• Acta Physica Sinica
  • Vol. 69, Issue 1, 014208-1 (2020)
Dan-Hong Xu and Sen-Yue Lou*
DOI: 10.7498/aps.69.20191347 Cite this Article
Dan-Hong Xu, Sen-Yue Lou. Dark soliton molecules in nonlinear optics[J]. Acta Physica Sinica, 2020, 69(1): 014208-1 Copy Citation Text show less

Abstract

The study on soliton molecules is one of the important topics in nonlinear science especially in nonlinear optics. The bright soliton molecules have been experimentally observed in optics, however, the dark soliton molecules have not yet been experimentally observed. Theoretically, the soliton molecules have been found for some coupled nonlinear systems. Nevertheless, the soliton molecules have not been obtained for non-coupled single component nonlinear models. In this paper, we first study the exact periodic waves (soliton lattices) and solitary waves for a nonlinear nonintegrable optical model with second and third order dispersions and high order nonlinear effects including self-steeping, Raman scattering and nonlinear dispersion. Two types of dark soliton lattice and three types of soliton lattice are explicitly exhibited for general nonintegrable system. Five types of bright (with and without gray background), dark and gray solitons can be obtained from the limit cases of the modules of the soliton lattices. For an integrable case, using a novel generalized bilinear form of a single component nonlinear system, the multi-soliton solutions are obtained and expressed by a completely new form which are invariant under the full reversal transformations such as the parity, the time reversal, the charge conjugate and the field reversal. To find soliton molecules, a novel mechanism, the velocity resonant, is proposed. Starting from the multi-soliton solutions and using the velocity resonance mechanism, the analytical expression of the dark soliton molecules can be readily obtained. For the model given in this paper, the integrable higher order nonlinear Schrodinger equation, one can proved that the interactions among the dark soliton molecules and the usually solitons are elastic. It is worth pointing out that soliton molecules can also exist in the case of nonintegrable systems.
$\begin{split} & \rm{i}q_t+d_{2}q_{xx}+\sigma r_2|q|^2q + \rm{i}\epsilon \big[\big(d_3q_{xxx}+\\ \, & 6\sigma r_3|q|^2q_x+\beta \sigma r_3(|q|^2)_xq\big)\big] = 0. \end{split}$(1)

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$ q = Q(\xi)\rm{e}^{\rm{i}(k_0x+\omega_0t+\phi_0)},\ \xi = kx+\omega t+\xi_0, $(2)

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$\begin{split} \,& k^3\epsilon d_3 Q_{\xi\xi\xi}+[\omega+2d_2kk_0 -3d_3\epsilon kk_0^2\\ &+2\sigma\epsilon r_3 k (\beta+3)Q^2 ] = 0, \end{split}$(3)

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$\begin{split} \, & k^2(d_2-3k_0d_3\epsilon)Q_{\xi\xi} -2\sigma (3\epsilon k_0 r_3-r_2)Q^3\\ & +(d_3\epsilon k_0^3-d_2k_0^2-\omega_0)Q = 0.\end{split} $(4)

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$ k_0 = \frac{d_2}{3d_3\epsilon} +\frac{\varDelta}{d_3\epsilon r_3\beta},\quad \varDelta\equiv d_2 r_3-d_3 r_2, $(5)

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$ \omega_0 = \frac{\varDelta (2 d_2^2 k+9 d_3\epsilon\omega)}{3r_3 d_3^2\epsilon^2 k\beta} -\frac{8\varDelta^3}{d_3^2\epsilon^2 r_3^3\beta^3} -\frac{2d_2^3}{27d_3^2\epsilon^2}. $(6)

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$\begin{split} Q_{\xi\xi} =\, & \frac1{\epsilon^2k^2d_3^2} \left(\frac{3\varDelta^2}{\beta^2r_3^2} -\frac{d_2^2}3-\frac{\epsilon d_3\omega}{k}\right)Q\\ & -\frac{2\sigma r_3(3+\beta)}{3d_3k^2}Q^3.\end{split}$(7)

