Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap

Abstract

We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant Hamiltonian system. In the framework of this system, we identify Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.

Bounds on quantum evolution complexity via lattice cryptography

“…It is known from extensive literature [40][41][42][43][44][45][46][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66] that, depending on the specific values of C, such systems of equations desplay a striking range of behaviors, some of which are at the forefront of contemporary PDE mathematics. Dynamical features that have been specifically studied are turbulent transfers of energy [64][65][66] including finite-time turbulent blow-up [40-42, 44, 46, 66], dynamical recurrence phenomena [40,45,62], integrability [64][65][66], extra conservation laws and solvability within restricted dynamically invariant manifolds [49,51,52,58,59,67], etc.…”

“…It is known from extensive literature [40][41][42][43][44][45][46][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66] that, depending on the specific values of C, such systems of equations desplay a striking range of behaviors, some of which are at the forefront of contemporary PDE mathematics. Dynamical features that have been specifically studied are turbulent transfers of energy [64][65][66] including finite-time turbulent blow-up [40-42, 44, 46, 66], dynamical recurrence phenomena [40,45,62], integrability [64][65][66], extra conservation laws and solvability within restricted dynamically invariant manifolds [49,51,52,58,59,67], etc. Thus, not only can one reliably connect the extremely simple quantum dynamics of (5.1) with its effectively finite-dimensional Hilbert spaces to classical Hamiltonian dynamics, but in addition this classical dynamics can display an array of rich and varied features depending on the choice of the couplings C. We proceed to present a few special cases of quantum resonant systems that will...…”

“…The Hamiltonian (1.1) may appear not-too-familiar to many readers, but, as a matter of fact, the corresponding classical Hamiltonian system frequently arises as a controlled approximation to weakly nonlinear partial differential equations (PDEs) in strongly resonant domains, originating from a number of branches of physics and mathematics. Specifically, classical systems corresponding to (1.1), together with some closely related variations, have been studied in the following contexts: gravitational dynamics in anti-de Sitter (AdS) spacetimes [40][41][42][43][44][45][46] (typically, motivated by the AdS instability conjecture [47,48]); related dynamical problems for classical relativistic fields [49][50][51][52][53][54]; nonrelativistic nonlinear Schrödinger equations describing, among other things, the dynamics of Bose-Einstein condensates in harmonic potentials [55][56][57][58][59][60][61][62][63]; and integrable models for turbulence [64][65][66]. These classical systems display, for different choices of C nmkl , a wide range of analytic and dynamical patterns ranging from full solvability [64] to Lax-integrability [64][65][66], partial solvability [49][50][51][52]58,59,67], turbulent cascades [42,…”