Modulational instability in asymmetric coupled wave functions

Kourakis, Ioannis; Shukła, P. K.

doi:10.1140/epjb/e2006-00106-1

Cited by 13 publications

(22 citation statements)

References 18 publications

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“…Note also that we reproduce the results of Kourakis and Shukla [17] in the local limit (N = κ −2 ). At the same time, the availability of an additional degree of freedom in the form of nonlocality enables us to manage the conditions of stability/instability.…”

Section: Summary Of the Analysissupporting

confidence: 81%

“…SinceR(κ) → 1 for σ → 0, the expression (6) reduces in this limit to the well-known dispersion relation for the local model [15,16,17].…”

Section: Modulational Instability: General Analysismentioning

confidence: 96%

“…will follow [32,17] and give a geometrical description of the nonlocality management in the case of the group-velocity mismatch. Let us write the dispersion relation (6) in the form…”

Section: Group-velocity Mismatch and Nonlocality Managementmentioning

confidence: 99%

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Modulational instability and nonlocality management in coupled NLS systems

Doktorov¹,

Molchan²

2007

Phys. Scr.

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Abstract. The modulational instability of two interacting waves in a nonlocal Kerrtype medium is considered analytically and numerically. For a generic choice of wave amplitudes, we give a complete description of stable/unstable regimes for zero groupvelocity mismatch. It is shown that nonlocality suppresses considerably the growth rate and bandwidth of instability. For nonzero group-velocity mismatch we perform a geometrical analysis of a nonlocality management which can provide stability of waves otherwise unstable in a local medium.

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Section: Summary Of the Analysissupporting

confidence: 81%

“…SinceR(κ) → 1 for σ → 0, the expression (6) reduces in this limit to the well-known dispersion relation for the local model [15,16,17].…”

Section: Modulational Instability: General Analysismentioning

confidence: 96%

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Modulational instability and nonlocality management in coupled NLS systems

Doktorov¹,

Molchan²

2007

Phys. Scr.

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“…Indeed, based on the results of Ref. 25 the coupled system exhibits MI much faster than its scalar counterpart, i.e. has higher growth rates.…”

mentioning

confidence: 98%

“…For the scalar NLS equation, when dispersion and nonlinearity share the same signs, the equation is said to be focusing, and the equation is modulationaly unstable; it is stable when the signs are opposite and the equation is said to be defocusing. The stability criteria are more involved for the coupled system 25 .…”

mentioning

confidence: 99%

Interacting nonlinear wave envelopes and rogue wave formation in deep water

Ablowitz

Horikis

2015

Physics of Fluids

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A rogue wave formation mechanism is proposed within the framework of a coupled nonlinear Schrödinger (CNLS) system corresponding to the interaction of two waves propagating in oblique directions in deep water. A rogue condition is introduced that links the angle of interaction with the group velocities of these waves: different angles of interaction can result in a major enhancement of rogue events in both numbers and amplitude. For a range of interacting directions it is found that the CNLS system exhibits significantly more extreme wave amplitude events than its scalar counterpart. Furthermore, the rogue events of the coupled system are found to be well approximated by hyperbolic secant functions; they are vectorial soliton-type solutions of the CNLS system, typically not considered to be integrable. Overall, our results indicate that crossing states provide an important mechanism for the generation of rogue water wave events.PACS numbers: 02.60.Cb Even in the age of modern ship building and innovative navigation equipment, ship accidents are an important area of maritime concern. In recent years researchers have been investigating a topic which heretofore had been reserved for marine folklore: giant waves that seem to appear from nowhere in high seas that can lead to disastrous outcomes. These waves have, nowadays, been documented to exist and are usually referred to as rogue or freak waves.The conditions that cause such waves to grow enormously in size are a topic of great interest with many different hypotheses being proposed 1,2 . An important one is the nonlinear mechanism of the self-wave interactions, such as modulation instability; in water wave physics this is called the Benjamin-Feir instability 3 and has been found under certain conditions to produce significant wave amplification.The envelope of nonlinear water waves, under suitable conditions, is modeled by a nonlinear Schrödinger equation (NLS) 4 . Interestingly enough, the NLS not only gives a suitable description of these waves 2,5-9 , it is also an important equation used to investigate propagation of pulses in many other physical systems such as nonlinear optical fibers 10-13 , Bose-Einstein condensates 14 , magnetic spin waves 15 and many others. The state of the sea in which rogue waves form is often complex 16-18 with certain key wave interactions dominating the wave structure. Here we focus on a coupled NLS (CNLS) system, derivable directly from the Euler equations. Underlying the wave phenomena are separate wave trains, propagating in different directions which interact at different angles; this provides a useful model of crossing sea states 16,[19][20][21][22] . The scalar NLS equation would be insufficient to describe these phenomena (rogue events associated with 2D scalar NLS equations are discussed a) Electronic mail: Corresponding author: horikis@uoi.gr in Ref.23 ). As with the scalar problem the CNLS system exhibits modulation instability (MI), which as mentioned above, is an important aspect of rogue wave dynamics. Indeed, rec...

