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A structure-preserving, second-order-in-time scheme for the von Neumann equation with power nonlinearity

Agissilaos Athanassoulis, Fotini Karakatsani, Irene Kyza

TL;DR

The paper develops a structure-preserving, second-order-in-time relaxation-Crank-Nicolson scheme with fourth-order spatial discretization for the Alber equation, enabling stable, large-scale simulations while preserving a discrete L2 balance law. It demonstrates that the advanced initialization of the auxiliary variable yields correct convergence orders and that the main solution is robust to auxiliary errors. Through linear stability analysis, nonlinear simulations, and Monte Carlo studies, it shows that linear instability predicts the onset and initial growth but not the maximum amplitude or coherent structure formation, revealing a second bifurcation around a background strength C ≈ 1.35 where inhomogeneous features become dominant. The amplification-factor study indicates that the magnitude of fully developed instability depends primarily on the homogeneous background, suggesting a practically relevant threshold for rogue-wave-like events in realistic sea states and highlighting the method’s capacity to probe long-time, large-domain wave dynamics.

Abstract

The von Neumann equation with power nonlinearity serves as a statistical model for nonlinear waves, and it exhibits a bifurcation between stable and unstable regimes. In ocean waves it is known as the Alber equation, and its bifurcation is considered important for understanding rogue waves, a key problem in marine safety. Despite this, only one first-order-in-time numerical method exists in the literature for its simulation. In this paper, we introduce a structure-preserving, linearly implicit, second-order-in-time scheme for its numerical solution. We employ fourth-order finite differences in space, allowing for much larger and faster simulations. As an illustrative example, we explore the onset of modulation instability. We verify that the linear stability analysis accurately predicts the initial growth phase, but find that it fails to forecast the maximum amplitude or the formation of a coherent structure in the nonlinear regime. Monte Carlo simulations with Gaussian background spectra reveal that the maximum amplitude depends solely on the homogeneous background, and not on the initial inhomogeneity. For weak instabilities, the inhomogeneity grows substantially from its initial condition, but it remains small compared to the background. On the other hand, past a critical threshold, the instability grows to yield dominant effects. Thus, beyond the onset of linear instability, the solution of the fully nonlinear problem presents a second bifurcation -- at the point which inhomogeneous features become dominant compared to the background.

A structure-preserving, second-order-in-time scheme for the von Neumann equation with power nonlinearity

TL;DR

The paper develops a structure-preserving, second-order-in-time relaxation-Crank-Nicolson scheme with fourth-order spatial discretization for the Alber equation, enabling stable, large-scale simulations while preserving a discrete L2 balance law. It demonstrates that the advanced initialization of the auxiliary variable yields correct convergence orders and that the main solution is robust to auxiliary errors. Through linear stability analysis, nonlinear simulations, and Monte Carlo studies, it shows that linear instability predicts the onset and initial growth but not the maximum amplitude or coherent structure formation, revealing a second bifurcation around a background strength C ≈ 1.35 where inhomogeneous features become dominant. The amplification-factor study indicates that the magnitude of fully developed instability depends primarily on the homogeneous background, suggesting a practically relevant threshold for rogue-wave-like events in realistic sea states and highlighting the method’s capacity to probe long-time, large-domain wave dynamics.

Abstract

The von Neumann equation with power nonlinearity serves as a statistical model for nonlinear waves, and it exhibits a bifurcation between stable and unstable regimes. In ocean waves it is known as the Alber equation, and its bifurcation is considered important for understanding rogue waves, a key problem in marine safety. Despite this, only one first-order-in-time numerical method exists in the literature for its simulation. In this paper, we introduce a structure-preserving, linearly implicit, second-order-in-time scheme for its numerical solution. We employ fourth-order finite differences in space, allowing for much larger and faster simulations. As an illustrative example, we explore the onset of modulation instability. We verify that the linear stability analysis accurately predicts the initial growth phase, but find that it fails to forecast the maximum amplitude or the formation of a coherent structure in the nonlinear regime. Monte Carlo simulations with Gaussian background spectra reveal that the maximum amplitude depends solely on the homogeneous background, and not on the initial inhomogeneity. For weak instabilities, the inhomogeneity grows substantially from its initial condition, but it remains small compared to the background. On the other hand, past a critical threshold, the instability grows to yield dominant effects. Thus, beyond the onset of linear instability, the solution of the fully nonlinear problem presents a second bifurcation -- at the point which inhomogeneous features become dominant compared to the background.

Paper Structure

This paper contains 17 sections, 3 theorems, 46 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Numerical methods for the von Neumann equation with power nonlinearity (Alber equation): state of the art
  3. Time semi-discrete scheme
  4. The balance laws
  5. The numerical method
  6. Fully discrete scheme
  7. Initialization
  8. Experimental order of convergence
  9. Qualitative behavior of the error
  10. Landau damping and modulation instability
  11. The bifurcation
  12. Numerical investigation of the bifurcation
  13. Monte Carlo investigation of the amplification factors
  14. Discussion and Outlook
  15. The problem and the new numerical method
  16. ...and 2 more sections

Key Result

Proposition 2.1

Consider equation eq:inhomalb1, assume that $u_0$ is a smooth function, and that the given autocorrelation function $\Gamma(x)$ is $\Gamma(x) = \mathcal{F}^{-1}_{k \to x}[P(k)],$ where $P(k)$ is a nonnegative smooth function of compact support. Then the potential $V(x,t)=u(x,x,t)$ is real valued for Equation eq:msbal21 holds when $(x,y) \in \mathbb{R}^{2n}$ and $u_0$ exhibits rapid decay, or when

Figures (10)

  • Figure 1: Evolution in time for the $l^\infty_{i,j}$ error for $u$ and $\phi,$ for the exact solution \ref{['eq:periodizedsoliton']}. The final time $T=11.1149$ used here amounts to a full lap of the computational domain of length $L=34.4562$, i.e. the soliton returns near the starting position. For this computation the mesh sizes used are $\tau=0.002$ and $h=0.12.$
  • Figure 2: Zoom in of the error in Figure \ref{['Fig:1']} at the initial stage. The oscillations relax after some time. The naive initialization produces substantially more pronounced oscillations.
  • Figure 3: Constraint error, as defined in equation \ref{['eq:constrerrdef']}, for the example of Figure \ref{['Fig:1']}.
  • Figure 4: This plot is the real part of $u(t^N)-U^N$ over $(x,y)\in[-\frac{L}{2},\frac{L}{2}]^2$ at final time, for the run described in Figure \ref{['Fig:1']}. The shape of the soliton is very well preserved; the bulk of the error is due to the phase of the numerical solution staying slightly behind that of the exact solution.
  • Figure 5: The curves $S_X(t)$ are plotted on the complex plane for various values of $X\in\mathbb{R}$ as in \ref{['eq:curveS']}. In the context of Section \ref{['subsec:42']}$p=q=1,$ hence the target point is $4\pi p /q =4\pi.$ For this particular case $C=1.6,$ the first four harmonics of the computational domain, $X=2\pi n / L,$$n\in\{1,2,3,4\},$ are unstable. The bandwidth of unstable wavenumbers for this case is approximately $1.13.$
  • ...and 5 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma A.1