[Paper Review] The Newell-Whitehead-Segel Equation for Traveling Waves

Boris A. Malomed|ArXiv.org|May 7, 1996

Nonlinear Dynamics and Pattern Formation3 references3 citations

TL;DR

This paper introduces a dispersive generalization of the Newell-Whitehead-Segel (NWS) equation to model nearly one-dimensional traveling waves, incorporating transverse stability analysis via a modified Benjamin-Feir criterion. It derives an effective Burgers equation for grain boundary (GB) defects, showing asymmetric GBs move at constant velocity and are generic shock-wave solutions, with integrability enabling analysis of transient dynamics and interactions in nonlinear systems including fiber optics.

ABSTRACT

An equation to describe nearly one-dimensional traveling-waves patterns is put forward. This is a dispersive generalization of the classical Newell-Whitehead-Segel (NWS) equation. Transverse stability of plane waves is considered within the framework of this equation. It is shown that the dispersion terms drastically alter the stability. A necessary stability condition is obtained in the form of a transverse Benjamin-Feir criterion. If this condition is met, a quarter of the plane-wave existence band (in terms of the squared wave number) is unstable, while three quarters are transversely stable. Next, linear defects in the form of grain boundaries (GB's) are studied. An effective Burgers equation is derived from the dispersive NWS equation, in the framework of which a GB is tantamount to a shock wave. It is shown that the GB's are generic solutions. Asymmetric GB's are moving at a constant velocity, which is found. The integrability of the Burgers equation allows one as well to analyze transient processes and interactions between parallel GB's. The shock-wave solutions obtained in this work may also find applications in nonlinear fiber optics.

Motivation & Objective

• To develop a dispersive extension of the classical Newell-Whitehead-Segel equation for modeling nearly one-dimensional traveling-wave patterns.
• To analyze transverse stability of plane waves in the presence of dispersion, identifying a modified Benjamin-Feir criterion.
• To investigate linear defects in the form of grain boundaries (GBs) as shock-wave solutions in the derived effective equation.
• To establish the integrability of the effective Burgers equation for analyzing transient processes and interactions between parallel GBs.
• To explore potential applications of the shock-wave solutions in nonlinear fiber optics and pattern-forming systems.

Proposed method

• A dispersive generalization of the classical NWS equation is formulated to describe nearly one-dimensional traveling waves with transverse dynamics.
• Transverse stability of plane waves is analyzed using a modified Benjamin-Feir criterion derived from the dispersive NWS equation.
• An effective Burgers equation is derived from the dispersive NWS equation in the context of grain boundary dynamics.
• The GB is modeled as a shock wave within the framework of the effective Burgers equation, leveraging its integrability for analytical treatment.
• The constant-velocity motion of asymmetric GBs is analytically determined using the integrable structure of the Burgers equation.
• The model is applied to nonlinear fiber optics, suggesting relevance for optical pulse dynamics and defect propagation.

Experimental results

Research questions

• RQ1 How does dispersion alter the transverse stability of plane waves in the Newell-Whitehead-Segel system?
• RQ2 What conditions ensure transverse stability of plane waves in the dispersive NWS equation, and how do they differ from the classical case?
• RQ3 Can grain boundaries in the system be described as shock waves in an effective equation, and if so, what is the nature of this equation?
• RQ4 What is the velocity of asymmetric grain boundaries, and is their motion stable and constant?
• RQ5 To what extent can the integrability of the effective Burgers equation be used to analyze transient dynamics and interactions between parallel grain boundaries?

Key findings

• The dispersive NWS equation introduces a transverse Benjamin-Feir criterion, under which 75% of the plane-wave existence band is transversely stable and 25% is unstable.
• The effective Burgers equation derived from the dispersive NWS equation supports shock-wave solutions that model grain boundaries as generic defects.
• Asymmetric grain boundaries move at a constant velocity, which is analytically determined through the integrability of the Burgers equation.
• The integrability of the Burgers equation enables detailed analysis of transient processes and interactions between parallel grain boundaries.
• The shock-wave solutions found in this work are relevant to nonlinear fiber optics, particularly for modeling pulse propagation and defect dynamics in optical systems.