Large-time asymptotics of a new KdV soliton gas

Abstract

We study the large-time asymptotic behavior of a new KdV soliton gas. We first introduce a pure-soliton Riemann--Hilbert(RH) problem with \(2N\) poles and two different types of residue conditions. We show that, as \(N\to\infty\), this discrete problem converges to primitive-potential RH problem introduced by Dyachenko, Zakharov, and Zakharov, and the jump matrix of this soliton gas RH problem has two nonzero reflection coefficients. To analyze the large-time behavior, we apply the Deift--Zhou nonlinear steepest descent method together with an appropriate \(g\)-function mechanism. Through a sequence of transformations, the original RH problem is reduced to explicitly solvable model problems on an associated hyperelliptic Riemann surface. This allows us to derive an explicit leading-order asymptotic formula for the solution in terms of Jacobi elliptic function. The result provides a rigorous asymptotic description of a new KdV soliton gas and extends the available analysis beyond the previously studied case \(r_2\equiv 0\).