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$\begin{split} & Q = \pm km\sqrt{\frac{-3\sigma d_3}{r_3(3+\beta)}}\rm{sn}(kx+\omega t+\xi_0,m),\\ & \omega = \epsilon(1+m^2)d_3k^3 -\frac{d_2^2k}{3d_3\epsilon}+\frac{3k\varDelta^2}{\epsilon d_3r_3^2\beta^2}. \end{split}$(8)

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$\begin{split} &Q = \pm km\sqrt{\frac{3\sigma d_3}{r_3(3+\beta)}}\rm{cn}(kx+\omega t+\xi_0,m),\\& \omega = \epsilon(1-2m^2)d_3k^3 -\frac{\big(d_2^2r_3^2\beta^2-9\varDelta^2\big)k}{3d_3\epsilon r_3^2\beta^2}.\end{split}$(9)

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$\begin{split} & Q = \pm k\sqrt{\frac{3\sigma d_3}{r_3(3+\beta)}}\rm{dn}(kx+\omega t+\xi_0,m),\\ & \omega = \epsilon(m^2-2)d_3k^3 -\frac{\big(d_2^2r_3^2\beta^2-9\varDelta^2\big)k}{3d_3\epsilon r_3^2\beta^2}. \end{split} $(10)

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$\begin{split} & Q = \pm k\sqrt{\frac{-3\sigma d_3}{r_3(3+\beta)}}\tanh(kx+\omega t+\xi_0),\\ & \omega = \epsilon2d_3k^3 -\frac{d_2^2k}{3d_3\epsilon}+\frac{3k\varDelta^2}{\epsilon d_3r_3^2\beta^2}. \end{split}$(11)

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$\begin{split} & Q = \pm k\sqrt{\frac{3\sigma d_3}{r_3(3+\beta)}}\rm{sech}(kx+\omega t+\xi_0),\\ & \omega = -\epsilon d_3k^3 -\frac{\big(d_2^2r_3^2\beta^2-9\varDelta^2\big)k}{3d_3\epsilon r_3^2\beta^2}. \end{split}$(12)

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$ k_0 = \frac{d_2}{3d_3\epsilon},\quad \omega_0 = -\frac{2d_2^3}{27d_3^2\epsilon^2},\quad \varDelta = 0. $(13)

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$ Q_{\xi\xi\xi}+\left[\frac{2 r_2\sigma (\beta+3)}{d_2k^2}Q^2 +\frac{d_2^2k+3d_3\epsilon\omega}{3d_3^2\epsilon^2 k^3}\right]Q_{\xi} = 0. $(14)

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$ \begin{split} & Q = \frac{b[a+\rm{sn}(\xi,\ m)]}{\alpha_1+\beta_1\rm{sn}(\xi,\ m)},\\& a = \frac{\beta_1(\alpha_1^2 m^2+\alpha_1^2-2\beta_1^2)} {\alpha_1(2\alpha_1^2 m^2-\beta_1^2 m^2-\beta_1^2)}, \\ & b^2 = \frac{3\sigma \alpha_1^2 d_2 k^2 (2\alpha_1^2 m^2-\beta_1^2 m^2-\beta_1^2)^2} {4r_2(\alpha_1^2-\beta_1^2) (\beta_1^2-\alpha_1^2 m^2) (\beta+3)}, \\ & \xi = kx+\omega t,\\& \omega \!=\! \frac{\epsilon k^3 d_3 [2 (m^2\!+\!1) (\alpha_1^4 m^2\!+\!\beta_1^4) \!+\!\beta_1^2 (m^4\!-\!10 m^2\!+\!1)\alpha_1^2]} {2(\alpha_1^2-\beta_1^2) (\alpha_1^2 m^2-\beta_1^2)}\\& \quad\;\; -\frac{d_2^2 k}{3d_3\epsilon}. \\[-15pt]\end{split} $(15)

