2019
Том 71
№ 10

# Fushchich V. I.

Articles: 34
Article (Ukrainian)

### Linear and nonlinear representations of Galilei groups in two-dimensional space-time

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 414-423

We study Galilei groups represented as groups of Lie transformations in the space of two independent variables and one dependent variable. We classify the representations of the groups A G 1(1,1), A G 2(1,1), A G 3(1,1), A ~G 1 (1,1), A ~G 2 (1,1), and A ~G3(1,1) in the class of Lie vector fields.

Article (Ukrainian)

### Conditional symmetry of the Navier-Stokes equations

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 806–813

We study the conditional symmetry of the Navier-Stokes equations and construct multiparameter families of exact solutions of the Navier-Stokes equations.

Article (Ukrainian)

### Symmetry of equations of linear and nonlinear quantum mechanics

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 164–176

We describe local and nonlocal symmetries of linear and nonlinear wave equations and present a classification of nonlinear multidimensional equations compatible with the Galilean principle of relativity. We propose new systems of nonlinear equations for the description of physical phenomena in classical and quantum mechanics.

Article (Ukrainian)

### Galilei-invariant higher-order equations of burgers and korteweg-de vries types

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1589-1601

We describe nonlinear Galilei-invariant higher-order equations of Burgers and Korteweg-de Vries types. We study symmetry properties of these equations and construct new nonlinear extensions for the Galilei algebra $AG(1, 1)$.

Article (English)

### Reduction of the self-dual Yang-Mills equations I. Poincaré group

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 456–462

For the vector potential of the Yang-Mills field, we give a complete description of ansatzes invariant under three-parameterP (1, 3) -inequivalent subgroups of the Poincaré group. By using these ansatzes, we reduce the self-dual Yang-Mills equations to a system of ordinary differential equations.

Article (English)

### Separation of variables in two-dimensional wave equations with potential

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1343–1361

The paper is devoted to solution of a problem of separation of variables in the wave equation $u_{tt} - u_{xx} + V(x)u = 0$. We give a complete classification of potentials $V(x)$ for which this equation admits a nontrivial separation of variables. Furthermore, we obtain all coordinate systems that provide separability of the equation considered.

Article (Russian)

### Reduction of the multidimensional d’Alembert equation to two-dimensional equations

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 651–662

We give a classification of the maximal subalgebras of rank $n - 1$ for the extended Poincare algebra $A\bar P (1.n)$, which is realized on the set of solutions of the d'Alembeit equation $\square u + \lambda u^k = 0$. These subalgebras are used for constructing the anzatses reducing this equation to differential equations with two invariant variables.

Article (Ukrainian)

### Conditional symmetry and new representations of the Galilean algebra for nonlinear parabolic equations

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1433–1443

An efficient method for finding operators of conditional symmetry, which form a basis of the Galilean algebra, is suggested for a class of Galilei noninvariant parabolic equations. Additional conditions, under which the symmetry can be extended, are described. For the nonlinear equation under consideration, the antireduction is realized and some exact solutions are found by using the conditional Galilei invariance of its differential consequences.

Article (Ukrainian)

### Symmetry and non-lie reduction of the nonlinear Schrödinger equation

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 539–551

The nonlinear Schrödinger-type equations invariant with respect to the extended Galilean group are described. We study the conditional symmetry of such equations, realize the reduction procedure, and construct the classes of exact solutions.

Article (Russian)

### Nonlocal ansatze and solutions of a nonlinear system of heat-conduction equations

Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 293–302

By a nonlocal substitution, a nonlinear system of heat-conduction equations is reduced to a scalar nonlinear heat-conduction equation. The Lie and conditional invariance of the scalar equation is used to find nonlocal ansatze which reduce the original system to systems of ordinary differential equations.

