Efficient Fourier-collocation method for full scattering data of Zakharov-Shabat periodic problem

2021 ◽  
Author(s):  
Ilya Mullyadzhanov ◽  
Rustam Mullyadzhanov ◽  
Andrey Gelash
Keyword(s):  
Wave Field ◽  
Collocation Method ◽  
Nonlinear Wave ◽  
Periodic Problem ◽  
Physical Review ◽  
Scattering Data ◽  
Eigenvalue Spectrum ◽  
Plane Wave Solution ◽  
Wave Fields ◽  
Fourier Collocation

<p>The one-dimensional nonlinear Schrodinger equation (NLSE) serves as a universal model of nonlinear wave propagation appearing in different areas of physics. In particular it describes weakly nonlinear wave trains on the surface of deep water and captures up to certain extent the phenomenon of rogue waves formation. The NLSE can be completely integrated using the inverse scattering transform method that allows transformation of the wave field to the so-called scattering data representing a nonlinear analogue of conventional Fourier harmonics. The scattering data for the NLSE can be calculated by solving an auxiliary linear system with the wave field playing the role of potential – the so-called Zakharov-Shabat problem. Here we present a novel efficient approach for numerical computation of scattering data for spatially periodic nonlinear wave fields governed by focusing version of the NLSE. The developed algorithm is based on Fourier-collocation method and provides one an access to full scattering data, that is main eigenvalue spectrum (eigenvalue bands and gaps) and auxiliary spectrum (specific phase parameters of the nonlinear harmonics) of Zakharov-Shabat problem. We verify the developed algorithm using a simple analytic plane wave solution and then demonstrate its efficiency with various examples of large complex nonlinear wave fields exhibiting intricate structure of bands and gaps. Special attention is paid to the case when the wave field is strongly nonlinear and contains solitons which correspond to narrow gaps in the eigenvalue spectrum, see e.g. [1], when numerical computations may become unstable [2]. Finally we discuss applications of the developed approach for analysis of numerical and experimental nonlinear wave fields data.</p><p>The work was supported by Russian Science Foundation grant No. 20-71-00022.</p><p>[1] A. A. Gelash and D. S. Agafontsev, Physical Review E 98, 042210 (2018).</p><p>[2] A. Gelash and R. Mullyadzhanov, Physical Review E 101, 052206 (2020).</p>

Nonlinear phase-resolved reconstruction of irregular water waves

2018 ◽  
Vol 838 ◽  
pp. 544-572 ◽  
Author(s):  
Yusheng Qi ◽  
Guangyu Wu ◽  
Yuming Liu ◽  
Moo-Hyun Kim ◽  
Dick K. P. Yue
Keyword(s):  
Wave Field ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Three Dimensional ◽  
Reconstruction Error ◽  
High Order ◽  
Order Effects ◽  
Wave Fields ◽  
Wave Measurements ◽  
Wave Nonlinearity

We develop and validate a high-order reconstruction (HOR) method for the phase-resolved reconstruction of a nonlinear wave field given a set of wave measurements. HOR optimizes the amplitude and phase of $L$ free wave components of the wave field, accounting for nonlinear wave interactions up to order $M$ in the evolution, to obtain a wave field that minimizes the reconstruction error between the reconstructed wave field and the given measurements. For a given reconstruction tolerance, $L$ and $M$ are provided in the HOR scheme itself. To demonstrate the validity and efficacy of HOR, we perform extensive tests of general two- and three-dimensional wave fields specified by theoretical Stokes waves, nonlinear simulations and physical wave fields in tank experiments which we conduct. The necessary $L$, for general broad-banded wave fields, is shown to be substantially less than the free and locked modes needed for the nonlinear evolution. We find that, even for relatively small wave steepness, the inclusion of high-order effects in HOR is important for prediction of wave kinematics not in the measurements. For all the cases we consider, HOR converges to the underlying wave field within a nonlinear spatial-temporal predictable zone ${\mathcal{P}}_{NL}$ which depends on the measurements and wave nonlinearity. For infinitesimal waves, ${\mathcal{P}}_{NL}$ matches the linear predictable zone ${\mathcal{P}}_{L}$, verifying the analytic solution presented in Qi et al. (Wave Motion, vol. 77, 2018, pp. 195–213). With increasing wave nonlinearity, we find that ${\mathcal{P}}_{NL}$ contains and is generally greater than ${\mathcal{P}}_{L}$. Thus ${\mathcal{P}}_{L}$ provides a (conservative) estimate of ${\mathcal{P}}_{NL}$ when the underlying wave field is not known.

