Abstract
The primary objective of the proposal for this PhD thesis is the theoretical and numerical study of lattice differential equations appearing as fundamental models in various nonlinear phenomena. This thesis uses techniques from nonlinear analysis, nonlinear physics, dynamical systems and the numerical analysis for the purpose of numerical simulations. Animportant general question for the asymptotic behavior of solutions of gradient systems, is if globally defined and bounded orbits converge to equilibrium as t converges to infinity. This simple stated question but of fundamental importance in theory and applications, remains open in its generality. Even for gradient systems in R^2, counter-examples have been constructed due to R. Palis and W. de Melo, showing that this convergence fails, and that La Salle’s invariance principle arguments are not applicable. On the other hand, even when convergence holds, other exciting situations may appear. Simple examples given by A. Haraux and M.A ...
The primary objective of the proposal for this PhD thesis is the theoretical and numerical study of lattice differential equations appearing as fundamental models in various nonlinear phenomena. This thesis uses techniques from nonlinear analysis, nonlinear physics, dynamical systems and the numerical analysis for the purpose of numerical simulations. Animportant general question for the asymptotic behavior of solutions of gradient systems, is if globally defined and bounded orbits converge to equilibrium as t converges to infinity. This simple stated question but of fundamental importance in theory and applications, remains open in its generality. Even for gradient systems in R^2, counter-examples have been constructed due to R. Palis and W. de Melo, showing that this convergence fails, and that La Salle’s invariance principle arguments are not applicable. On the other hand, even when convergence holds, other exciting situations may appear. Simple examples given by A. Haraux and M.A Jendoubi, demonstrate the convergence to a continuum of equilibria (e.g., the equilibrium set is not discrete), and a nontrivial structure of the ω-limit set of the flow appears. The derivation of global stability results and convergence rates to nonlinear excited states, for DNLS and DKG lattices involving dissipative effects, is of primary interest in this PhD thesis. In Section2, we present some characteristics of the Klein-Gordon model. Next we prove global existence of solutions, while we present the analytical considerations, concerning the convergence to a single equilibrium. Then the nest paragraph is devoted to the analytical results, concerning the global bifurcation of nonlinear equilibria. Finally we report the results of our numerical simulations. In section 3, we begin with the applications of the discrete nonlinear Schr\"odinger to various physical phenomena. Concerning mechanisms of nonlinear gain or loss we study the so-called defocusing case for s=1, while for s = −1 the focusing case. Together with the initial conditions, we supplement the lattice, with either periodic boundary conditions or Dirichlet boundary conditions . In the case of an infinite lattice, we consider vanishing boundary conditions . For simplicity, in what follows, the periodic initial-boundary value problem will be called as ( P), while the Dirichlet or vanishing initial-boundary value problem, will be called as ( D). Then, we prove the analytical arguments on the finite-time collapse for its solutions. The methods are motivated from the results and questions posed on the dynamics of the continuous counterpart. Extending the energyarguments from the continuum to the discrete ambient space, and by using a spatially averaged power energy functional, we prove analytical estimates for the blow-up time. The values of the real effect coefficients, γ (linear) and δ (nonlinear), define areas of different dynamics. As we will see, for γ,δ>0 we have the blow-up of solutions in finite time, while for γ,δ<0, decay of solutions. In addition, we derive the existence of a critical value γ∗ on the linear loss, separating the finite-time collapse (when γ> γ∗) from the decay (when γ< γ∗) of solutions, as in the continuous case. Next, we present the results of numerical simulations. In this study we establish the validity of the analytical estimates. The numerical findings revealed that the analytical estimates can be used in order to classify distinct types of collapse: extended, localized, or collapse dynamics which is combination of the previous types. We explore the role of the discreteness, the amplitude of initial condition and the defocusing/focusing nature of the lattice in the dynamics. The numerical blow-up times are close (and in some cases where found to be in excellent agreement), with theanalytical upper or lower bound. They correspond to the extended or localized type of collapse. On the other hand, when these times lie between the analytical bounds, the system has a non-trivial dynamical transition towards collapse. The results of this study, aspire to reveal substantial differences in the mathematical treatment of the above questions, between discrete and continuous models and their observed dynamics. The role of the discreteness is proved to be of fundamental importance.
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