Localization and coherence in nonintegrable systems

Abstract

We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian oscillator chains approaching their statistical asymptotic states. In systems constrained by more than one conserved quantity, the partitioning of the conserved quantities leads naturally to localized and coherent structures. If the phase space is compact, the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps. In systems where the phase space is not compact, the coherent structures can be collapses represented in phase space by a heteroclinic connection of some unstable saddle to infinity.

Emergence of rogue waves from optical turbulence

“…However, we underline that, in general, the formation of a large scale coherent structure is only possible if the amount of incoherence in the system is not too large [31,32], a feature that was also observed in the highly incoherent regime of supercontinuum generation [33][34][35]. This aspect has been the subject of a detailed study in the context of wave condensation, where it was shown that the emergence of a large-scale coherent structure (a plane wave) only occurs below some critical 'energy' [36][37][38].…”

Stable and metastable states and the formation and destruction of breathers in the discrete nonlinear Schrödinger equation

“…Fluctuations near the minimum energy state R n = (−1) n R lead to the partition function (17) and (18), which is valid for β…”

“…de. system is high-dimensional and nonintegrable, and the trajectory may reach any point at the shells of fixed values of A = A and H = E. This has led to the question of whether breathers are associated with almost all microstates on the shell of fixed energy and waveaction, so that breathers should be considered a statistical phenomenon. A number of papers have studied the statistical mechanics of nonlinear Schrödinger models [7][8][9][10][11][12] and other lattices that support breathers [13][14][15][16][17][18]. The direct approach of computing the microcanonic ensemble would require us to solve integrals over the δ-functions of A and H. One alternative approach is to compute the grand canonical distribution function y(α, β, N) = exp(−αA − βH)dΓ , where the parameters α and β are inverse temperatures that control the content of the quantities A and H. dΓ is a phase space volume element.…”

“…(iii) from the partition function(17),(18) near the state R n = (−1) n R, which corresponds to energies slightly above the minimum energy E min .…”

Self-Organization, Coherence and Turbulence in Laser Optics

“…The dispersion relation providing the resonance with linear waves is k(ω)=βω 2 /2.T obestable(i.e.,nonradiating),theDSspectrumhas to be localized within a frequency window AEΔ: k(AEΔ)=q,w h e r eΔ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2γP 0 =β p ( Figure 3). 2 Formation of these "domain walls" [24][25][26] due to phase effects in a dissipative system results in natural frequency cut-off, which is essential for inherent analogy between DS and a turbulent entity.…”