Prediction of Oceanic Rogue Waves Through Tracking Energy Fluxes

2017 ◽  
Author(s):  
Qiuchen Guo ◽  
Mohammad-Reza Alam
Keyword(s):  
Spatial Distribution ◽  
Wave Field ◽  
Spectral Method ◽  
Nonlinear Wave ◽  
Statistical Approach ◽  
Preventive Measures ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Wave Fields ◽  
Weakly Nonlinear

Here, we show that location of an upcoming rogue wave can be inferred, well in advance, from spatial distribution of energy flux across the ocean surface. We use a statistical approach, and by investigating hundreds of numerical rogue wave realizations in weakly nonlinear wave fields establish a quantitative metric via which predictions can be made. Direct simulations are performed by a higher-order spectral method (HOS), and JONSWAP distribution is used to initialize the wave field. The presented metric may establish a readily achievable measure to identify turbulent locations within a sea, through which timely preventive measures can be taken to minimize damages to lives and properties.

scholarly journals Null modes effect in Rossby wave model

2004 ◽  
Vol 11 (3) ◽  
pp. 281-293
Author(s):  
V. Goncharov ◽  
V. Pavlov
Keyword(s):  
Nonlinear Wave ◽  
Rossby Wave ◽  
Wave Model ◽  
Wave Fields

Abstract. The problem of the null-modes existence and some particularities of their interaction with nonlinear vortex-wave-like structures is discussed. We show that the null-modes are fundamental elements of nonlinear wave fields. The conditions under which null-modes can manifest themselves are elucidated. The Rossby-Hasegawa-Mima (RHM) model is used for the illustration of features of null-modes-waves interactions.

Interaction of High Frequency Lamb Waves with Surface-Mount Sensor Adhesives

2013 ◽  
Vol 558 ◽  
pp. 489-500 ◽  
Author(s):  
Patrick Norman ◽  
Claire Davis ◽  
Cédric Rosalie ◽  
Nik Rajic
Keyword(s):  
Wave Field ◽  
Lamb Waves ◽  
Base Material ◽  
Adhesive Layer ◽  
Scattered Wave ◽  
Phase Lag ◽  
Wave Fields ◽  
Surface Mount ◽  
Frequency Regime ◽  
Fibre Bragg Gratings

The application of Lamb waves to damage and/or defect detection in structures is typicallyconfined to lower frequencies in regimes where only the lower order modes propagate in order to simplifyinterpretation of the scattered wave-fields. Operation at higher frequencies offers the potentialto extend the sensitivity and diagnostic capability of this technique, however there are technical challengesassociated with the measurement and interpretation of this data. Recent work by the authorshas demonstrated the ability of fibre Bragg gratings (FBGs) to measure wave-fields at frequencies inexcess of 2 MHz [1]. However, when this work was extended to other thinner plate specimens it wasfound that at these higher frequencies, the cyanoacrylate adhesive (M-Bond 200) used to attach theFBG sensors to the plate was significantly affecting the propagation of the waves. Laser vibrometrywas used to characterise the wave-field in the region surrounding the adhesive and it was found that theself-adhesive retro-reflective tape applied to aid with this measurement was also affecting the wavefieldin the higher frequency regime. This paper reports on an experimental study into the influence ofboth of these materials on the propagating wave-field. Three different lengths of retro-reflective tapewere placed in the path of Lamb waves propagating in an aluminium plate and laser vibrometry wasused to measure the wave-field upstream and downstream of the tape for a range of different excitationfrequencies. The same experiment was conducted using small footprint cyanoacrylate film samplesof different thickness. The results show that both of these surface-mount materials attenuate, diffractand scatter the incoming waves as well as introducing a phase lag. The degree of influence of thesurface layer appears to be a function of its material properties, the frequency of the incoming waveand the thickness and footprint of the surface layer relative to the base material thickness. Althoughfurther work is required to characterise the relative influence of each of these variables, investigationsto date show that for the measurement of Lamb Waves on thin structures, careful considerationshould be given to the thickness and footprint of the adhesive layer and sensor, particularly in the highfrequency regime, so as to minimise their effect on the measurement.

