Pages: 1 - 23
As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting
ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0,
is fully analyzed, including symmetries (nonclassical and contact transformatons),
Pages: 24 - 39
The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik Novikov
Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation,...
Pages: 40 - 50
In 1807 Fourier suggested a original method of solving partial differential equations. The
method is known to lead to ordinary differential equations containing some arbitrary
Pages: 51 - 62
A brief review is presented of the two recent perturbation algorithms. Their common
idea lies in a not quite usual treatment of linear Schrödinger equations via nonlinear
The first approach (let us call it a quasi-exact perturbation theory, QEPT) tries to
get the very zero-order...
Pages: 63 - 67
We investigate conditional symmetry in three directions. The first direction is a research
of the Q-conditional symmetry. The second direction is studying conditional symmetry
when an algebra of invariance is known and an additional condition is unknown. The
third direction is...
Pages: 63 - 67
We apply the similarity method based on a Lie group to a nonlinear model of the heat
equation and find its Lie algebra.The optimal system of the model is contructed from
the Lie algebra. New classes of similarity solutions are obtained.
Pages: 68 - 84
Pages: 85 - 89
Computer-aided symbolic and graphic computation allows to make significantly easier
both theoretical and applied symmetry analysis of PDE. This idea is illustrated by
applying a special "Mathematica" package for obtaining conditional symmetries of
the nonlinear wave equation ut = (u ux)x invariant...
Pages: 90 - 95
We search for hidden symmetries of two-particle equations with oscillator-equivalent
potential proposed by Moshinsky with collaborators. We proved that these equations
admit hidden symmetries and parasupersymmetries which enable easily to find the
Hamiltonian spectra using algebraic methods.
Pages: 90 - 95
The generalization of parasupersymmetric quantum mechanics generated by an arbitrary number of parasupercharges and characterized by an arbitrary order of paraquantization is given. The relations for parasuperpotentials are obtained. It is shown that
parasuperpotentials can be explicitly expressed via...
Pages: 96 - 101
Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A
concept of approximate invariants is introduced and the algorithm of their calculating
Pages: 102 - 110
In recent years T.A. Osborn and his coworkers at the University of Manitoba have extensively developed the well known connected graph expansion and applied it to a wide
variety of problems in semiclassical approximation to quantum dynamics [2, 5, 7, 19, 21,
22, 26, 27]. The work I am reporting on attempts...
Pages: 111 - 129
The idea of introducing coordinate transformations to simplify the analytic expression
of a general problem is a powerful one. Symmetry and differential equations have been
close partners since the time of the founding masters, namely, Sophus Lie (18421899),
and his disciples. To this days, symmetry...
Pages: 130 - 138
The method of one parameter, point symmetric, approximate Lie group invariants is
applied to the problem of determining solutions of systems of pure one-dimensional,
diffusion equations. The equations are taken to be non-linear in the dependent variables but otherwise homogeneous. Moreover, the matrix...
Pages: 139 - 146
Pages: 147 - 151
The problem of construction of boundary conditions for nonlinear equations compatible with their higher symmetries is considered. Boundary conditions for the sineGordon, ZhiberShabat and KdV equations are discussed. New examples are found
for the JS equation.
Pages: 152 - 155
The singular manifold expansion of Weiss, Tabor and Carnevale  has been successfully applied to integrable ordinary and partial differential equations. They yield information such as Lax pairs, Bäcklund transformations, symmetries, recursion operators,
pole dynamics, and special solutions. On the...
Pages: 156 - 159
The different second-order nonlinear partial equations are found that are invariant
under the representation D(1
2, 0) D(0, 1
2) of the Poincaré group P(1, 3) and also
under conformal group C(1, 3). The some exact solutions are constructed for the one
of these equations.
Pages: 160 - 163
Pages: 164 - 169
The paper presents a survey of some new results concerning the approach to construction
of explicit solutions for nonlinear evolution equations
du/dt = F[u],
proposed in [1, 2].
Pages: 170 - 174
This report is devoted to generalization of the equivalence transformations. Let a system of
differential equations be given. Almost all systems of differential equations have arbitrary
elements: arbitrary functions or arbitrary constants.
Pages: 175 - 180
Pages: 186 - 195
It is well-known that symmetry properties are extremely important for choosing differential equations which can be suitable for description of real physical processes.
We present functional bases of second-order differential invariants for various representations of some extensions of the Poincaré group...
Pages: 196 - 201
The transition from Eulerian to Lagrangian coordinates is a nonlocal transformation.
In general, isomorphism should not take place between basic Lie groups of studied
equations. Besides, in the case of plane and rotational symmetric motion hydrodynamic equations in Lagrangian coordinates are partially...
Pages: 210 - 213
The Hilbert space representations of a class of commutation relations associated with
a Möbius transformation is studied using results on convergence of continued fractions.
Pages: 214 - 218
Let us consider the multidimensional nonlinear system of heat equations
u0 = f(v)u;
v0 = u,
where u = u(x) R1, v = v(x) R1, x = (x0, x) R1+3, is the Laplace operator, f(v)
is an arbitrary differentiable function.
In this paper the classification of symmetry properties of equations (1) is...
Pages: 226 - 235
We study the general applicability of the ClarksonKruskal's direct method, which
is known to be related to symmetry reduction methods, for the similarity solutions
of nonlinear evolution equations (NEEs). We give a theorem that will, when satisfied, immediately simplify the reduction procedure or...
Pages: 236 - 240
Quasiclassic method of solving of the Schrödinger equation with quadratic Hamiltonian
is used to derive solutions of Klein-Fock equation for the particle in the constant
magnetic field and the jumping magnetic field.