arXiv subjects

C. M. Schober

Publications and source records attributed to C. M. Schober.

6 recordsLinked to original sources

arXiv2026

Interaction-Phase Dynamics and Spectral Organization in Damped Higher-Order Nonlinear Schrödinger Models

We investigate the dynamical mechanisms underlying contrasting nonlinear Floquet spectral evolutions in viscously damped and nonlinear mean-flow damped higher-order nonlinear Schrödinger models. A reduced five-mode carrier-sideband truncation is derived in amplitude-phase variables to isolate principal interaction phases associated with dominant four-wave interaction products. Within this framework, viscous damping acts modewise without explicitly contributing to leading interaction-phase equations, whereas nonlinear mean-flow damping acts through coupled mean-flow interactions and generates terms of the form $-κ_j \sin(ψ_j)$ in the carrier-sideband regime. The reduced system serves as a mechanism-identification model. To interpret the interaction-phase evolution, we examine recurrent NLS benchmark solutions whose modulation dynamics and Floquet spectrum are independently characterized, showing that substantial interaction-phase evolution can coexist with recurrent modulation and invariant Floquet-band structure. Full-PDE diagnostics compare phase dynamics within spectral regimes identified for both damped systems. Under nonlinear mean-flow damping, interaction phases develop persistent large-scale evolution, with larger phase changes remaining tied to recurrent focusing events while the Floquet-band configuration persists. Under viscous damping, large-scale phase trends change repeatedly without systematic association with recurrent focusing, while the Floquet spectrum undergoes repeated critical-point crossings and band reconfiguration. A Fourier-energy assessment confirms this contrast is not simply due to broader Fourier mode excitation in the viscous system, supporting a structural distinction in how the two dissipative mechanisms act on dominant carrier-sideband interactions.

nlin.PS↗

arXiv2026

Higher-Order Extensions of Weakly Viscous Dysthe Theory and a Phase-Lag Model for Nonlinear Mean-Flow Damping

We extend the Carter-Govan multiple-scales analysis of weakly viscous, narrowband deep-water wave packets beyond Dysthe order within the potential-flow reduction of Dias, Dyachenko, and Zakharov (DDZ). We seek to determine whether this framework generates the complex multiplier $(1 + iβ)$ used phenomenologically to modify the nonlocal Dysthe mean-flow interaction. Although order counting places a direct viscous carrier-mean interaction at sixth order, it does not exclude an indirect fifth-order contribution arising from viscosity dependent lower-order harmonics and nonlinear interactions. We therefore derive the first correction to the induced mean flow and the complete fifth-order first-harmonic solvability condition. The resulting nonlocal terms are derivative--dependent and contain no explicit viscosity, excluding the proposed indirect mechanism within the DDZ framework. At sixth order, a restricted calculation of the nonlinear viscous block isolates a direct carrier-mean contribution with the same operator structure as the imaginary component of the prescribed mean-flow correction. Independently, a finite-adjustment-time model yields an exact, frequency dependent mean-flow response. Its low-frequency expansion produces the multiplier $ 1 + iβ_{\mathrm{eff}}(Ω)$, with $β_{\mathrm{eff}}(Ω) =Ωτ.$ When $Ωτ= \mathcal O(ε)$, the resulting phase-lag correction enters at fifth order, one order beyond the leading Dysthe mean-flow interaction.

physics.flu-dyn↗

arXiv2025

Soliton-like Rogue Wave Dynamics in Dissipative Higher-Order NLS Models: A Floquet Spectral Perspective

We investigate rogue wave formation and spectral downshifting in the higher-order nonlinear Schrödinger (HONLS) equation and its dissipative extensions: the nonlinear mean-flow damping model (NLD-HONLS) and the viscous damping model (V-HONLS). By applying Floquet spectral analysis, we characterize i) the structural organization of the dynamical background and ii) the nature of the rogue waves that appear, distinguishing sharply localized, soliton-like structures from more diffuse, spatially extended waveforms with mixed mode characteristics. In the conservative HONLS, soliton-like rogue waves (SRWs) arise only for steep initial data, with the dynamics intermittently switching between periods of SRW formation and periods dominated by a disordered multi-mode background. For moderately steep initial data, only broader, less coherent rogue waves form. Nonlinear damping in the NLD-HONLS model suppresses disorder and supports a stable, well-organized Floquet spectra that reflects a sustained soliton-like state from which SRWs emerge, along with strong phase coherence. In contrast, viscous damping in the V-HONLS model leads to a disordered Floquet spectral evolution with broader, less localized rogue waves and increased phase variability. Furthermore, the NLD-HONLS model shows a close link between rogue wave events and the time of permanent downshift, whereas these phenomena appear decoupled in the V-HONLS model. These results clarify how dissipation type and wave steepness interact to shape extreme events in near-integrable wave systems and highlight the value of spectral diagnostics for studying nonlinear wave dynamics.

nlin.PS↗

arXiv2023

The Effects of Viscosity on the Linear Stability of Damped Stokes Waves, Downshifting, and Rogue Wave Generation

We investigate a higher order nonlinear Schrödinger equation with linear damping and weak viscosity, recently proposed as a model for deep water waves exhibiting frequency downshifting. Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift in the spectral peak. The novel results in this work include the analysis of the transition from the initial Benjamin-Feir instability to a predominantly oscillatory behavior, which takes place in a time interval when most rogue wave activity occurs. In addition, we propose new criteria for downshifting in the spectral peak and determine the relation between the time of permanent downshift and the location of the global minimum of the momentum and the magnitude of its second derivative.

nlin.PS↗

arXiv2022

Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger equation, prominent in modeling rogue waves, are unstable. In this paper we numerically investigate the effects of nonlinear dissipation and higher order nonlinearities on the routes to stability of the SPBs in the framework of the nonlinear damped higher order nonlinear Schrödinger (NLD-HONLS) equation. The initial data used in the experiments are generated by evaluating exact SPB solutions at time $T_0$. The number of instabilities of the background Stokes wave and the damping strength are varied. The Floquet spectral theory of the NLS equation is used to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. Significantly, as $T_0$ is varied, tiny bands of complex spectrum are observed to pinch off in the Floquet decomposition of the NLD-HONLS data, reflecting the breakup of the SPB into a waveform that is close to either a one or two "soliton-like" structure. For wide ranges of $T_0$, i.e. for solutions initialized in the early to middle stage of the development of the MI, all rogue waves are observed to occur when the spectrum is close to a one or two soliton-like state. When the solutions are initialized as the MI is saturating, rogue waves also can occur after the spectrum has left a soliton-like state. Other novel features arise due to nonlinear damping: enhanced asymmetry, two timescales in the evolution of the spectrum and a delay in the growth of instabilities due to frequency downshifting.

nlin.PS↗

arXiv2004

Local Lagrangian Formalism and Discretization of the Heisenberg Magnet Model

In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of Nöther's theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle $N = M\times S^2$ over an appropriate space-time $M$. Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.

physics.comp-ph↗