Efficient computation of soliton gas primitive potentials

TL;DR

The paper develops an efficient, RH-based framework to compute soliton gas primitive potentials for the KdV equation $u_t + 6 u u_x + u_{xxx} = 0$, leveraging RH steepest-descent deformations with a g-function and endpoint-aware basis functions to handle many disjoint bands. By reformulating the RH problem as a stable singular integral equation and discretizing with weighted Chebyshev polynomials, the method avoids high-precision arithmetic while remaining accurate across large-band configurations. The authors extend the approach to nonlinear superpositions with finitely many solitons by incorporating pole residues into the RH problem, and they provide a broad set of numerical experiments demonstrating convergence, performance, and the capability to compute gas-only and gas-plus-soliton solutions. The framework is highly parallelizable over (x,t) and capable of covering substantial portions of the (x,t)-plane, with ongoing work to extend to broader soliton-gas classes and entire-plane computations.

Abstract

We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $-\infty$. This accumulation results in an associated Riemann--Hilbert problem on a number of disjoint intervals. In the case where the jump matrices have specific square-root behavior, we describe an efficient and accurate numerical method to solve this Riemann--Hilbert problem and extract the potential. The keys to the method are, first, the deformation of the Riemann--Hilbert problem, making numerical use of the so-called $g$-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation.

Figures (12)

• Figure 1: KdV soliton gas with $r_1(\lambda)$ supported on five pairs of bands, nonlinearly superimposed with five solitons. In the notation of \ref{['r1-practice']}, $I_1 = (0.25,0.5)$, $I_2 = (0.8,1.2)$, $I_3=(1.5,2)$, $I_4=(2.5,3)$, $I_5=(4,5)$, with $f_1(z)=1$, $f_2(z)=1/2$, $f_3(z)=1/4$, $f_4(z)=1/8$, $f_5(z)=1/16$ and $\alpha_j=\beta_j=\frac{1}{2}$ for $j=1,\ldots,5$. The solitons are associated with (see Riemann--Hilbert Problem \ref{['rhp:gas-soliton']}) $\kappa_1=0.1$, $\kappa_2 = 0.7$, $\kappa_3=2.25$, $\kappa_4=3.5$, $\kappa_5 = 5.5$ with the norming constants $\chi_1=10^5$, $\chi_2=1000$, $\chi_3=100$, $\chi_4=10$, and $\chi_5=10^{-6}$.
• Figure 2: A pure KdV soliton gas with $r_1(\lambda)$ supported on five pairs of bands. In the notation of \ref{['r1-practice']}, $I_1=(0.25,0.5)$, $I_2=(0.8,1.2)$, $I_3=(1.5,2)$, $I_4=(2.5,3)$, $I_5=(4,5)$ with $f_1(z)=1$, $f_2(z)=1/2$, $f_3(z)=1/4$, $f_4(z)=1/8$, $f_5(z)=1/16$ and $\alpha_j=\beta_j=\frac{1}{2}$ for $j=1,\ldots,5$.
• Figure 3: Density plot of the computed soliton gas with $r_1(\lambda)$ supported on a single pair of bands with two solitons. In the notation of \ref{['r1-practice']}, $I_1=(1.5,2.5)$, $f_1(z)=1$, and $\alpha_1=\beta_1=\frac{1}{2}$. The solitons are associated with the eigenvalue parameters $\kappa_1=1$, $\kappa_2=4$ and the norming constants $\chi_1=10$, $\chi_2=10^{-10}$. Outside of the wedge region, the numerical method presented here is seen to be uniformly accurate with a computational cost that is independent of $(x,t)$. Inside the wedge, the numerical method begins to break down, and additional RH deformations will need to be incorporated.
• Figure 4: A pure KdV soliton gas with $r_1(\lambda)$ supported on two pairs of bands. In the notation of \ref{['r1-practice']}, $I_1=(1,2)$, $I_2=(2.5,3)$ with $f_1(z)=100$, $f_2(z)=1$ and $\alpha_{j}=\beta_{j}=\frac{1}{2}$, $j=1,2$. Bottom panel: The same solution plotted along the ray $x/t = -32$.
• Figure 5: Contour plot showing the sign of $\mathrm{Re}(\varphi(z;x,t))$ in the complex plane for $t>0$ in the unmodulated region treated in Section \ref{['s:unmodulated']}. Here, $x=-2$, $t=0.01$, and in the notation of \ref{['r1-practice']}, $I_1=(1,2)$, $I_2=(2.5,3)$.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2