Energy Content Characterization of Water Waves Using Linear and Nonlinear Spectral Analysis

2021 ◽  
pp. 1-30
Author(s):  
Ali Mohtat ◽  
Solomon Yim ◽  
Alfred R. Osborne
Keyword(s):  
Wave Field ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Ad Hoc ◽  
Energy Content ◽  
Energy Calculation ◽  
Linear And Nonlinear ◽  
Definition Of ◽  
Exact Energy ◽  
Wave Modes

Abstract The survivability, safe operation, and design of marine vehicles and wave energy converters are highly dependent on accurate characterization and estimation of the energy content of the ocean wave field. In this study, analytical solutions of the nonlinear Schrödinger equation (NLS) using periodic inverse scattering transformation (IST) and its associated Riemann spectrum are employed to obtain the nonlinear wave modes (eigen functions of the nonlinear equation consisting of multiple phase-locked harmonic components). These nonlinear wave modes are used in two approaches to develop a more accurate definition of the energy content. First, in an ad hoc approach, the amplitudes of the nonlinear wave modes are used with a linear energy calculation resulting in a semi-linear energy estimate. Next, a novel, mathematically exact definition of the energy content taking into account the nonlinear effects up to fifth order is introduced in combination with the nonlinear wave modes, the exact energy content of the wave field is computed. Experimental results and numerical simulations were used to compute and analyze the linear, ad hoc, and exact energy contents of the wave field, using both linear and nonlinear spectra. The ratio of the ad hoc and exact energy estimates to the linear energy content were computed to examine the effect of nonlinearity on the energy content. In general, an increasing energy ratio was observed for increasing nonlinearity of the wave field, with larger contributions from higher-order harmonic terms. It was confirmed that the significant increase in nonlinear energy content with respect to its linear counterpart is due to the increase in the number of nonlinear phase-locked (bound wave) modes.

scholarly journals Statistical properties of nonlinear one-dimensional wave fields

2005 ◽  
Vol 12 (5) ◽  
pp. 671-689 ◽  
Author(s):  
D. Chalikov
Keyword(s):  
Surface Waves ◽  
Nonlinear Wave ◽  
High Accuracy ◽  
Statistical Properties ◽  
Statistical Characteristics ◽  
Small Scale ◽  
Sea Waves ◽  
Wave Fields ◽  
Linear Waves ◽  
Stokes Waves

Abstract. A numerical model for long-term simulation of gravity surface waves is described. The model is designed as a component of a coupled Wave Boundary Layer/Sea Waves model, for investigation of small-scale dynamic and thermodynamic interactions between the ocean and atmosphere. Statistical properties of nonlinear wave fields are investigated on a basis of direct hydrodynamical modeling of 1-D potential periodic surface waves. The method is based on a nonstationary conformal surface-following coordinate transformation; this approach reduces the principal equations of potential waves to two simple evolutionary equations for the elevation and the velocity potential on the surface. The numerical scheme is based on a Fourier transform method. High accuracy was confirmed by validation of the nonstationary model against known solutions, and by comparison between the results obtained with different resolutions in the horizontal. The scheme allows reproduction of the propagation of steep Stokes waves for thousands of periods with very high accuracy. The method here developed is applied to simulation of the evolution of wave fields with large number of modes for many periods of dominant waves. The statistical characteristics of nonlinear wave fields for waves of different steepness were investigated: spectra, curtosis and skewness, dispersion relation, life time. The prime result is that wave field may be presented as a superposition of linear waves is valid only for small amplitudes. It is shown as well, that nonlinear wave fields are rather a superposition of Stokes waves not linear waves. Potential flow, free surface, conformal mapping, numerical modeling of waves, gravity waves, Stokes waves, breaking waves, freak waves, wind-wave interaction.

Determination of the Wave Field from Scattering Data

1986 ◽  
Vol 57 (7) ◽  
pp. 783-786 ◽  
Author(s):  
James H. Rose ◽  
Margaret Cheney ◽  
Brian DeFacio
Keyword(s):  
Wave Field ◽  
Scattering Data
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