Whitham modulation theory for the Davey-Stewartson system and stability analysis of its periodic traveling wave solutions
Abstract
The Davey-Stewartson (DS) system is a canonical model for a weakly nonlinear wave packet in two spatial dimensions that is resonantly coupled to the slowly varying mean flow it generates, and as such it governs modulated wave trains in settings as different as finite-depth water waves, quadratic optical media and magnetic films. Here we develop the Whitham modulation theory for the DS system and use it to study the stability of its periodic traveling wave solutions. Specifically: 1. We perform a detailed study of the parametric dependence of the periodic solutions of all four variants of the DS system, including their harmonic and soliton limits. 2. We present the formulation of Whitham modulation theory for the DS system and write down the resulting Whitham modulation equations for all four variants of the DS system. 3. We use the resulting modulation equations to study the stability of the periodic traveling wave solutions of all four variants of the DS system. 4. We validate the theoretical predictions by comparing them with a direct linearization of the DS system as well as with the direct numerical simulations of the DS system, showing excellent agreement. The results indicate that the periodic solutions of the defocusing DSII system are linearly stable, whereas those of all other variants of the DS system are unstable. The calculations also evidence many structural features (non-evolutionary modulation equations, constraints on the admissible initial data, and auxiliary mean-field constants absent from the traveling wave solutions yet indispensable to the modulation system) are expected to recur in any two-dimensional envelope equation with mean-field coupling.
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BibTeX
@misc{biondini2026whitham,
title = {{Whitham modulation theory for the Davey-Stewartson system and stability analysis of its periodic traveling wave solutions}},
author = {Gino Biondini and Alexander Chernyavsky and Haodong Lin and John Ringland},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06403},
eprint = {2610.06403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06403},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Whitham modulation theory for the Davey-Stewartson system and stability analysis of its periodic traveling wave solutions
AU - Gino Biondini
AU - Alexander Chernyavsky
AU - Haodong Lin
AU - John Ringland
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06403
UR - https://arxiv.org/abs/2610.06403
AB - The Davey-Stewartson (DS) system is a canonical model for a weakly nonlinear wave packet in two spatial dimensions that is resonantly coupled to the slowly varying mean flow it generates, and as such it governs modulated wave trains in settings as different as finite-depth water waves, quadratic optical media and magnetic films. Here we develop the Whitham modulation theory for the DS system and use it to study the stability of its periodic traveling wave solutions. Specifically: 1. We perform a detailed study of the parametric dependence of the periodic solutions of all four variants of the DS system, including their harmonic and soliton limits. 2. We present the formulation of Whitham modulation theory for the DS system and write down the resulting Whitham modulation equations for all four variants of the DS system. 3. We use the resulting modulation equations to study the stability of the periodic traveling wave solutions of all four variants of the DS system. 4. We validate the theoretical predictions by comparing them with a direct linearization of the DS system as well as with the direct numerical simulations of the DS system, showing excellent agreement. The results indicate that the periodic solutions of the defocusing DSII system are linearly stable, whereas those of all other variants of the DS system are unstable. The calculations also evidence many structural features (non-evolutionary modulation equations, constraints on the admissible initial data, and auxiliary mean-field constants absent from the traveling wave solutions yet indispensable to the modulation system) are expected to recur in any two-dimensional envelope equation with mean-field coupling.
ER -