Large-space and long-time asymptotic behaviors of $N_{\infty }$-soliton solutions (soliton gas) for the focusing Hirota equation

Abstract

The Hirota equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation, and can describe the ultra-short optical pulse propagation in the form $i{q}_t+\alpha (q_{xx}+2|q{|}^{2}q)+i\beta (q_{xxx}+6|q{|}^2{q}_x)=0,(x,t)\in {\mathbb{R}}^2(\alpha ,\beta \in \mathbb{R})$. In this paper, we analytically explore the asymptotic behaviors of a soliton gas for the Hirota equation including the complex modified KdV equation, in which the soliton gas is regarded as the limit $N\to \infty$ of $N$-soliton solutions, and characterized using the Riemann-Hilbert problem with discrete spectra restricted in the intervals $(ia,ib)\cup (-ib,-ia)(0<a<b)$. We find that this soliton gas tends slowly to the Jaocbian elliptic wave solution with an error $\mathcal{O}(|x{|}^{-1})$ (zero exponentially quickly ) as $x\to -\infty$ ($x\to +\infty$). We also present the long-time asymptotics of the soliton gas under the different velocity conditions: $x/t>4\beta b^2,{\xi }_c<x/t<4\beta b^2,x/t<{\xi }_c$. Moreover, we analyze the property of the soliton gas for the case of the discrete spectra filling uniformly a quadrature domain.

39 pages, 8 figures,