Article (Ukrainian)

### Non-Lie ansatzen and conditional symmetry of the nonlinear Schrodinger equation

Ukr. Mat. Zh. - 1991. - 43, № 12. - pp. 1620–1628

Anniversaries (Ukrainian)

### Parasyuk Ostap Stepanovich (his 70th birthday)

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1443-1444

Article (Ukrainian)

### Conditional symmetry of the equations of nonlinear mathematical physics

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1456–1470

Article (Ukrainian)

### General solutions of the nonlinear wave equation and of the eikonal equation

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1471–1487

Article (Ukrainian)

### Reduction of a multidimensional poincare-invariant nonlinear equation to two-dimensional equations

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1311–1323

Article (Ukrainian)

### Connected subgroups of the conformal group C(1,4)

Ukr. Mat. Zh. - 1991. - 43, № 7-8. - pp. 870–884

A method is given for describing maximal subalgebras of rank r, 1 ≤r ≤ 4, of the conformal algebra AC(1,4), which is the maximal invariance algebra of the eikonal equation. With the help of this method a classification is made up to C(1,4)-equivalence of all maximal subalgebras L of rank 1, 2, 3, and 4 of the algebra AC(1,4) satisfying the condition L ∩ V ⊂〈p1, P2, P3, p4〉, where V is the space of translations.

Article (Ukrainian)

### Qualitative analysis of families of bounded solutions of the multidimensional nonlinear Schrodinger equation

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 821-829

Article (Ukrainian)

### Reduction and exact solutions of the eikonal equation

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 461-474

Article (Ukrainian)

### Maximal subalgebras of rank n?1 of the algebra AP(1, n) and the reduction of nonlinear wave equations. II

Ukr. Mat. Zh. - 1990. - 42, № 12. - pp. 1693–1700

Article (Ukrainian)

### Qualitative analysis of families of bounded solutions of the nonlinear three-dimensional schrodinger equation

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1344–1349

Article (Ukrainian)

### A new mathematical model of heat conduction processes

Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 237–245

Article (Ukrainian)

### Galilei invariant nonlinear equations of Schrodinger type and their exact solutions. II

Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1687–1694

Article (Ukrainian)

### Galilei invariant non-linear equations of schrodinger type and their exact solutions. I

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1349–1357

Article (Ukrainian)

### Subalgebras of the poincare algebra AP (2, 3) and the symmetric reduction of the nonlinear ultrahyperbolic d'Alembert equation. I

Ukr. Mat. Zh. - 1988. - 40, № 4. - pp. 411-416

Article (Ukrainian)

### Symmetry and exact solutions of multidimensional nonlinear wave equations

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 116-123

Article (Ukrainian)

### Continuous subgroups of a generalized Euclidean group

Ukr. Mat. Zh. - 1986. - 38, № 1. - pp. 67–72

Article (Ukrainian)

### Ostap Stepanovich Parasiuk (on his sixtieth birthday)

Ukr. Mat. Zh. - 1981. - 33, № 6. - pp. – C 800-801

Article (Ukrainian)

### A method for investigating the group properties of integrodifferential equations

Ukr. Mat. Zh. - 1981. - 33, № 6. - pp. 834-838

Article (Ukrainian)

### Invariant systems of equations in generalized mechanics

Ukr. Mat. Zh. - 1980. - 32, № 4. - pp. 569–576

Article (Ukrainian)

### Algebra-theoretic analysis of Lame's equation

Ukr. Mat. Zh. - 1980. - 32, № 2. - pp. 267 - 273

Article (Ukrainian)

### Invariance groups of certain equations of relativistic quantum mechanics

Ukr. Mat. Zh. - 1976. - 28, № 6. - pp. 844–849

Brief Communications (Russian)

### On the analytical properties of some apical amplitudes in the theory of perturbations

Ukr. Mat. Zh. - 1965. - 17, № 3. - pp. 137-141

Article (Russian)

### Analytical properties of the scattering amplitude, corresponding to a class of Feinman diagrams

Ukr. Mat. Zh. - 1964. - 16, № 5. - pp. 610-623

Brief Communications (Russian)

### Analytical properties of the amplitudes of generation in uniparticular approximation as a functicn of two variables

Ukr. Mat. Zh. - 1963. - 15, № 2. - pp. 227-232