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$\begin{split} & k = \beta_1 = \sigma = 1,\ m = 0.99,\\ & \alpha_1 = \frac{199}{198},\ \xi_0 = 0. \end{split}$(16)

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$\begin{split} & k = \beta_1 = -\sigma = 1,\ m = 0.98, \\ & \alpha_1 = \frac{50}{49},\ \xi_0 = 0. \end{split} $(17)

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$\begin{split} & Q = \frac{b[a+\rm{cn}(\xi,\ m)]}{\alpha_1+\beta_1\rm{cn}(\xi,\ m)},\\ & \xi = kx+\omega t+\xi_0, \end{split}$(18)

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$\begin{split}& a = \frac{\beta_1 [2 m^2(\alpha_1^2-\beta_1^2) -\alpha_1^2+2\beta_1^2]} {\alpha_1 (2\alpha_1^2 m^2-2\beta_1^2 m^2+\beta_1^2)}, ~~ b^2 = \frac{3\sigma\alpha_1^2 d_2 k^2 (2\alpha_1^2 m^2-2\beta_1^2 m^2+\beta_1^2)^2} {4r_2 (\alpha_1^2-\beta_1^2) (\alpha_1^2 m^2-\beta_1^2 m^2+\beta_1^2) (\beta+3)}, \\ & \omega = \frac{\epsilon d_3 k^3 [2 (1-2 m^2) (\alpha_1^4 m^2+\beta_1^4 m^2-\beta_1^4)+\alpha_1^2 \beta_1^2(8 m^4-8 m^2-1)] }{2(\alpha_1^2-\beta_1^2) (\alpha_1^2 m^2-\beta_1^2 m^2+\beta_1^2)} -\frac{d_2^2 k}{3d_3\epsilon}. \end{split}$(19)

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$ k = \beta_1 = \sigma = 1,\ m = 0.9,\ \alpha_1 = \frac{41}{40},\ \xi_0 = 0. $(20)

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$ Q = \frac{b[a+\rm{dn}(\xi,\ m)]}{\alpha_1+\beta_1\rm{dn}(\xi,\ m)},\quad \xi = kx+\omega t+\xi_0, $(21)

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$\begin{split} &a = \frac{\beta_1 (\alpha_1^2 m^2-2\beta_1^2 m^2-2\alpha_1^2+2\beta_1^2)} {\alpha_1 (2\beta_1^2 -\beta_1^2 m^2-2\alpha_1^2)}, ~~ b^2 = \frac{ 3\sigma\alpha_1^2 d_2 k^2 (\beta_1^2 m^2+2\alpha_1^2-2\beta_1^2)^2} {4r_2 (\beta_1^2 m^2+\alpha_1^2-\beta_1^2) (\beta+3) (\alpha_1^2-\beta_1^2)},\\ &\omega = \frac{d_3 k^3\epsilon[2(m^2-2) (\beta1^4-m^2\beta_1^4 +\alpha_1^4)-\alpha_1^2 (m^4+8 m^2-8)\beta_1^2]} {2(\alpha_1^2-\beta_1^2) (\beta_1^2 m^2+\alpha_1^2-\beta_1^2)} -\frac{d_2^2 k}{3d_3\epsilon}. \end{split}$(22)

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$\begin{split} & k = \beta_1 = \sigma = 1,\ m = 0.99999,\\ & \alpha_1 = {41}/{40},\ \xi_0 = 0. \end{split}$ (23)

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$\begin{split} & k = \beta_1 = -\sigma = 1,\ m = 0.99999,\\ &\alpha_1 = 1/2,\ \xi_0 = 0. \end{split} $(24)

