scholarly journals Rogue waves and downshifting in the presence of damping

2011 ◽  
Vol 11 (2) ◽  
pp. 383-399 ◽  
Author(s):  
A. Islas ◽  
C. M. Schober
Keyword(s):  
Initial Data ◽  
Rogue Waves ◽  
Nonlinear Damping ◽  
Damping Term ◽  
Nls Equation ◽  
Inverse Spectral Theory ◽  
Damping Parameter ◽  
Spectral Splitting ◽  
Linear Damping ◽  
Further Development

Abstract. Recently Gramstad and Trulsen derived a new higher order nonlinear Schrödinger (HONLS) equation which is Hamiltonian (Gramstad and Trulsen, 2011). We investigate the effects of dissipation on the development of rogue waves and downshifting by adding an additonal nonlinear damping term and a uniform linear damping term to this new HONLS equation. We find irreversible downshifting occurs when the nonlinear damping is the dominant damping effect. In particular, when only nonlinear damping is present, permanent downshifting occurs for all values of the nonlinear damping parameter β. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. We examine the generation of rogue waves in the presence of damping for sea states characterized by JONSWAP spectrum. Using the inverse spectral theory of the NLS equation, simulations of the NLS and damped HONLS equations using JONSWAP initial data consistently show that rogue wave events are well predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. We define δcutoff by requiring that 95% of the rogue waves occur for δ < δcutoff. We find that δcutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur. As a result when damping is present the proximity to homoclinic data and instabilities is more crucial for the development of rogue waves.

scholarly journals Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

2014 ◽  
Vol 14 (6) ◽  
pp. 1431-1440 ◽  
Author(s):  
A. Calini ◽  
C. M. Schober
Keyword(s):  
Initial Data ◽  
Numerical Investigation ◽  
Rogue Waves ◽  
Random Perturbations ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Spatial Modes ◽  
Large Ensembles ◽  
Wave Forms ◽  
The One

Abstract. In this article we conduct a broad numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, a widely used model of rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently used to model rogue waves. However, the issue of whether these solutions are robust with respect to the kind of random perturbations occurring in physical settings and laboratory experiments has just recently begun to be addressed. Numerical experiments for spatially periodic breathers with one or two modes involving large ensembles of perturbed initial data for six typical random perturbations suggest interesting conclusions. Breathers over an unstable background with N unstable modes are generally unstable to small perturbations in the initial data unless they are "maximal breathers" (i.e., they have N spatial modes). Additionally, among the maximal breathers with two spatial modes, the one of highest amplitude due to coalescence of the modes appears to be the most robust. The numerical observations support and extend to more realistic settings the results of our previous stability analysis, which we hope will provide a useful tool for identifying physically realizable wave forms in experimental and observational studies of rogue waves.

Routes to stability for spatially periodic breather solutions of a damped NLS equation.

2020 ◽  
Author(s):  
Constance Schober
Keyword(s):  
Nonlinear Damping ◽  
Heteroclinic Orbits ◽  
Mean Flow ◽  
Nls Equation ◽  
Wave Dynamics ◽  
Finite Dimensional ◽  
Stokes Waves ◽  
Spatially Periodic ◽  
Dimensional Dynamical System ◽  
Linear Damping

&lt;p&gt;&amp;#160;The spatially periodic breather solutions (SPBs) of the nonlinear Schr&amp;#246;dinger (NLS) equation, i.e. the heteroclinic orbits of unstable Stokes waves, are typically unstable. In this talk&amp;#160; we examine&amp;#160; the effects of dissipation on the &amp;#160;one- mode SPBs&amp;#160; U&lt;sup&gt;(j)&lt;/sup&gt;(x,t) as well as multi-mode SPBs U&lt;sup&gt;(j,k)&lt;/sup&gt;(x,t)&amp;#160;using a damped&amp;#160; NLS equation which incorporates both uniform linear damping and nonlinear damping&amp;#160; of the mean flow,&lt;br&gt;for a range of parameters typically encountered in experiments. The damped wave dynamics is viewed as near integrable, allowing one to use the spectral theory of the NLS equation to interpret the perturbed flow. A broad categorization of how the route to stability for the SPBs&amp;#160; depends on the mode structure of the SPB and whether the damping is linear or nonlinear is obtained&amp;#160;&lt;br&gt;as well as the distinguishing features of the stabilized state.&amp;#160; Time permitting, a reduced, finite dimensional dynamical system that goverms the linearly damped SPBs will be presented&amp;#160;&lt;/p&gt;

The Damping Term, from First Principles

2017 ◽  
Author(s):  
Olle Eriksson ◽  
Anders Bergman ◽  
Lars Bergqvist ◽  
Johan Hellsvik
Keyword(s):  
Density Functional Theory ◽  
First Principles ◽  
Density Functional ◽  
Spin Dynamics ◽  
Damping Term ◽  
Functional Theory ◽  
Basic Principles ◽  
Sham Equation ◽  
Damping Parameter ◽  
Simulation Cell

In the previous chapters we described the basic principles of density functional theory, gave examples of how accurate it is to describe static magnetic properties in general, and derived from this basis the master equation for atomistic spin-dynamics; the SLL (or SLLG) equation. However, one term was not described in these chapters, namely the damping parameter. This parameter is a crucial one in the SLL (or SLLG) equation, since it allows for energy and angular momentum to dissipate from the simulation cell. The damping parameter can be evaluated from density functional theory, and the Kohn-Sham equation, and it is possible to determine its value experimentally. This chapter covers in detail the theoretical aspects of how to calculate theoretically the damping parameter. Chapter 8 is focused, among other things, on the experimental detection of the damping, using ferromagnetic resonance.

Experimental investigation on the evolution of the modulation instability with dissipation

2012 ◽  
Vol 711 ◽  
pp. 101-121 ◽  
Author(s):  
Y. Ma ◽  
G. Dong ◽  
M. Perlin ◽  
X. Ma ◽  
G. Wang
Keyword(s):  
Experimental Investigation ◽  
Modulational Instability ◽  
Experimental Results ◽  
Damping Term ◽  
Wave Flume ◽  
Perturbation Frequency ◽  
Linear Damping ◽  
Wave Trains ◽  
Momentum Integral ◽  
The Waves

AbstractAn experimental investigation focusing on the effect of dissipation on the evolution of the Benjamin–Feir instability is reported. A series of wave trains with added sidebands, and varying initial steepness, perturbed amplitudes and frequencies, are physically generated in a long wave flume. The experimental results directly confirm the stabilization theory of Segur et al. (J. Fluid Mech., vol. 539, 2005, pp. 229–271), i.e. dissipation can stabilize the Benjamin–Feir instability. Furthermore, the experiments reveal that the effect of dissipation on modulational instability depends strongly on the perturbation frequency. It is found that the effect of dissipation on the growth rates of the sidebands for the waves with higher perturbation frequencies is more evident than on those of waves with lower perturbation frequencies. In addition, numerical simulations based on Dysthe’s equation with a linear damping term included, which is estimated from the experimental data, can predict the experimental results well if the momentum integral of the wave trains is conserved during evolution.

A variable coefficient nonlinear Schrödinger equation: localized waves on the plane wave background and their dynamics

2019 ◽  
Vol 33 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Variable Coefficient ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Plane Wave Background ◽  
Localized Waves ◽  
Similarity Reductions

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.

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