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The linearized Korteweg-de Vries equation on the line with metric graph defects

Dave Smith

TL;DR

The paper extends the unified transform method to the linearized Korteweg–de Vries equation posed on real-line segments with defect regions modeled as metric graphs attached at a single point. It develops a three-stage framework (global relation, linear boundary-value system, and rigorous stage-3 verification) tailored to graph domains with leads and bonds, and derives explicit contour-integral solution formulas for four defect types: mismatch, loop, source, and sink. Each defect is handled via determinant-based representations involving Δ(λ) and δ_k(λ), with careful asymptotic analysis ensuring existence, uniqueness, and satisfaction of PDEs, initial data, and vertex conditions. The results provide a rigorous, scalable path for solving higher-order linear PDEs on graphs, illuminating how interface conditions influence well-posedness and solution structure in dispersive wave contexts such as metamaterials.

Abstract

We study the small amplitude linearization of the Korteweg de Vries equation on the line with a local defect scattering waves represented by a metric graph domain adjoined at one point. For a representative collection of examples, we derive explicit solution formulae expressed as contour integrals and obtain existence and unicity results for piecewise absolutely continuous data. In so doing, we implement the unified transform method on metric graphs comprising both bonds and leads for a third order differential operator.

The linearized Korteweg-de Vries equation on the line with metric graph defects

TL;DR

The paper extends the unified transform method to the linearized Korteweg–de Vries equation posed on real-line segments with defect regions modeled as metric graphs attached at a single point. It develops a three-stage framework (global relation, linear boundary-value system, and rigorous stage-3 verification) tailored to graph domains with leads and bonds, and derives explicit contour-integral solution formulas for four defect types: mismatch, loop, source, and sink. Each defect is handled via determinant-based representations involving Δ(λ) and δ_k(λ), with careful asymptotic analysis ensuring existence, uniqueness, and satisfaction of PDEs, initial data, and vertex conditions. The results provide a rigorous, scalable path for solving higher-order linear PDEs on graphs, illuminating how interface conditions influence well-posedness and solution structure in dispersive wave contexts such as metamaterials.

Abstract

We study the small amplitude linearization of the Korteweg de Vries equation on the line with a local defect scattering waves represented by a metric graph domain adjoined at one point. For a representative collection of examples, we derive explicit solution formulae expressed as contour integrals and obtain existence and unicity results for piecewise absolutely continuous data. In so doing, we implement the unified transform method on metric graphs comprising both bonds and leads for a third order differential operator.

Paper Structure

This paper contains 22 sections, 28 theorems, 223 equations, 12 figures.

Table of Contents

  1. Introduction
  2. UTM for lKdV on metric graphs
  3. Overview of UTM
  4. UTM stage 1 for lKdV on metric graphs
  5. Sketch of UTM stage 2
  6. Definitions of contours for UTM stage 3
  7. Mismatch defect
  8. Solution representation: proof of proposition \ref{['prop:mismatchSolution']}
  9. Solution satisfies the problem: proof of proposition \ref{['prop:mismatchExistence']}
  10. Loop defect
  11. Solution representation: proof of proposition \ref{['prop:loopSolution']}
  12. Solution satisfies the problem: proof of proposition \ref{['prop:loopExistence']}
  13. Source defect
  14. Solution representation: proof of proposition \ref{['prop:sourceSolution']}
  15. Solution satisfies the problem: proof of proposition \ref{['prop:sourceExistence']}
  16. ...and 7 more sections

Key Result

Lemma 1

As $\lambda\to\infty$, $\nu(\lambda) = \lambda+\mathcal{O}\left(\lambda^{-1}\right)$, and the asymptotic term has a control uniform in $\arg(\lambda)$. Outside $B(0,R)$, $\nu(\mathbb R^\pm),\nu^{-1}(\mathbb R^\pm)\subset\mathbb R^\pm$. Outside $B(0,R)$, $\nu(\mathbb C^\pm),\nu^{-1}(\mathbb C^\pm)\su

Figures (12)

  • Figure 1: The contour $\partial D$.
  • Figure 2: The contours $\Gamma$ and $\gamma$ and the closed sets $S$ and $s$.
  • Figure 3: The metric graph domain with a mismatch defect.
  • Figure 4: The solution of the mismatch problem at time $t=0.014$ with vertex coefficients $(B_0,B_1,B_2)=(2.2,2.0,1.1)$, with initial data shown in yellow.
  • Figure 5: The metric graph domain with a loop defect.
  • ...and 7 more figures

Theorems & Definitions (57)

  • Lemma 1
  • proof : Proof of lemma \ref{['lem:generalNu']}
  • Proposition 2
  • proof : Proof of proposition \ref{['prop:UTM']}
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • proof
  • proof : Proof that each $u_e$ satisfies its PDE
  • proof : Proof that each $u_e$ satisfies its initial condition
  • ...and 47 more