On multi-solitons for coupled Lowest Landau Level equations

Thomann, Laurent

doi:10.48550/arxiv.2111.05035

Cited by 1 publication

(2 citation statements)

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“…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.…”

Section: Classical Dynamics and Integrabilitymentioning

confidence: 99%

“…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,…”

Section: Introductionmentioning

confidence: 99%

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Bounds on quantum evolution complexity via lattice cryptography

Craps¹,

Clerck²,

Evnin³

et al. 2022

Preprint

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We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries. (An appropriate 'complexity metric' must be used that takes into account the relative difficulty of performing 'nonlocal' operations that act on many degrees of freedom at once.) While simply formulated and geometrically attractive, this notion of complexity is numerically intractable save for toy models with Hilbert spaces of very low dimensions. To bypass this difficulty, we trade the exact definition in terms of geodesics for an upper bound on complexity, obtained by minimizing the distance over an explicitly prescribed infinite set of curves, rather than over all possible curves. Identifying this upper bound turns out equivalent to the closest vector problem (CVP) previously studied in integer optimization theory, in particular, in relation to lattice-based cryptography. Effective approximate algorithms are hence provided by the existing mathematical considerations, and they can be utilized in our analysis of the upper bounds on quantum evolution complexity. The resulting algorithmically implemented complexity bound systematically assigns lower values to integrable than to chaotic systems, as we demonstrate by explicit numerical work for Hilbert spaces of dimensions up to ∼ 10 4 .

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