# Samoilenko V. G.

### Asymptotic $Σ$-solutions to singularly perturbed Benjamin – Bona – Mahony equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 236-254

We study the problem of construction of asymptotic $\Sigma$ -solutions to the singularly perturbed Benjamin – Bona – Mahony equation with variable coefficients. An algorithm for the construction of solutions is described. We determine main and first terms of the asymptotic solution. The theorems on the accuracy with which the indicated asymptotic solution satisfies the considered equation are also proved.

### Oleksandr Mykolaiovych Sharkovs’kyi (on his 80th birthday)

Fedorenko V. V., Ivanov А. F., Khusainov D. Ya., Kolyada S. F., Maistrenko Yu. L., Parasyuk I. O., Pelyukh G. P., Romanenko O. Yu., Samoilenko V. G., Shevchuk I. A., Sivak A. G., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk A. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Asymptotic Multiphase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg–de-Vries Equation with Variable Coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1640–1657

We describe the set of initial conditions under which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic multiphase solitonlike solution. The notion of manifold of initial values for which the above-mentioned solution exists is proposed for the analyzed Cauchy problem. The statements on the estimation of the difference between the exact and constructed asymptotic solutions are proved for the Cauchy problem.

### Two-Phase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg-De-Vries Equation with Variable Coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1515–1530

We describe a set of initial conditions for which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic two-phase solitonlike solution. The notion of the manifold of initial data of the Cauchy problem for which this solution exists is proposed.

### Asymptotic *m*-phase soliton-type solutions of a singularly perturbed Korteweg?de Vries equation with variable coefficients. II

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1089-1105

We consider the problem of the construction of higher terms of asymptotic many-phase soliton-type solutions of the singular perturbed Korteweg – de Vries equation with variable coefficients. The accuracy with which the obtained asymptotic solution satisfies the original equation is determined.

### Asymptotic *m*-phase soliton-type solutions of a singularly perturbed Korteweg?de Vries equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 970-87

We propose an algorithm for the construction of asymptotic *m*-phase soliton-type solutions of a singularly perturbed
Korteweg – de Vries equation with varying coefficients and establish the accuracy with which the main term asymptotically satisfies the considered equation.

### Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 388–397

We propose an algorithm of the construction of asymptotic two-phase soliton-type solutions of the Korteweg - de Vries equation with a small parameter at the higher derivative.

### Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 122–132

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients and prove a theorem on the estimation of its precision.

### Asymptotic Expansions for One-Phase Soliton-Type Solutions of the Korteweg-De Vries Equation with Variable Coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 111–124

We construct asymptotic expansions for a one-phase soliton-type solution of the Korteweg-de Vries equation with coefficients depending on a small parameter.

### V. G. Georgii Mykolaiovych Polozhyi (on his 90th birthday)

Glushchenko A. A., Lyashko I. I., Mitropolskiy Yu. A., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 560-561

### Structure of Binary Darboux-Type Transformations for Hermitian Adjoint Differential Operators

Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 271-275

For Hermitian adjoint differential operators, we consider the structure of Darboux–Bäcklund-type transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux–Bäcklund-type transformations.

### On Some Spectral Properties of the Energy Operator for an Infinite System in a Magnetic Field

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 284-289

For systems in a magnetic field, we investigate the form sum of an infinite-dimensional energy operator perturbed by a potential. We also investigate changes in the spectrum of the energy operator in the case of its perturbation by a potential.

### Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory

Prikarpatskii Ya. A., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1704-1719

On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.

### On Extendability of Solutions of Differential Equations to a Singular Set

Kaplun Yu. I., Samoilenko V. G.

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 373-378

We consider the problem of the extendability of solutions of differential equations to a singular set that consists of points at which the right-hand side of the equation considered is undefined.

### The Reduction Method in the Theory of Lie-Algebraically Integrable Oscillatory Hamiltonian Systems

Prikarpatskii A. K., Samoilenko V. G., Taneri U.

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 232-240

We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra \(G = K \oplus P{\text{, where }}K\) is the Lie algebra of a fixed subgroup \(K \subset {\text{G}}\) with respect to an involution σ : *G* → *G* on the Lie group *G*, and with a Poisson action of special type on a symplectic matrix manifold.