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$\begin{split} & Q = \frac{b[a+\tanh(\xi)]}{\alpha_1+\beta_1\tanh(\xi)},~~~~ a = \frac{\beta_1} {\alpha_1},\\ & b^2 = -\frac{3\sigma \alpha_1^2 d_2 k^2} {r_2 (\beta+3)},~~~~ \xi = kx+\omega t,\\ & \omega = {2\epsilon k^3 d_3 } -\frac{d_2^2 k}{3d_3\epsilon}. \end{split}$(25)

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$ \delta\equiv \frac{\sigma d_2}{r_2(3+\beta)} < 0,\quad \alpha_1^2 > \beta_1^2. $()

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$ \alpha_1 = \frac{51}{49}, \beta_1 = k = m = -\sigma = 1,\ \xi_0 = 0. $(26)

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$ Q = \frac{b[a+ \rm{sech}(\xi)]}{\alpha_1+\beta_1\rm{sech}(\xi)},\quad \xi = kx+\omega t+\xi_0, $(27)

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$\begin{split}& a = \frac{\beta_1 \alpha_1 } {2\alpha_1^2-\beta}, \\ & b^2 = \frac{3\sigma d_2 k^2 (2\alpha_1^2-\beta_1^2 )^2} {4r_2 (\alpha_1^2-\beta_1^2) (\beta+3)},\\ &\omega = -\frac{\epsilon d_3 k^3[2 \alpha_1^2+ \beta_1^2] }{2(\alpha_1^2-\beta_1^2)} -\frac{d_2^2 k}{3d_3\epsilon}. \end{split}$(28)

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$ \{\delta > 0,\ \beta_1^2 < \alpha_1^2\},\ \rm{or}\ \{\delta < 0,\ \beta_1^2 > \alpha_1^2\} $()

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$ \alpha_1 = \frac{41}{40}, \beta_1 = k = m = \sigma = 1,\ \xi_0 = 0. $(29)

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$ \{\delta < 0,\ \beta_1 < \alpha_1 < 0 \},\ \rm{or}\ \{\delta < 0,\ 0 < \alpha_1 < \beta_1 \} $()

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$ 2\alpha_1 = \beta_1 = k = m = -\sigma = 1. $(30)

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$\begin{split} & m = 1 = -\sigma = \beta_1 = k = 1, \\ & \alpha_1 = 0.001,\ \xi_0 = 0. \end{split} $(31)

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$ \beta = 0,\ \varDelta = 0, $(32)

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$ \rm{i}q_t+d_{2}q_{xx}+\sigma r_2|q|^2q +\rm{i}\epsilon_1 \left(d_2q_{xxx}+6\sigma r_2|q|^2q_x\right) = 0. $(33)

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$ q = {g}/f $(34)

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$\begin{split} & (2\rm{i}D_t-2d_2D_x^2+\rm{i} \epsilon_1 d_2D_x^3\\ \, & -3\kappa^2 d_2\rm{i} \epsilon_1D_x +\kappa^2 d_2)f\cdot g = 0, \end{split}$(35)

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$ \left(2d_2D_x^2-\kappa^2 d_2\right)f\cdot f-4\sigma r_2 g\cdot g^* = 0. $(36)

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$\begin{split} & f = \sum\limits_{\{\nu\}}K_{\{\nu\}}\cosh\left(\frac12 \sum\limits_{i = 1}^N\nu_i\xi_i\right),\\ & \xi_i = k_ix+\omega_it+\xi_{i0},\end{split}$(37)

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$\begin{split} & g = g_0 \sum\limits_{\{\nu\}}K_{\{\nu\}}\cosh\left(\frac12 \sum\limits_{i = 1}^N\nu_i\eta_i\right),\\ & \eta_i = \xi_i+2\rm{i}\phi_i,\\ & g_0\equiv \frac12\kappa\sqrt{\frac{-\sigma d_2}{r_2}}\rm{e}^{\rm{i}(k_0x+\omega_0t+\phi_0)}, \end{split}$(38)

