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The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves

Elliot Blackstone, Louise Gassot, Patrick Gérard, Peter D. Miller

Abstract

The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\mathrm{ZD}(t,x;ε)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;ε)$ of the Benjamin-Ono equation with dispersion parameter $ε>0$ and initial data $u_0$ satisfies $u(t,x;ε)-u^\mathrm{ZD}(t,x;ε)\to 0$ in the locally uniform sense as $ε\to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases this convergence implies strong $L^2(\mathbb{R})$ convergence. The asymptotic profile $u^\mathrm{ZD}(t,x;ε)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever.

The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves

Abstract

The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in is obtained explicitly for generic rational initial data . An explicit asymptotic wave profile is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data , such that the solution of the Benjamin-Ono equation with dispersion parameter and initial data satisfies in the locally uniform sense as , provided a discriminant inequality holds implying that certain caustic curves in the -plane are avoided. In some cases this convergence implies strong convergence. The asymptotic profile is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever.

Paper Structure

This paper contains 33 sections, 25 theorems, 212 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Background: weak dispersion and dispersive shock waves
  3. The Benjamin-Ono equation
  4. Main result: strong zero-dispersion asymptotics
  5. Plan of the paper
  6. The solution of Benjamin-Ono with rational initial data
  7. The case of one pole
  8. Determination of suitable branch cuts and integration contours
  9. Steepest descent analysis
  10. Case $J=0$
  11. Case $J=1$
  12. Extended level curves
  13. Extended level curves as trajectories of a quadratic differential
  14. Extended level curves as integral curves of an analytic vector field
  15. Conditions guaranteeing typical behavior of extended level curves
  16. ...and 18 more sections

Key Result

Theorem 1.1

\newlabelthm:J-phase0 The $J$-phase solutions are characterized by real constant parameters $R_0<R_1<R_2<\cdots<R_{2J}$, and nonzero complex parameters $\gamma_1,\dots,\gamma_J$ satisfying They are given by in which $\mathbf{M}(t,x;\epsilon)$ is a $J\times J$ matrix with elements and $\theta^\mathrm{L}_j(t,x)$ is a linear phase given by

Figures (18)

  • Figure 1: The plots in this figure correspond to the initial condition $u_0(x)$ of the form \ref{['eq:rationalIC']} for $N=2$ with $c_1=1-\mathrm{i}$, $c_2=1+\mathrm{i}/\sqrt{2}$, $p_1=\mathrm{i}$, $p_2=16+\mathrm{i}$. Left: the caustic curves dividing the plane into regions where the solution of Burgers' equation is $2J+1=2J(t,x)+1$-valued, as shown. Right: the evolution of $u_0(x)$ under Burgers' equation at $t=3$. The blue, green, red, purple, black curves are $u^{\mathrm{B}}_k(3,x)$, $k=0,\ldots,4$, respectively. The dashed black lines in the right pane are placed at the $x$ values where the dashed blue line intersects the caustic curves in the left pane.
  • Figure 1: Left: admissible branch cuts of $h(z)$ in the $z$-plane for a rational initial condition with $N=5$. Right: corresponding contours $C_1,C_2,C_3,C_4,C_5$ and $C_0$ in the general case that $c_n\not\in\mathrm{i}\epsilon \mathbb{N}$ for each $n$.
  • Figure 1: The union of the red curves is the branch cut $\Gamma_1$, emanating from the pole $p_1$ in a $J(t,x)=1$ scenario. The black curves are the critical trajectories $\mathrm{Re}(-\mathrm{i} h(z))=0$ oriented according to increasing parametrization by $s$, and the blue curves $W_0$, $W_1$ are the integration contours. The dashed blue line segments passing $y_2$, $y_1$, $y_0$ are to be understood as local steepest descent contours passing through their respective saddle points.
  • Figure 1: Possible faces of an abstract Stokes graph $\mathcal{S}$ representing end domains. $\mathcal{S}$ contains precisely one of the left two faces and precisely one of the right two faces.
  • Figure 1: A heatmap of $\mathrm{Re}(-\mathrm{i} h(z;1.75,6))$. The black contours are the level set $\mathrm{Re}(-\mathrm{i} h(z;1.75,6))=0$, the white lines are the branch cuts of $h(z;t,x)$, and the blue path is $C_{0,1}$, a truncation of $C_0$, which passes through the relevant saddle points $y_k$, $k=0,1,2$.
  • ...and 13 more figures

Theorems & Definitions (64)

  • Theorem 1.1: Multi-phase solutions of Dobrokhotov, Krichever DobrokhotovK91
  • Definition 1.2: Zero-dispersion approximation
  • Remark 1.3
  • Remark 1.4: Local wavenumber and frequency
  • Theorem 1.5: Strong zero-dispersion asymptotics
  • Corollary 1.6: Convergence in $L^2$
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1: Branch cuts of $h$
  • Definition 2.2
  • ...and 54 more