The dissertation is devoted to the construction of existence classes for solutions of discrete infinite-dimensional Hamiltonian systems on a two-dimensional lattice.
In particular, the dissertation investigates infinite systems of coupled nonlinear 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 potential 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 traveling 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 using the mountain pass theorem. The existence of homoclinic traveling waves is proved using the method of periodic approximations, and heteroclinic ones, using 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 Schrödinger 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 Schrödinger 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.