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$ K_{\{\nu\}} = \prod\limits_{i < j} \sin\left(\frac{\phi_i-\nu_i\nu_j\phi_j}2\right), $(39)

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$ \begin{split} & k_i = \kappa\sin(\phi_i),\\ & \omega_0 = \frac12d_2(2 k_0^3\epsilon_1+3 k_0\kappa^2\epsilon_1-2 k_0^2-\kappa^2), \end{split} $(40)

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$\begin{split} \omega_i = & \frac12k_id_2 \big[\epsilon_1 (6 k_0^2+3\kappa^2-2 k_i^2)-4 k_0\\ & +2\kappa (1-3 k_0\epsilon_1) \cos(\phi_i)\big],\ i = 1,\ \cdots,\ N, \end{split}$(41)

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$\begin{split} \; & q = {g_2}/{f_2},\\ \; & f_2 = \sin\left(\frac{\phi_1-\phi_2}2\right) \cosh\left(\frac{\xi_1+\xi_2}2\right) \\ & \qquad +\sin\left(\frac{\phi_1+\phi_2}2\right) \cosh\left(\frac{\xi_1-\xi_2}2\right),\end{split} $(42)

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$\begin{split} g_2 = &g_0\left[ \sin\left(\frac{\phi_1-\phi_2}2\right) \cosh\left(\frac{\eta_1+\eta_2}2\right)\right. \\ &\left. +\sin\left(\frac{\phi_1+\phi_2}2\right) \cosh\left(\frac{\eta_1-\eta_2}2\right)\right]. \end{split}$(43)

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$\begin{split} & \kappa = -15,\ k_0 = 0,\ \phi_1 = {2{\text{π}}}/3,\\ & \phi_2 = {{\text{π}}}/3,\ \xi_{20} = -\xi_{10} = 20. \end{split}$(44)

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$ \frac{k_1}{k_2} = \frac{\omega_1}{\omega_2} = \frac{k_1\big[\epsilon_1 (6 k_0^2+3\kappa^2-2 k_1^2)-4 k_0+2\kappa (1-3 k_0\epsilon_1) \cos(\phi_1)\big]}{k_2\big[\epsilon_1 (6 k_0^2+3\kappa^2-2 k_2^2)-4 k_0+2\kappa (1-3 k_0\epsilon_1) \cos(\phi_2)\big]}, $(45)

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$ \cos(\phi_2) = -\cos(\phi_1)+ \frac{3 k_0}{\kappa}-\frac1{\kappa\epsilon_1}. $(46)

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$\begin{split} & \kappa = -15,\ k0 = 0,\ \phi_1 = \arccos (1/6),\\ & \phi_2 = {{\text{π}}}/3,\ \xi_{20} = -\xi_{10} = 20. \end{split} $(47)

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$ \cos(\phi_j) = -\cos(\phi_i)+ \frac{3 k_0}{\kappa}-\frac1{\kappa\epsilon_1}, $(48)

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$\begin{split} & \kappa = -15,\ k_0 = 0, \phi_1 = \arccos\left(1/6\right),\\ & \phi_2 = {{\text{π}}}/3,\ \phi_3 = {{\text{π}}}/4,\\ & \xi_{20} = -\xi_{10} = 10,\ \xi_{30} = 0. \end{split}$(49)

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$\begin{split} & \kappa = -15, \ k_0 = 0,\\ & \phi_1 = {\text{π}}-\arccos\left(\frac12\sqrt{2}-\frac23\right),\\ & \phi_2 = \arccos\left( 1/6\right),\ \phi_3 = {{\text{π}}}/3,\ \phi_4 = {{\text{π}}}/4,\\ & \xi_{20} = -\xi_{10} = 10,\ \xi_{30} = \xi_{40} = 0.\end{split}$(50)

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Dan-Hong Xu, Sen-Yue Lou. Dark soliton molecules in nonlinear optics[J]. Acta Physica Sinica, 2020, 69(1): 014208-1
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