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Coupled nonlinear Schrödinger (CNLS) equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

Lazarides,

Kourakis

2023

Nonlinear Dyn

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The nonlinear dynamics of two co-propagating electrostatic wavepackets in a one-dimensional non-magnetized plasma fluid model is considered, from first principles. The coupled waves are characterized by different (carrier) wavenumbers and amplitudes. A plasma consisting of non-thermalized ($$\kappa $$ κ -distributed) electrons evolving against a cold (stationary) ion background is considered. The original model is reduced, by means of a multiple-scale perturbation method, to a pair of coupled nonlinear Schrödinger (CNLS) equations for the dynamics of the wavepacket envelopes. For arbitrary wavenumbers, the resulting CNLS equations exhibit no known symmetry and thus intrinsically differ from the Manakov system, in general. Exact analytical expressions have been derived for the dispersion, self-modulation (nonlinearity) and cross-modulation (coupling) coefficients involved in the CNLS equations, as functions of the wavenumbers ($$k_1$$ k 1 , $$k_2$$ k 2 ) and of the spectral index $$\kappa $$ κ characterizing the electron profile. An analytical investigation has thus been carried out of the modulational instability (MI) properties of this pair of wavepackets, focusing on the role of the intrinsic (variable) parameters. Modulational instability is shown to occur in most parts of the parameter space. The instability window(s) and the corresponding growth rate are calculated numerically in a number of case studies. Two-wave interaction favors MI by extending its range of occurrence and by enhancing its growth rate. Growth rate patterns obtained for different $$\kappa $$ κ index (values) suggest that deviation from thermal (Maxwellian) equilibrium, for low $$\kappa $$ κ values, leads to enhance MI of the interacting wave pair. To the best of our knowledge, the dynamics of two co-propagating wavepackets in a plasma described by a fluid model with $$\kappa $$ κ -distributed electrons is investigated thoroughly with respect to their MI properties as a function of $$\kappa $$ κ for the first time, in the framework of an asymmetric CNLS system whose coefficients present no obvious symmetries for arbitrary $$k_1$$ k 1 and $$k_2$$ k 2 . Although we have focused on electrostatic wavepacket propagation in non-thermal (non-Maxwellian) plasma, the results of this study are generic and may be used as basis to model energy localization in nonlinear optics, in hydrodynamics or in dispersive media with Kerr-type nonlinearities where MI is relevant.

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Modulational instability in asymmetric coupled wave functions

Cited by 13 publications

References 18 publications

Modulational instability and nonlocality management in coupled NLS systems

Modulational instability and nonlocality management in coupled NLS systems

Interacting nonlinear wave envelopes and rogue wave formation in deep water

Coupled nonlinear Schrödinger (CNLS) equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model

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