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 Schr\"odinger equation, and can describe the ultra-short optical pulse propagation in the form $iq_t+\alpha(q_{xx}+ 2|q|^2q)+i\beta (q_{xxx}+ 6|q|^2q_x)=0,\, (x,t)\in\mathbb{R}^2\, (\alpha,\,\beta\in\mathbb{R})$. 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.