图 1  取亮孤子为初态, 淬火交叉相互作用$ \sigma_{2} $产生的冲击波　(a)参数取$ \sigma_{2}=1 $时, 亮孤子的轮廓图; (b)冲击波波速$ v_{\eta} $与声速$ C_{\eta} $随交叉相互作用强度$ \sigma_{2} $的变化曲线; (c)和(d)分别为$ \sigma_{2}=4,\; s=18 $和$ \sigma_{2}=5.8, \;s=16.5 $时冲击波的轮廓图, 二者波速分别为$ v_{\eta}=1.57 $和$ v_{\eta}=1.63 $, 此时选取(b)图中黑点对应的参数. 图中的插图均是取$ \xi=0 $的截面图

Fig. 1.  A bright soliton is chosen as the initial incident wave, shock waves are generated through quenching the cross-interaction strength $ \sigma_{2} $: (a) Profile of the bright soliton with $ \sigma_{2}=1 $; (b) curves illustrating the variation of shock wave velocity and sound speed as a function of $ \sigma_{2} $; (c) and (d) represent the profiles of shock waves at $ \sigma_{2}=4,\; s = 18,\; v_{\eta}=1.57 $ and $ \sigma_{2}=5.8,\; s = 16.5,\; v_{\eta}=1.63 $, respectively, with corresponding parameters marked by the black dots in panel (b). The insets are cross-sections of the bright soliton and shock waves at $ \xi=0 $.

图 2  取类暗孤子为初态, 淬火交叉相互作用强度$ \sigma_{2} $产生的冲击波　(a)参数取$ \sigma_{2}=1 $时, 类暗孤子的轮廓图; (b)冲击波波速$ v_{\eta} $与声速$ C_{\eta} $随交叉相互作用强度$ \sigma_{2} $的变化曲线; (c)和(d)分别为$ \sigma_{2}=4.5,\; s=18,\;v_{\eta}=1.5 $和$ \sigma_{2}=6.5,\; s=18,\; v_{\eta}= $$ 1.5 $时冲击波的轮廓图, 此时选取(b)图中黑点对应的参数. 图中的插图均是取$ \xi=0 $的截面图

Fig. 2.  A dark-like soliton is chosen as the initial incident wave, shock waves are generated through quenching the cross-interaction strength $ \sigma_{2} $: (a) Profile of the dark-like soliton with $ \sigma_{2}=1 $; (b) curves illustrating the variation of shock wave velocity and sound speed as a function of $ \sigma_{2} $; (c) and (d) represent the profiles of shock waves at $ \sigma_{2}=4.5,\; s=18,\; v_{\eta}=1.5 $ and $ \sigma_{2}=6.5,\; s=18,\; v_{\eta}=1.5 $, respectively, with corresponding parameters marked by the black dots in panel (b). The insets are cross-sections of the dark-like soliton and shock waves at $ \xi=0 $.

图 3  以类暗孤子为初态, 淬火外势产生的冲击波　(a) 冲击波的存在区域图. (b)—(c) 分别为 $ \sigma_{11}=3,\; v_{\eta}=1.42,\; s=11 $ 和$ \sigma_{11}=7,\; v_{\eta}=1.42,\; s=11 $的冲击波轮廓图, 其中 $ d_{\xi}=8, d_{\eta}=1 $. 右上插图给出冲击波的投影图, 右下插图给出$ \xi=0 $ ($ \eta=0 $) 时的截面图, 用蓝色实线 (红色虚线) 表示. (d) $ \sigma_{11}=3 $, $ d_{\xi}=\infty $, $ d_{\eta}=1, \; s=8 $ 时冲击波的投影图, 此时, 冲击波速度为$ v_{\eta}=1.42 $, 图3(e) 和 3(f) 分别给出 3(d) 中 $ \xi=-16.16 $ 与$ \xi=0 $ 处两条黑色虚线对应的截面图

Fig. 3.  A dark-like soliton is chosen as the initial incident wave, shock waves are produced by quenching the external potential. (a) Existence interval of the shock wave as a function of $ d_{\eta} $ and $ \sigma_{12} $; (b)–(c) Profiles of shock waves with $ \sigma_{11}=3, \, 7,\; s=11 $, here, $ d_{\xi}=8,\; d_{\eta}=1,\; v_{\eta}=1.42 $, projections and cross-sections along the η (blue solid line)and ξ (red dashed line)directions are displayed on the right side of (b) and (c); (d) Projection of the shock wave, here, $ \sigma_{11}=3,\; d_{\xi}=\infty, \; d_{\eta}=1,\; s=8,\; v_{\eta}=1.42 $; The cross-sections corresponding to the two black dashed lines for $ \xi=-16.16 $ and $ \xi=0 $ are illustrated in (e) and (f), respectively.

图 4  通过调控外势在均匀凝聚体中激发的冲击波　(a) 冲击波的存在区域图. (b)—(c) $ \sigma_{11}= 3$和$ \sigma_{11}= 7 $时对应的冲击波轮廓图, 其中 $ d_{\xi}=8, \; d_{\eta}=1,\; v_{\eta}=1.72, s=11 $. 右上插图给出冲击波的投影图, 右下插图给出$ \xi=0 $ ($ \eta=0 $) 时的截面图, 用蓝色实线 (红色虚线) 表示

Fig. 4.  Shock waves induced by potential in a homogeneous condensate. (a) Existence interval of the shock wave as a function of $ d_{\eta} $ and $ \sigma_{12} $; (b)–(c) Profiles of shock waves with $ \sigma_{11}=3, \, 7 $, here, $ d_{\xi}=8, \;d_{\eta}=1,\; v_{\eta}=1.72,\; s=11 $, projections and cross-sections along the η (blue solid line)and ξ (red dashed line) directions are displayed on the right side of (b) and (c).

图 5  通过调控泵浦在均匀凝聚体中激发的冲击波　(a) 冲击波的存在区域图. (b)—(c) $ \sigma_{10}=3 $ 和$ \sigma_{10}=7 $时对应的冲击波轮廓图, 其中 $ d_{\xi}=10,\; d_{\eta}=2, \; v_{\eta}=1.57, \; s=17 $. 右上插图给出冲击波的投影图, 右下插图给出$ \xi=0 $ ($ \eta=0 $) 时的截面图, 用蓝色实线 (红色虚线) 表示.

Fig. 5.  Shock waves induced by incoherent pump in a homogeneous condensate. (a) Existence interval of the shock wave as a function of $ d_{\eta} $ and $ \sigma_{12} $; (b)–(c) Profiles of shock waves with $ \sigma_{10}=3, \, 7 $, here, $ d_{\xi}=10, \; d_{\eta}=2, \; v_{\eta}=1.57, \; s=17 $, projections and cross-sections along the η (blue solid line)and ξ (red dashed line) directions are displayed on the right side of (b) and (c).