The Small Dispersion Limit of the Intermediate Long Wave Equation via Semiclassical Soliton Ensembles

TL;DR

This work analyzes the small dispersion limit of the ILW equation $0 = u_t + 2 u u_x + \epsilon \mathpzc{T}_{\delta\epsilon}[u_{xx}]$ for admissible initial data with many solitons. It develops a formal WKB direct scattering theory to obtain approximate eigenvalues and norming constants, then constructs a modified scattering data and studies the inverse problem via a semiclassical soliton ensemble (SSE). The main results include an ILW Weyl law for eigenvalues, a distributional limit of the SSE in terms of an equilibrium measure solving a Greens-energy minimization, and an $L^2$ convergence of the SSE to the inviscid Burgers' solution for $0<t< t_\mathrm{c}$, with a Burgers-like variational structure guiding the limiting densities. The paper also discusses how the post-catastrophe regime may be described by modulated multi-periodic ILW solutions, connecting the spectral data to DSW dynamics and offering a Lax–Levermore–style framework for ILW. Overall, the approach unites semiclassical spectral analysis, variational potential theory, and inverse scattering to illuminate the ILW small-dispersion dynamics and the emergence of dispersive shocks.

Abstract

We study the small dispersion limit of the intermediate long wave (ILW) equation, specifically on a class of well-behaved initial conditions $u_0$ where the number of solitons in the solution increases without bound. First, we conduct a formal WKB-style analysis on the ILW direct scattering problem, generating approximate eigenvalues and norming constants. We then use this to define a modified set of scattering data and rigorously analyze the associated inverse scattering problem. The main results include demonstrating $L^2$-convergence of the solution at $t = 0$ to the original initial condition $u_0$ and for $0 < t < t_\mathrm{c}$ to the associated solution of invicid Burgers' equation, where $t_\mathrm{c}$ is the time of gradient catastrophe.

Figures (8)

• Figure 1: Schematic for the two dimensional incompressible and irrotational fluid system from which the ILW equation is derived. The equilibrium position (dashed black) of the thin pycnocline (solid black) between the two fluid densities (yellow and blue) are depicted above. We require the density of the top fluid $\rho_1$ to be less than that of the lower fluid $\rho_2$ so the fluids are stably separated by gravity. The system is confined between two rigid and flat surfaces, a ceiling and floor. Labeled in the schematic, $h$ and $\delta$ are the equilibrium depths of the fluids, $a$ is the maximum amplitude of the disturbance of the pycnocline and $\lambda$ is a characteristic wavelength of the disturbance. We characterize the nonlinearity governing the dynamics of the system by the parameter $\alpha = a / h$ as well as that for the linear dispersion by $\epsilon = h / \lambda$. The ILW equation obtains in the asymptotic limit where $h \to 0^+$, $\delta$ and $\epsilon / \alpha$ fixed with $\epsilon$ and $\alpha$ individually bounded. Here, the top layer is depicted with vanishing depth $h$ while the bottom layer has fixed depth $\delta$. However, the model is independent of which layer has the fixed depth $\delta$ and which has the vanishing depth $h$KubotaKoDobbs_1978.
• Figure 2: Split-step simulation of the ILW equation in the small dispersion limit with initial condition $u_0(x) = \mathop{\mathrm{sech}}\nolimits^2(x)$ on a symmetric periodic domain of length $L=12$, $N_x=2000$ sample points in $x$ and time step $\Delta t = 0.0001$. The first column is at a fixed $\epsilon = 0.05$ and for different times between $t=0.00$ and the gradient catastrophe time $t_\mathrm{c} = 3\sqrt{3}/8 \approx 0.650$. The second column shows the effect of decreasing $\epsilon$ at fixed time $t = 1.500$.
• Figure 3: The strip domain of analyticity of the wavefunction $\psi(z)$. Also shown are the definitions of the continuous boundary values $\psi^\pm(x)$ via limits from inside the strip.
• Figure 4: The proposed ILW scattering equation spectrum in the $\zeta$ plane. The blue contours are the visible portion of the quadratrix of Hippias, solid for the quadratrix and dashed for the other components. The discrete spectrum shows up on the quadratrix and the continuous spectrum is labeled in solid yellow.
• Figure 5: $z \mapsto W_n(z)$ demonstrated for three characteristic branches: $n = -2, -1,$ and $0$. The colored $z$-planes map into the same colored branch ranges indicated below. Additionally, the colored edges in the $z$-planes indicate the direction that the same colored range boundaries are approached from each side of the branch cut. The edge of the same color as the plane are the ones that are included in that branch definition. The $n = 1$ maps similar to the $n = -1$ branch while all others are similar to the $n = -2$ branch.

Theorems & Definitions (31)

• Definition 1.1: Admissible initial condition.
• Conjecture 1.1: Semiclassical asymptotics of the ILW scattering data
• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Lemma 2.2: Some properties of the ILW Weyl Law
• proof
• Definition 2.3: The Modified Scattering Data
• Definition 3.1: The ILW semiclassical soliton ensemble