On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models

In this paper we consider a class of evolution equations up to fifth-order containing many arbitrary smooth functions from the point of view of nonlinear self-adjointness. The studied class includes many important equations modeling different phenomena. In particular, some of the considered equations were studied previously by other researchers from the point of view of quasi self-adjointness or strictly self-adjointness. Therefore we find new local conservation laws for these equations invoking the obtained results on nonlinearly self-adjointness and the conservation theorem proposed by Nail Ibragimov.     

Attractors for certain nonlinear problems of mathematical physics

One proves the existence of a compact, connected global attractor for the Navier-Stokes-Voigt equations, for pseudoparabolic equations, and for quasilinear wave equations with a strongly dissipative term.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 50–54, 1986.     

Third-order scalar evolution equations with conservation laws

Associated to the family of third-order quasilinear scalar evolution equations is the geometry of point transformations. This geometry provides a framework from which to study the structure of conservation laws of the equation, and to study the special nature of the geometry of those equations which do possess conservation laws. There is an easy and obvious normal form for equations which possess at least one conservation law. The geometric structure of the equation gives rise to a simple yet much less obvious normal form for equations which possess at least two conservation laws.     

Smooth measures and nonlinear equations of mathematical physics

We present explicit equations for the logariphmic derivatives of smooth measures (or their densities) and indicate the connection of these equations with the well-known equations of mathematical physics. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 70–83.     

Existence of infinitely many large solutions for the nonlinear Schrödinger-Maxwell equations

In this paper, we study the nonlinear stationary Schrödinger-Maxwell equations

(∗)     

Conditional symmetry of the equations of nonlinear mathematical physics

We present a survey of results regarding the investigation of the conditions of the symmetry of nonlinear equations of mathematical and theoretical physics: the wave equation, the Schrödinger, Boussinesq, Korteweg-de Vries, Maxwell, and Dirac equations. We construct families of exact solutions that cannot be obtained by the classical Lie approach.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1456–1470, November, 1991.     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.     

Remarks on proofs of conservation laws for nonlinear Schrödinger equations

Conservation laws of the charge and of the energy are proved for nonlinear Schrödinger equations with nonlinearities of gauge invariance in a way independent of approximate solutions.     

On conservation laws for hyperbolized equations

We carry out an analysis of hyperbolized equations of diffusion type convenient for modeling on high-performance computer systems; in particular, we study conservation laws for these equations.     

Existence of infinitely many solutions for fourth-order impulsive differential equations

The aim of this paper is to study the existence of infinitely many solutions for fourth-order impulsive differential equations involving oscillatory behaviors of nonlinearity at infinity. The result is proved by using critical point theory and variational approach.     

On the evolution operators for some equations of mathematical physics with variable coefficients

It is shown that, with the help of a relatively simple operator technique, it is possible to solve, from a common point of view, the Cauchy problem for many important equations of mathematical physics with variable coefficients. This result is applied to the equations of kinetic theory, and diffusion and heat conduction equations. We discuss the problem of equivalence of different schemes of expansion according to the Hausdorff formula.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 857–869, July, 1994.     

Stochasticp-adic equations of mathematical physics

Properties and probability characteristics of solutions of the stochastic p-adic equations of mathematical physics containing the Vladimirov pseudodifferential operators and their generalizations are discussed. The p-adic analogues of the Brownian sheet and the Levy Brownian motion are constructed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 249–263, May, 1999.     

Systems of singular integral equations of certain mixed boundary-value problems of mathematical physics

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 43, pp. 33–42, 1985.     

Conservation laws for nonlinear telegraph equations

A complete conservation law classification is given for nonlinear telegraph (NLT) systems with respect to multipliers that are functions of independent and dependent variables. It turns out that a very large class of NLT systems admits four nontrivial local conservation laws. The results of this work are summarized in tables which display all multipliers, fluxes and densities for the corresponding conservation laws. A physical example is considered for possible applications.