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Dynamics of the Drinfeld-Sokolov-Wilson system: well-posedness and (in)stability of the traveling waves

Ognyan Christov, Sevdzhan Hakkaev, Seungly Oh, Atanas G. Stefanov

TL;DR

This work analyzes the periodic Drinfeld-Sokolov-Wilson system, proving local well-posedness at low regularity and global existence via $L^2$ conservation, plus persistence in $H^1\times L^2$ through a normal-form transformation. It constructs an explicit one-parameter family of $L$-periodic traveling waves using Jacobi elliptic functions and derives their spectral instability, showing exactly one unstable real mode and a Hamiltonian index of 1. The methodology blends Bourgain-space techniques for dispersive PDE with a normal-form approach to gain regularity, and couples analytical construction with a detailed spectral index computation (via operators $\mathcal{L}_+$, $\mathcal{H}$ and the matrix $\mathcal{D}$). The findings illuminate the delicate balance between dispersion and a dispersionless constraint in multi-component wave systems, with implications for the long-time dynamics and stability of coupled KdV-type models on periodic domains.

Abstract

We analyze the Drinfeld-Sokolob-Wilson system, which features a dispersive, KdV type evolution with a dispersionless conservation law. We establish well-posedness with low regularity initial data $L^2({\mathbb T})\times L^2({\mathbb T})$ for the Cauchy problem on periodic background, which is then extrapolated to global solutions, due to $L^2$ conservation law. We also establish a dynamically more relevant result, namely a global persistence of solutions with (large) initial data in $H^1({\mathbb T})\times L^2({\mathbb T})$. This is obtained by following a more sophisticated approach, specifically the method of normal forms. Finally, for a fixed period $L$, we construct an explicit one parameter family of periodic waves, see \eqref{2.16} below. We show that they are all spectrally unstable with respect to co-periodic perturbations. Specifically, we show that the Hamiltonian instability index is equal to one, which identifies the instability as a single positive growing mode.

Dynamics of the Drinfeld-Sokolov-Wilson system: well-posedness and (in)stability of the traveling waves

TL;DR

This work analyzes the periodic Drinfeld-Sokolov-Wilson system, proving local well-posedness at low regularity and global existence via conservation, plus persistence in through a normal-form transformation. It constructs an explicit one-parameter family of -periodic traveling waves using Jacobi elliptic functions and derives their spectral instability, showing exactly one unstable real mode and a Hamiltonian index of 1. The methodology blends Bourgain-space techniques for dispersive PDE with a normal-form approach to gain regularity, and couples analytical construction with a detailed spectral index computation (via operators , and the matrix ). The findings illuminate the delicate balance between dispersion and a dispersionless constraint in multi-component wave systems, with implications for the long-time dynamics and stability of coupled KdV-type models on periodic domains.

Abstract

We analyze the Drinfeld-Sokolob-Wilson system, which features a dispersive, KdV type evolution with a dispersionless conservation law. We establish well-posedness with low regularity initial data for the Cauchy problem on periodic background, which is then extrapolated to global solutions, due to conservation law. We also establish a dynamically more relevant result, namely a global persistence of solutions with (large) initial data in . This is obtained by following a more sophisticated approach, specifically the method of normal forms. Finally, for a fixed period , we construct an explicit one parameter family of periodic waves, see \eqref{2.16} below. We show that they are all spectrally unstable with respect to co-periodic perturbations. Specifically, we show that the Hamiltonian instability index is equal to one, which identifies the instability as a single positive growing mode.

Paper Structure

This paper contains 22 sections, 23 theorems, 176 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Well-posedness of the DSW system
  3. Periodic solutions
  4. On the stability of the periodic TW solutions
  5. Construction of the solutions
  6. Well-posedness of the DSW system
  7. Preliminaries
  8. Reduction to the case of mean-zero initial data
  9. Bilinear estimates in Bourgain spaces
  10. Proof of Theorem \ref{['th:wp']}
  11. Normal form transformation
  12. Remaining multilinear estimates in Bourgain spaces
  13. Proof of Theorem \ref{['prop:109']}
  14. Spectral stability - preparations
  15. Spectral properties of ${\mathcal{L}}_+$
  16. ...and 7 more sections

Key Result

Theorem 1

Let $0\leq s \leq \frac{1}{2}$. Given $u_0 \in H^s_{per.}(0,L)$ and $v_0\in L^2_{per.}(0,L)$, we can find $T>0$ depending only on ${\left\| u_0 \right\|}_{H^s}+{\left\| v_0 \right\|}_{L^2}$ so that there exist a solution for eq:1, $(u, v) \in C^0_t H^s_x \times C^0_t L^2_x$ on where $t\in [0,T]$. Th

Figures (3)

  • Figure 1: The positive periodic solution of (\ref{['2.7']}).
  • Figure 2: Graph of $\kappa\to \det(\mathcal{D}(\kappa))$
  • Figure 3: Graph of $A_2(\kappa)$

Theorems & Definitions (32)

  • Theorem 1: Local well-posedness in $L^2$
  • Corollary 1
  • Theorem 2
  • Proposition 1
  • Definition 1
  • Theorem 3
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 1
  • ...and 22 more