Justification of the KP-II approximation in dynamics of two-dimensional FPU systems

Hristov, Nikolay; Pelinovsky, Dmitry E.

doi:10.1007/s00033-022-01846-1

Cited by 9 publications

(6 citation statements)

References 33 publications

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“…The approximation of a 2D Fermi-Pasta-Ulam (FPU) lattice by the Kadomtsev-Petviashvili II (KP-II) equation has been justified in Hristov and Pelinovsky. 6 for waves of long length, traveling in certain directions. A justification of the KP-II approximation for general directions is given.…”

Section: Depicts the Latticementioning

confidence: 99%

“…For lattices where the interaction among neighboring masses is nonlinear, approximations by nonlinear PDEs have been used to investigate behavior for longer time scales than those considered here. The approximation of a 2D Fermi–Pasta–Ulam (FPU) lattice by the Kadomtsev–Petviashvili II (KP‐II) equation has been justified in Hristov and Pelinovsky 6 . for waves of long length, traveling in certain directions.…”

Section: Introductionmentioning

confidence: 99%

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Macroscopic wave propagation for 2D lattice with random masses

McGinnis

2023

Stud Appl Math

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We consider a simple two-dimensional harmonic lattice with random, independent, and identically distributed masses. Using the methods of stochastic homogenization, we prove that solutions with initial data, which varies slowly relative to the lattice spacing, converge in an appropriate sense to solutions of an effective wave equation. The convergence is strong and almost sure. In addition, the role of the lattice's dimension in the rate of convergence is discussed. The technique combines energy estimates with powerful classical results about sub-Gaussian random variables.

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Section: Depicts the Latticementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Macroscopic wave propagation for 2D lattice with random masses

McGinnis

2023

Stud Appl Math

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“…The system of ( 12), (13), and ( 15) is a closed system. Once 𝑛 (1) 𝑖 is solved from (15), we have all the other profiles 𝑛 12) and (13). From (11a) and (11c), we may assume…”

Section: Theorem 2 the Cauchy Problem For The Dkp Equation (mentioning

confidence: 99%

“…In this paragraph, we make some remarks on the existing work that is closely related to our present work. Indeed, in the past one or two decades, many efforts have been made to rigorously justify various equations, such as the nonlinear Schrödinger equation, 15,18,22 the KdV equation, 2,5,8,21,25,32 the KP equation, 5,13,14,19,24,31,33,36,37 the Zakharov-Kuznetsov (ZK) equation, 20,31 and the Ostrovsky equation. 28 Whether the KP equation provides a good approximation to the 2D RQEP system is not known.…”

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confidence: 99%

The KP limit of a reduced quantum Euler–Poisson equation

Liu

2023

Stud Appl Math

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In this paper, we consider the derivation of the Kadomtsev–Petviashvili (KP) equation for cold ion‐acoustic wave in the long wavelength limit of a two‐dimensional reduced quantum Euler–Poisson system under different scalings for varying directions in the Gardner–Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter . The KP‐I is derived for , KP‐II for , and the dispersiveless KP (dKP) equation for the critical case . The rigorous proof for these limits is given in the well‐prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner–Morikawa transform. The results can be generalized in several directions.

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“…A justification of the asymptotic reduction to the KP-II equation for the two-dimensional Boussinesq equation was done by Gallay and Schneider [14]. In the recent series of papers [15,19,35], the KP-II equation was justified as the leading model for a two-dimensional Fermi-Pasta-Ulam system on a square lattice. See also [1,2] for recent work on transverse stability of line solitary waves in other generalizations of the KP equation.…”

Section: Introductionmentioning

confidence: 99%

Stability of smooth periodic travelling waves in the Camassa–Holm equation

et al. 2021

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We solve the open problem of spectral stability of smooth periodic waves in the Camassa-Holm equation. The key to obtaining this result is that the periodic waves of the Camassa-Holm equation can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg-de Vries equation. The standard formulation has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We prove that the nonstandard formulation has the advantage that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.

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Justification of the KP-II approximation in dynamics of two-dimensional FPU systems

Cited by 9 publications

References 33 publications

Macroscopic wave propagation for 2D lattice with random masses

Macroscopic wave propagation for 2D lattice with random masses

The KP limit of a reduced quantum Euler–Poisson equation

Stability of smooth periodic travelling waves in the Camassa–Holm equation

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