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

<p> The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation, i.e. the heteroclinic orbits of unstable Stokes waves, are typically unstable. In this talk  we examine  the effects of dissipation on the  one- mode SPBs  U<sup>(j)</sup>(x,t) as well as multi-mode SPBs U<sup>(j,k)</sup>(x,t) using a damped  NLS equation which incorporates both uniform linear damping and nonlinear damping  of the mean flow,<br>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  depends on the mode structure of the SPB and whether the damping is linear or nonlinear is obtained <br>as well as the distinguishing features of the stabilized state.  Time permitting, a reduced, finite dimensional dynamical system that goverms the linearly damped SPBs will be presented </p>

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.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis

1993 ◽  
Vol 251 ◽  
pp. 21-53 ◽  
Author(s):  
Sergei I. Badulin ◽  
Victor I. Shrira
Keyword(s):  
Shear Flow ◽  
Internal Wave ◽  
Wave Field ◽  
Large Scale ◽  
Spatial Scales ◽  
Energy Concentration ◽  
Local Analysis ◽  
Mean Flow ◽  
Wave Trapping ◽  
Wave Dynamics

The propagation of guided internal waves on non-uniform large-scale flows of arbitrary geometry is studied within the framework of linear inviscid theory in the WKB-approximation. Our study is based on a set of Hamiltonian ray equations, with the Hamiltonian being determined from the Taylor-Goldstein boundary-value problem for a stratified shear flow. Attention is focused on the fundamental fact that the generic smooth non-uniformities of the large-scale flow result in specific singularities of the Hamiltonian. Interpreting wave packets as particles with momenta equal to their wave vectors moving in a certain force field, one can consider these singularities as infinitely deep potential holes acting quite similarly to the ‘black holes’ of astrophysics. It is shown that the particles fall for infinitely long time, each into its own ‘black hole‘. In terms of a particular wave packet this falling implies infinite growth with time of the wavenumber and the amplitude, as well as wave motion focusing at a certain depth. For internal-wave-field dynamics this provides a robust mechanism of a very specific conservative and moreover Hamiltonian irreversibility.This phenomenon was previously studied for the simplest model of the flow non-uniformity, parallel shear flow (Badulin, Shrira & Tsimring 1985), where the term ‘trapping’ for it was introduced and the basic features were established. In the present paper we study the case of arbitrary flow geometry. Our main conclusion is that although the wave dynamics in the general case is incomparably more complicated, the phenomenon persists and retains its most fundamental features. Qualitatively new features appear as well, namely, the possibility of three-dimensional wave focusing and of ‘non-dispersive’ focusing. In terms of the particle analogy, the latter means that a certain group of particles fall into the same hole.These results indicate a robust tendency of the wave field towards an irreversible transformation into small spatial scales, due to the presence of large-scale flows and towards considerable wave energy concentration in narrow spatial zones.

Zero-dimensional covers of finite dimensional dynamical systems

1995 ◽  
Vol 15 (5) ◽  
pp. 939-950 ◽  
Author(s):  
John Kulesza
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

AbstractIf (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

scholarly journals Detiding Shipboard ADCP Data in Eastern Boundary Current

2011 ◽  
Vol 28 (1) ◽  
pp. 94-103 ◽  
Author(s):  
Heriberto Jesus Vazquez ◽  
Jose Gomez-Valdes ◽  
Modesto Ortiz ◽  
Juan Adolfo Dworak
Keyword(s):  
Least Squares ◽  
Baja California ◽  
Gaussian Function ◽  
Kelvin Wave ◽  
Boundary Current ◽  
Mean Flow ◽  
Wave Dynamics ◽  
Eastern Boundary ◽  
Eastern Boundary Current ◽  
Adcp Data

Abstract Spatiotemporal fitting by the least squares method is commonly applied to distinguish the mean flow from the tidal current from shipboard ADCP data in coastal ocean. To analyze this technique in a pelagic region of an eastern boundary current system, a 6-yr period of shipboard ADCP data off Baja California is examined. A diverse set of basis functions is studied and a global tidal model is used for comparison purposes. The Gaussian function together with a nodal configuration of one node in the middle and close to the coast of the region is the best option. However, to obtain the optimal solution, the geostrophic flow, which is stronger than the tidal flow and highly variable off Baja California, might be removed prior to fitting the data. In general, the semimajor axis of the tidal ellipse (M2) is parallel to the coast and the phase speed is poleward and parallel to the coast, in agreement with Kelvin wave dynamics. Because the tides in eastern boundary currents are explained by Kelvin wave dynamics, the use of both the velocity field without geostrophic variability and the Gaussian function in the spatiotemporal fitting by least squares technique is a promising tool for detiding shipboard ADCP data from these systems.

Large-scale flow in turbulent convection: a mathematical model

1986 ◽  
Vol 170 ◽  
pp. 385-410 ◽  
Author(s):  
L. N. Howard ◽  
R. Krishnamurti
Keyword(s):  
Mathematical Model ◽  
Large Scale ◽  
Laboratory Experiments ◽  
Boussinesq Equations ◽  
Time Dependent ◽  
Heteroclinic Orbits ◽  
Turbulent Convection ◽  
Mean Flow ◽  
Dependent Flow ◽  
Large Scale Flow

A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.

scholarly journals Absolute instability in shock-containing jets

2021 ◽  
Vol 930 ◽  
Author(s):  
Petrônio A.S. Nogueira ◽  
Peter Jordan ◽  
Vincent Jaunet ◽  
André V.G. Cavalieri ◽  
Aaron Towne ◽  
...  
Keyword(s):  
Cell Structure ◽  
Saddle Points ◽  
Flow Analysis ◽  
Mode Shapes ◽  
Absolute Instability ◽  
Mean Flow ◽  
Shock Cell ◽  
Spatially Periodic ◽  
Spatio Temporal ◽  
Parallel Case

We present an analysis of the linear stability characteristics of shock-containing jets. The flow is linearised around a spatially periodic mean, which acts as a surrogate for a mean flow with a shock-cell structure, leading to a set of partial differential equations with periodic coefficients in space. Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. The characteristics of the solution are directly compared with the locally parallel case, and some of the features are similar. The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. Analysis of the spatio-temporal growth rates led to the identification of a saddle point between the Kelvin–Helmholtz mode and the guided jet mode, characterising an absolute instability mechanism. Frequencies and mode shapes related to the saddle points for two conditions (associated with axisymmetric and helical modes) are compared with screech frequencies and the most energetic coherent structures of screeching jets, resulting in a good agreement for both. The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. The results suggest that screech can occur in the absence of a nozzle, and that the upstream reflection condition is not essential for screech frequency selection. Connections to previous models are also discussed.

Sign in / Sign up

Export Citation Format

Share Document