On the stability of nonlinear waves in integrable models

Кузнецов, Е. А.; Spector, M.D.; Falkovich, Gregory

doi:10.1016/0167-2789(84)90186-6

Cited by 50 publications

(46 citation statements)

References 2 publications

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“…Using this technique, below we will give an exact solution of the one-dimensional soliton stability problem for the two-dimensional KP equation following ref. [67],…”

Section: Atj=iwac!i/ax-3k2cljmentioning

confidence: 99%

Soliton stability in plasmas and hydrodynamics

1986

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“…Using this technique, below we will give an exact solution of the one-dimensional soliton stability problem for the two-dimensional KP equation following ref. [67],…”

Section: Atj=iwac!i/ax-3k2cljmentioning

confidence: 99%

Soliton stability in plasmas and hydrodynamics

1986

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“…Less is known about such instability for the cnoidal wave [17]. The growth rate in this case can be found exactly by means of the linearized dressing procedure [18] and expressed in terms of σ-and ζ-Weierstrass functions. When the distance between solitons becomes large enough, the maximal instability growth rate turns out to be exponentially small, but increases with the distance decrease [17].…”

mentioning

confidence: 99%

“…This scheme is nothing more than the linearized version of the Zakharov-Shabat dressing procedure [26]. This version, in fact, was introduced for a first time in 1974 [25] and developed later in [18].…”

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confidence: 99%

Fermi–Pasta–Ulam recurrence and modulation instability

Kuznetsov

2017

Jetp Lett.

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We give a qualitative explanation of the analog of the Fermi-Pasta-Ulam (FPU) recurrence in a one-dimensional focusing nonlinear Schrodinger equation (NLSE). That recurrence can be considered as a result of the nonlinear development of modulation instability. All known exact localized solitonstype solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. This is the analog of the FPU recurrence for the NLSE. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate but for more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. This instability at the linear stage does not provide the cnoidal wave recurrence. The recurrence happens at the nonlinear stage of the modulation instability. From the practical point of view the latter property is very important, especially for the fiber communication systems which use soliton as an information carrier.1. The phenomenon of recurrence in nonlinear systems with many degrees of freedom was first observed in numerical experiment by Fermi, Pasta and Ulam [1] in 1954. The idea of Fermi was to address how randomization due to the nonlinear interaction leads to the energy equipartition between large number of degrees of freedom in the mechanical chain. The chain in [1] had a quadratic nonlinearity and included 64 oscillators supplemented with long-wave initial conditions. Instead of the energy equipartition, numerical experiments showed that after a finite time a recurrence to the initial data was achieved accompanied by a quasi-periodic energy exchange between several initially exited modes. That recurrence phenomenon became known as the Fermi-PastaUlam (FPU) problem and has been one of the most attractive subjects for numerous investigations. Later, mainly by efforts of N. Zabusky, these results were reproduced by means of more powerful computers. Besides, there were observed many other peculiarities in this problem (for details, see the original papers by Zabusky Since the discovery of the Inverse Scattering Transform (IST), which was first applied to the KDV equation by Gardner, Greene, Kruskal and Miura [5], and later to the nonlinear Schrodinger equation (NLSE) by Zakharov and Shabat [6], many aspects of the FPU recurrence became more clear. In 1971 Zakharov and Faddeev [7] proved that the KDV equation, which, in particular, can be obtained from the FPU system in the continuous limit for waves propagated in one direction...

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“…Thus, for fixed values of the gain and loss parameters α, γ and Γ, the stability of the periodic waves is determined by u 0 and m, the latter constant being related to the period L, see eqs. (8). This suggests to display stability regions for the CnW solutions in the (L, u 0 ) parametric plane, if the values of α, γ and Γ are fixed, which will be done below.…”

Section: §1 Introductionmentioning

confidence: 99%

“…In particular, the stability of periodic waves in perturbed Korteweg -de Vries (KdV) and Benjamin-Ono equations was investigated, respectively, in refs. [8]- [11] and [12,13].…”

Section: §1 Introductionmentioning

confidence: 99%

Stable Periodic Waves in Coupled Kuramoto–Sivashinsky–Korteweg–de Vries Equations

2002

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Periodic waves are investigated in a system composed of a Kuramoto -Sivashinsky -Korteweg -de Vries (KS-KdV) equation linearly coupled to an extra linear dissipative one. The model describes, e.g., a two-layer liquid film flowing down an inclined plane. It has been recently shown that the system supports stable solitary pulses. We demonstrate that a perturbation analysis, based on the balance equation for the net field momentum, predicts the existence of stable cnoidal waves (CnWs) in the same system. It is found that the mean value u 0 of the wave field u in the main subsystem, but not the mean value of the extra field, affects the stability of the periodic waves. Three different areas can be distinguished inside the stability region in the parameter plane (L,u 0 ), where L is the wave's period. In these areas, stable are, respectively, CnWs with positive velocity, constant solutions, and CnWs with negative velocity. Multistability, i.e., the coexistence of several attractors, including the waves with several maxima per period, appears at large value of L. The analytical predictions are completely confirmed by direct simulations. Stable waves are also found numerically in the limit of vanishing dispersion, when the KS-KdV equation goes over into the KS one.

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On the stability of nonlinear waves in integrable models

Cited by 50 publications

References 2 publications

Soliton stability in plasmas and hydrodynamics

Soliton stability in plasmas and hydrodynamics

Fermi–Pasta–Ulam recurrence and modulation instability

Stable Periodic Waves in Coupled Kuramoto–Sivashinsky–Korteweg–de Vries Equations

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