Large-space and long-time asymptotic behaviors of $N_{\infty}$-soliton solutions (soliton gas) for the focusing Hirota equation
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arXiv
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2024
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| author | Weng, Weifang Yan, Zhenya |
| author_facet | Weng, Weifang Yan, Zhenya |
| contents | 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 $iq_t+α(q_{xx}+ 2|q|^2q)+iβ(q_{xxx}+ 6|q|^2q_x)=0,\, (x,t)\in\mathbb{R}^2\, (α,\,β\in\mathbb{R})$. 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: $x/t>4βb^2,\, ξ_c<x/t<4βb^2,\, x/t<ξ_c$. Moreover, we analyze the property of the soliton gas for the case of the discrete spectra filling uniformly a quadrature domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_08924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large-space and long-time asymptotic behaviors of $N_{\infty}$-soliton solutions (soliton gas) for the focusing Hirota equation Weng, Weifang Yan, Zhenya Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Optics 35Q51, 35Q15, 37K15, 35C20 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 $iq_t+α(q_{xx}+ 2|q|^2q)+iβ(q_{xxx}+ 6|q|^2q_x)=0,\, (x,t)\in\mathbb{R}^2\, (α,\,β\in\mathbb{R})$. 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: $x/t>4βb^2,\, ξ_c<x/t<4βb^2,\, x/t<ξ_c$. Moreover, we analyze the property of the soliton gas for the case of the discrete spectra filling uniformly a quadrature domain. |
| title | Large-space and long-time asymptotic behaviors of $N_{\infty}$-soliton solutions (soliton gas) for the focusing Hirota equation |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Optics 35Q51, 35Q15, 37K15, 35C20 |
| url | https://arxiv.org/abs/2401.08924 |