scholarly journals An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

2015 ◽  
Vol 22 (3) ◽  
pp. 313-324 ◽  
Author(s):  
I. V. Shugan ◽  
H. H. Hwung ◽  
R. Y. Yang
Keyword(s):  
Schrödinger Equation ◽  
Spatial Scale ◽  
Schrodinger Equation ◽  
Wave Interaction ◽  
Experimental Results ◽  
Stokes Wave ◽  
Initial State ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Cubic Schrödinger Equation

Abstract. An analytical weakly nonlinear model of the Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly varying steady current. In contrast to the models based on versions of the cubic Schrödinger equation, the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. The spatial scale of the current variation is assumed to have the same order as the spatial scale of the Benjamin–Feir (BF) instability. The model includes wave action conservation law and nonlinear dispersion relation for each of the wave's triad. The effect of nonuniform current, apart from linear transformation, is in the detuning of the resonant interactions, which strongly affects the nonlinear evolution of the system. The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, resulting in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the waves accompanied by energy income from sufficiently strong opposing current. We employ the model for the estimation of the maximum amplification of wave amplitudes as a function of opposing current value and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation.

Benjamin-Feir Wave Instability on the Opposite Current

2016 ◽  
Author(s):  
Igor V. Shugan ◽  
Hwung-Hweng Hwung ◽  
Ray-Yeng Yang
Keyword(s):  
Wave Interaction ◽  
Stokes Wave ◽  
Initial State ◽  
Weakly Nonlinear ◽  
Initial Wave ◽  
Steady Current ◽  
Cubic Schrödinger Equation ◽  
Energy Peak ◽  
Waves Interaction ◽  
Weakly Nonlinear Model

An analytical weakly nonlinear model of Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly-varying steady current. In contrast to the models based on versions of the cubic Schrodinger equation the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, results in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current.

On the modulation of water waves in the neighbourhood of kh ≈ 1.363

1977 ◽  
Vol 357 (1689) ◽  
pp. 131-141 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Multiple Scales ◽  
Wave Interaction ◽  
Periodic Wave ◽  
Stokes Wave ◽  
Modulation Equation ◽  
Wave Trains ◽  
The Stability

In 1967, T. Brooke Benjamin showed that periodic wave-trains on the surface of water could be unstable. If the undisturbed depth is h , and k is the wavenumber of the fundamental, then the Stokes wave is unstable if kh ≥ σ 0 , where σ 0 ≈ 1.363. The instability is provided by the growth of waves with a wavenumber close to k . This result is associated with an almost resonant quartet wave interaction and can be obtained by examining the cubic nonlinearity in the nonlinear Schrodinger equation for the modulation of harmonic water waves: this term vanishes at kh = cr0. In this paper the multiple-scales technique is adapted in order to derive the appropriate modulation equation for the amplitude of the fundamental when kh is near to σ 0 . The resulting equation takes the form i A T - a 1 A ζζ - a 2 A | A | 2 + a 3 A | A | 4 + i( a 4 | A | 2 A ζ - a 5 A (| A | 2 ) ζ ) - a 6 Aψ T = 0 where ψ ζ = | A | 2 , and the a i are real numbers. [Coefficients a 3 - a 6 are given on kh ≈ 1.363 only.] This equation is uniformly valid in that it reduces to the classical non-linear Schrödinger equation in the appropriate limit and is correct when a 2 = 0, i.e. at kh = σ 0 . The equation is used to examine the stability of the Stokes wave and the new inequality for stability is derived: this now depends on the wave amplitude. If the wave is unstable then it is expected that soli to ns will be produced: the simplest form of soliton is therefore examined by constructing the corresponding ordinary differential equation. Some comments are made concerning the phase-plane of this equation, but more analytical details are extracted by treating the new terms as perturbations of the classical Schrodinger soliton. It is shown that the soliton is both flatter (symmetrically) and skewed forward, although the skewing eventually gives way to an oscillation above the mean level.

A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation

1985 ◽  
Vol 150 ◽  
pp. 395-416 ◽  
Author(s):  
Edmond Lo ◽  
Chiang C. Mei
Keyword(s):  
Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Numerical Study ◽  
Spectral Band ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
Cubic Schrödinger Equation ◽  
Slow Evolution ◽  
Fourth Order Equations

In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrödinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

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