Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of T = ∞, we identify a phase transition, through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breather-like localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.The pioneering studies of Fermi, Pasta and Ulam [1] (FPU) showed that energy exchange between coupled systems may be suppressed in the presence of nonlinearity; instead a type of behavior that severely contrasts equipartition among the linear modes is observed. The question of whether equipartition of excitation energy always appears is a contemporary issue in various fields of physics. Many manifestations of nonequilibrium and non-equipartition phenomena equivalent to the dynamical behavior of systems with few degrees of freedom contrasting statistical mechanics expectations have been observed. Some of these phenomena, and therefore the absence of immediate equipartition expressed in terms of self-trapping of energy, play an important role for optical storage patterns in nonlinear fibers, condensed matter physics, and biophysics [2].A particularity of discrete nonlinear systems is their ability to sustain strong localization of energy [3]. This is accomplished via intrinsic localized modes (breathers) which are modes that remain stable for extremely long times. So far it is a largely unaddressed problem how to handle and describe these excitations in a statistical mechanics framework although it has been argued that breathers may act as virtual bottlenecks [4] delaying the thermalization process.In this work, we develop a statistical understanding of the dynamics, including the breathers, in a discrete nonlinear Schrödinger (DNLS) equation. The DNLS equation plays a significant role in several branches nonlinear physics as a simple physical model because it may approximate many of the above mentioned nonlinear systems. We study analytically and numerically the thermalization of the lattice for T ≥ 0. We identify the regime in phase space wherein regular statistical mechanics considerations apply, and hence, thermalization is observed numerically and explored analytically using regular, grand-canonical, Gibbsian equilibrium measures.However, the nonlinear dynamics of the problem renders permissible the realization of regimes of phase space which would formally correspond to "negative temperatures" [5] in the sense of statistical mechanics. The novel feature of these states is that the energy tends to be localized in certain lattice sites forming breather-like excitations. Returning to statistical mechanics, such realizations, which would formally correspond to negative temperatures, are not possible (since the Hamil...

The evolution towards equipartition in the β-FPU chain is studied considering as initial condition the highest frequency mode. Above an analytically derived energy threshold, this zone-boundary mode is shown to be modulationally unstable and to give rise to a striking localization process. The spontaneously created excitations have strong similarity with moving exact breathers solutions. But they have a finite lifetime and their dynamics is chaotic. These chaotic breathers are able to collect very efficiently the energy in the chain. Therefore their size grows in time and they can transport a very large quantity of energy. These features can be explained analyzing the dynamics of perturbed exact breathers of the FPU chain. In particular, a close connection between the Lyapunov spectrum of the chaotic breathers and the Floquet spectrum of the exact ones has been found. The emergence * INFN and INFM, Firenze (Italy) † INFM, Firenze (Italy) 1 of chaotic breathers is convincingly explained by the absorption of high frequency phonons whereas a breather's metastability is for the first time identified. The lifetime of the chaotic breather is related to the time necessary for the system to reach equipartition. The equipartition time turns out to be dependent on the system energy density ε only. Moreover, such time diverges as ε −2 in the limit ε → 0 and vanishes as ε −1/4 for ε → ∞.

We develop an analytical approach for describing a birth of internal modes of solitary waves in nonintegrable nonlinear models. We show that a small perturbation of a proper sign to an integrable model can create a soliton internal mode bifurcating from the continuous wave spectrum. The theory is applied to the double sine-Gordon and discrete nonlinear Schrödinger equations, and an excellent agreement with numerical data is demonstrated. [S0031-9007(98)06329-7] PACS numbers: 03.40. Kf, 42.65.Tg, 52.35.Mw, 63.20.Pw As is well known, different nonlinear models can possess spatially localized solutions for solitary waves [1]. In many cases, the solitary waves are analyzed in the framework of integrable models which, however, describe realistic physical systems only with certain approximation [2]. Therefore, the fundamental question is the following: What kind of novel physical effects can be expected for solitary waves in nonintegrable models? It is commonly believed that solitary waves of nonintegrable models differ from solitons of integrable models only in the character of the soliton interactions: unlike proper solitons, interaction of solitary waves is accompanied by radiation [2]. In this Letter we demonstrate the existence of nontrivial effects of different nature, generic for nearly integrable and nonintegrable models. We show that a small perturbation to an integrable model may create an internal mode of a solitary wave. This effect is beyond a regular perturbation theory, because solitons of integrable models do not possess internal modes. But in nonintegrable models such modes introduce qualitatively new features into the system dynamics being responsible for long-lived oscillations of the solitary wave shape and resonant soliton interaction.Until now, internal modes have been analyzed only for the so-called kink solitons, topological solitary waves of the Klein-Gordon type models (see, e.g., Refs. [3,4]). The internal modes of kinks, usually called "shape modes," are known to modify drastically the kink dynamics because they can temporarily store energy taken away from the kink's translational motion and later restore the energy back. This mechanism gives rise to resonant structures in the kink-antikink collisions [3] and kink-impurity interactions [5]. In spite of many (basically numerical) results obtained for the kink's internal modes, the important problem still remains unsolved: What is the analytical criterion for creating the kink's internal mode?As a matter of fact, this problem is much more general, and it can be formulated for many soliton bearing physical models. For example, nonlinear propagation of modulated wave packets is often described with the help of envelope solitons of the integrable cubic nonlinear Schrödinger (NLS) equation. However, the analysis of self-focusing and propagation of self-guided spatially localized beams (spatial solitons) in plasmas and optical non-Kerr materials requires one to employ the (usually nonintegrable) models more general than the cubic NLS equation [6...

Breathers may be mobile close to an instability threshold where the frequency of a pinning mode vanishes. The translation mode is a marginal mode that is a solution of the linearized (Hill) equation of the breather which grows linearly in time. In some cases, there are exact mobile breather solutions (found numerically), but these solutions have an infinitely extended tail which shows that the breather motion is nonradiative only when it moves (in equilibrium) with a particular phonon field.More generally, at any instability threshold, there is a marginal mode. There are situations where excitations by marginal modes produce new type of behaviors such as the fission of a breather. We may also have fusion. This approach suggests that breathers (which can be viewed as cluster of phonons) may react by themself or one with each others as well as in chemistry for atoms and molecules, or in nuclear physics for nuclei.

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We present a continuum phase-field model of crack propagation. It includes a phase-field that is proportional to the mass density and a displacement field that is governed by linear elastic theory. Generic macroscopic crack growth laws emerge naturally from this model. In contrast to classical continuum fracture mechanics simulations, our model avoids numerical front tracking. The added phase-field smooths the sharp interface, enabling us to use equations of motion for the material (grounded in basic physical principles) rather than for the interface (which often are deduced from complicated theories or empirical observations). The interface dynamics thus emerges naturally. In this paper, we look at stationary solutions of the model, mode I fracture, and also discuss numerical issues. We find that the Griffith's threshold underestimates the critical value at which our system fractures due to long wavelength modes excited by the fracture process.

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