Fund Project:National Natural Science Foundation of China (Nos. 11975131, 11435005)
Received Date:06 September 2019
Accepted Date:25 October 2019
Available Online:12 December 2019
Published Online:05 January 2020
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. Keywords:dark soliton molecules/ higher order nonlinear effects/ elastic interactions/ integrable systems
对于小的$ m $, Jacobi椭圆函数sn接近于三角函数sin. 而随着模m 越来越接近于1, sn越来越接近于双曲tanh函数, 而相应的周期解就看起来像是孤立子的周期性排列. 因此对于接近于m = 1的周期解, 也可以称之为孤子晶格解. 图1 展示了用(15) 式描述的周期波解、亮孤子晶格解. 其相应的参数为(本节所有的图中的模型参数固定为$ \beta = d_2 = d_3 = r_2 = 10\epsilon = 1 $) 图 1 由(15) 式描述的亮孤子晶格, 其中参数由(16)式给定 Figure1. Bright soliton lattice described by Eq.(15) with the parameter selected from Eq. (16)
$\begin{split} & k = \beta_1 = \sigma = 1,\ m = 0.99,\\ & \alpha_1 = \frac{199}{198},\ \xi_0 = 0. \end{split}$
图2展示了用同一表达式(15)式 描述的暗孤子晶格解. 其相应的参数为 图 2 由(15)式描述的暗孤子晶格, 其中参数由 (17)式给定 Figure2. Dark soliton lattice described by Eq. (15) with the parameter selected from Eq. (17)
图3 展示了第二种类型的亮孤子晶格结构. 这类亮孤子晶格由表达式(18) 式描述. 与图3对应的参数为 图 3 由(18)式描述 第二类亮孤子晶格, 其中参数由 (20)式给定 Figure3. Second type of bright soliton lattice described by Eq. (18) with the parameter selected from Eq. (20)
$ k = \beta_1 = \sigma = 1,\ m = 0.9,\ \alpha_1 = \frac{41}{40},\ \xi_0 = 0. $
由于Jacibi椭圆函数dn的恒正性, 这类函数描述的孤子晶格既可以是亮孤子晶格, 也可以是暗孤子晶格. 图4 展示了第三种类型的亮孤子晶格结构. 这类亮孤子晶格由表达式(21)式 描述. 与图4对应的参数为 图 4 第三类亮孤子晶格. 由(21)式描述, 其中参数由 (23)式给定 Figure4. Third type of bright soliton lattice described by Eq. (21) with the parameter selected from Eq. (23)
$\begin{split} & k = \beta_1 = \sigma = 1,\ m = 0.99999,\\ & \alpha_1 = {41}/{40},\ \xi_0 = 0. \end{split}$
这一类亮孤子也具有灰色背景, 但不同于第二类亮孤子晶格, 灰背景和亮孤子之间没有暗区隔离. 图5展示了第二类暗孤子晶格结构. 这类暗孤子晶格由表达式(21) 式描述. 与图5对应的参数为 图 5 第二类暗孤子晶格由(21)式描述, 其中参数由 (24)式给定 Figure5. Second type of dark soliton lattice described by Eq. (21) with the parameter selected from Eq. (24)
$\begin{split} & k = \beta_1 = -\sigma = 1,\ m = 0.99999,\\ &\alpha_1 = 1/2,\ \xi_0 = 0. \end{split} $
在一般情况下双孤子解(42)式和(43)式是具有弹性相互作用的双孤子态. 图10展示了这样一个典型的双孤子作用图像, 其中采用的参数为(本节中统一采用的模型参数是$ d_2 = r_2 = -\sigma = 10\epsilon_1 = 1 $) 图 10 由(42)?(43)式描述的二暗孤子相互作用的密度图, 图中参数由 (44)式给定 Figure10. Density plot of the interaction between two dark solitons described by Eq. (42)and Eq. (43) with the parameter selected from Eq. (44)
图11展示了孤子分子对应的光强的密度图(图11(a))和立体图(图11(b)), 与图对应的参数为 图 11 (a) 由(42)式和(43)式描述的暗孤子分子密度图, 图中参数由 (47)式给定; (b) 与图(a)对应的三维立体图 Figure11. (a) Density plot of the dark soliton molecule described by Eq. (42) and Eq.(43) with the parameter selected from Eq. (47); (b) three dimensional plot related to Fig. (a)
则解(34)式, (37)式和(38)式表示了$ n $暗孤子分子和$ N-2 n $ 暗孤子的混合相互作用解. 图12展示的是一个暗孤子分子和一个暗孤子的相互作用. 图中对应的参数为 图 12 由(34)式,(37)式和(38)式描述的暗孤子分子和暗孤子的弹性相互作用的密度图, 图中参数由 (49)式给定 Figure12. Density plot of the interaction between a dark soliton molecule and a dark soliton described by Eq. (34), Eq. (37) and Eq. (38) with the parameter selected from Eq. (49)
图13展示了二暗孤子分子的相互作用. 图中对应的参数为 图 13 由(34)式,(37)式和(38)式描述的二暗孤子分子的弹性相互作用的密度图, 图中参数由 (50)式给定 Figure13. Density plot of the interaction between two dark soliton molecules described by Eq. (34), Eq. (37) and Eq. (38) with the parameter selected from Eq. (50)