–26, July, 1999.     

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.     

Integrals,Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems

We generalize the notions of Laplace transformations and Laplace invariants for systems of hyperbolic equations and study conditions for their existence. We prove that a hyperbolic system admits the Laplace transformation if and only if there exists a matrix of rank k mapping any vector whose components are functions of one of the independent variables into a solution of this system, where k is the defect of the corresponding Laplace invariant. We show that a chain of Laplace invariants exists only if the hyperbolic system has a entire collection of integrals and the dual system has a entire collection of solutions depending on arbitrary functions. An example is given showing that these conditions are not sufficient for the existence of a Laplace transformation.