### Lyapunov–Schmidt Approach to Studying Homoclinic Splitting in Weakly Perturbed Lagrangian and Hamiltonian Systems

Prikarpatskii A. K., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 66-74

We analyze the geometric structure of the Lyapunov–Schmidt approach to studying critical manifolds of weakly perturbed Lagrangian and Hamiltonian systems.

### Singularly Perturbed Equations with Impulse Action

Kaplun Yu. I., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1089-1009

We propose and justify an algorithm for the construction of asymptotic solutions of singularly perturbed differential equations with impulse action.

### Existence and Extendability of Solutions of the Equation $g(t, x) = 0$

Kaplun Yu. I., Samoilenko V. G.

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 372-382

We consider the problem of extendability and existence of solutions of the equation *g*(*t*, *x*) = 0 on the maximum interval of their definition.

### Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

Brzychczy S., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 220-228

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.

### On periodic solutions of the equation of a nonlinear oscillator with pulse influence

Samoilenko A. M., Samoilenko V. G., Sobchuk V. S.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 827–834

We study periodic solutions and the behavior of phase trajectories of the differential equation of a nonlinear oscillator with pulse influence at unfixed moments of time.

### Kato inequality for operators with infinitely many separated variables

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 718–720

We find conditions under which the Kato inequality is preserved in the case where, instead of an operator with finitely many variables, an operator with infinitely many separated variables is taken. We use the inequality obtained to study both self-adjointness of the perturbed operator with infinitely many separated variables and the domain of definition of the form-sum of this operator and a singular potential.

### Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system

Samoilenko A. M., Samoilenko V. G., Sidorenko Yu. M.

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 78–97

We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensional*k*-constrained KP-hierarchy, briefly called the 2*d k-c*-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2*d k-c* KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2*d k-c* KP-hierarchy is proposed.

### On periodic solutions of linear differential equations with pulsed influence

Elgondyev K. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 141–148

We study periodic solutions of ordinary linear second-order differential equations with publsed influence at fixed and nonfixed times.

### Quasiinvariant deformations of invariant submanifolds of Hamiltonian dynamical systems

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1043–1054

We study quasiinvariant deformations of invariant submanifolds of nonlinear Hamiltonian dynamical systems and their small perturbations.

### The complete integrability analysis of the inverse Korteweg-de Vries equation

Pritula N. N., Samoilenko V. G., Suyarov U. S.

Ukr. Mat. Zh. - 1991. - 43, № 9. - pp. 1239–1248

### Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel'nikov type

Ukr. Mat. Zh. - 1990. - 42, № 5. - pp. 655–659

### Abelian integrals, integrable dynamic systems of the Neumann-Rosochatius type, and the Lax representation

Mikityuk I. V., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1094–1100

### Algebraic scheme of discrete approximations of linear and nonlinear dynamical systems of mathematical physics

Mitropolskiy Yu. A., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1988. - 40, № 4. - pp. 453-458

### Modification of the spectrum of the second-quantization operator under its perturbation by a potential

Ukr. Mat. Zh. - 1987. - 39, № 3. - pp. 396-397

### Bogolyubov's functional equation and the lie-poisson-lasov simplectic structure associated with it

Bogolyubov N. N., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1986. - 38, № 6. - pp. 774–778

### Complete integrability of the differential equations connected with the problem of nonlinear oscillations of a homogeneous beam compressed longitudinally

Mitropolskiy Yu. A., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1985. - 37, № 6. - pp. 727–729

### Backlund transformations for the nonlinear Korteweg-De Vries equation from the algebrogeometric point of view

Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1985. - 37, № 5. - pp. 670–673

### Differential-difference dynamical systems associated with the Dirac difference operator and their total integrability

Mitropolskiy Yu. A., Samoilenko V. G.

Ukr. Mat. Zh. - 1985. - 37, № 2. - pp. 180 – 186

### Integrability of ideals in Grassman algebras on differentiable manifolds and some of its applications

Mitropolskiy Yu. A., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 451 – 456

### Self-adjointness of a second-order elliptic operator with infinite number of variables

Ukr. Mat. Zh. - 1982. - 34, № 5. - pp. 647—650

### Periodic problem for a Toda chain

Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 469—475

### Inverse periodic problem for nonlinear Langmuir chain equations

Ukr. Mat. Zh. - 1982. - 34, № 3. - pp. 322—327

### Elliptic operators of second order with an infinite number of variables

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 405 – 409

### A difference operator with infinitely many variables

Ukr. Mat. Zh. - 1979. - 31, № 5. - pp. 586–588