Information on the thesis

The dissertation is devoted to the construction of existence classes for soluti-
ons of discrete infinite-dimensional Hamiltonian systems on a two-dimensional latti-
ce.
In particular, the dissertation investigates infinite systems of coupled nonli-
near oscillators on a two-dimensional lattice. Sufficient conditions for the existence
and uniqueness of local and global solutions are obtained for systems of oscillators
with linear coupling. The conditions for the boundedness of the global solution are
also established. Conditions for the non-existence of a global solution in the case
of power potentials are obtained. For this, the classical theorems of existence and
uniqueness in Banach spaces and the representation of the system in Hamiltonian
form are used.
Also, conditions for the existence of periodic solutions in a time variable are
established for such systems. For this, the method of critical points and the method
of periodic approximations were used. It is shown that in the case of a power potenti-
al function, the constrained minimization method can be used to construct periodic
solutions.
Using the mountain pass theorem and the method of periodic approximations,
the existence of non-constant supersonic periodic and solitary traveling waves for
systems of oscillators with linear and nonlinear coupling is established. It is proved
that the profile of a solitary traveling wave decreases exponentially at infinity. Using
the linking theorem, the results on the existence of subsonic periodic traveling waves
are obtained.
In addition, the dissertation established the existence of non-constant traveli-
ng waves in discrete sine-Gordon type equations on a two-dimensional lattice. Three
types of traveling waves are considered: periodic, homoclinic and heteroclinic. To
prove the existence of periodic traveling waves, a variational method is used usi-
ng the mountain pass theorem. The existence of homoclinic traveling waves is
proved using the method of periodic approximations, and heteroclinic ones, usi-
ng the concentration compactness principle.
By the variational technique, the existence of traveling waves in Fermi-Pasta-
Ulam type systems on a two-dimensional lattice is established. In particular, the
existence of monotonic and not necessarily monotonic traveling waves has been
established. First, traveling waves of two types are considered. In the first case, the
derivative of the profile is a 2k-periodic function, and in the second — the derivative
of the profile vanishes at infinity. Further, the existence of traveling waves with
similar conditions was established, which are imposed on the wave profile itself, and
not on its derivative.
The dissertation also investigates the existence of standing waves in discrete
nonlinear Schr¨odinger type equations on a two-dimensional lattice. Such equations
with cubic and saturable nonlinearities are studied. Two types of standing waves
are considered: with a periodic amplitude (periodic solutions) and an amplitude
that converges to zero (localized solutions). First, the question of the existence of
nontrivial standing waves in discrete nonlinear Schr¨odinger type equations on a two-
dimensional lattice with cubic nonlinearity is studied. The existence of nontrivial
periodic and localized solutions is established. Here, as in previous chapters, we used
the linking theorem for periodic solutions and the periodic approximation method
for localized solutions. Next, the question of the existence of nontrivial standing
waves in such equations with saturable nonlinearity is studied. To obtain the main
results, the critical points method and Nehari manifolds are used.
The results of the dissertation are theoretical. They can be used in the theory
of ordinary differential equations and in nonlinear physics.

Key words: Hamiltonian systems, nonlinear oscillators, Fermi-Pasta-Ulam
type systems, discrete sine-Gordon type equations, discrete nonlinear Schr¨odinger
type equations, two-dimensional lattice, Cauchy problem, periodic solutions, traveli-
ng waves, standing waves, critical points, mountain pass theorem, linking theorem,
concentration compactness principle, periodic approximations, Nehari manifold.