1. Introduction

In recent years, soliton propagation dynamics in birefringent optical fibers has become a widely studied topic [1–3] because of its important potential applications in optical communication systems [4]. It relevant to mention that due to the presence of birefringence [5], no mode is single in realistic nonlinear fibers and single mode fibers are normal bimodal [6]. Noting that the birefringence as a natural phenomenon in optical fibers can lead to pulse splitting, which has increased researchers’ interests in studying the propagation and interaction of nonlinear waves in its presence.

For monomode optical fibers, the cubic nonlinear Schrö dinger (NLS) equation can be used to describe the transmission of an optical soliton within the picosecond duration range [7]. Such key model contains principal physical effects the pulse propagation, namely, the group velocity dispersion and Kerr nonlinearity (self-phase modulation). The counterbalancing between these two processes leads to the formation of a bright soliton pulse in the anomalous dispersion regime [8] and a fundamental dark soliton in the normal dispersion regime [9]. It is interesting to note that such localized pulses can propagate in the transmitting medium without changing their forms, which makes them of great importance in optical communication systems as they are expected to be appropriate information carriers.

However, the interaction of field components with different polarizations or frequencies in optical media such as multimode fibers and birefringent fibers, and the propagation of vector solitons should consider the couple nonlinear Schrödinger (CNLS) equation [9,10]. This significant promotion of the scalar NLS equation constitute the well known integrable Manakov system [11,12], which incorporates nonlinear terms depicting the cross-interaction between the fields in addition to the group velocity dispersion and self-phase modulation effects.

Due to the extremely rich dynamics of multicomponent nonlinear waves, coupled NLS equations are under continuous exploration [13–16]. This is not surprising because these model equations are widely applied in many physical fields, ranging from optical communication, plasma physics, to Bose-Einstein condensates, and biophysics, etc [17–24]. Regarding their applications in the context of nonlinear optics, such CNLS equations arise in studying soliton exchange in the birefringent fibers, the propagation and collision of the temporal vector solitons in birefringent fibers, multi-channel bit parallel-wavelength fiber networks, and soliton wavelength division multiplexing [25–28], etc. On the other hand, recent research on the propagation of coherent and incoherent beams in photorefractive media requires in-depth research on the CNLS equation.

Nowadays, considerable interest is still paid to search for new kinds of localized pulses, uncover their optical transmission properties, and study the interaction between them. In addition, as the advancement of optical communication, research on the propagation dynamics and interaction properties of optical soliton switches becomes active both theoretically and experimentally [29–31]. In order to improve the extinction ratio and optical exchanging speed of the optical communication, the all-optical switching of optical solitons has been studied with emphasis [32,33]. Fiber optic coupler is a fibre-optical equipment applied to all-optical exchanging [34–36]. The exchanging dynamics of optical solitons in a fibre-optical coupler can be described by the following CNLS equations [37,38]

Via the dependent variable transformation

Theoretical and experimental researches have indicated that the strength redistribution of the bright solitons in the integrable CNLS systems with focused nonlinearity is characterized by the strength redistribution among interacting solitons in all components [40–42]. In such a process of multi-soliton interaction, after the interaction, intensity suppression or enhancement occurs in some components, while intensity enhancement or suppression occurs in the remaining components [40–42]. This interesting interaction behavior has also been experimentally verified in photorefractive media and birefringent fibers [28,43,44].

In recent research, a new type of the multimodal solitons has been discovered under different physical conditions [45–50]. Asymmetric bimodal-unimodal frozen states have been derived in birefringent dispersion nonlinear media [45]. When the self trapping incoherent wave packets propagate in dispersive nonlinear media, multi-hump solitons are observed for the first time in experiments [46,47]. In addition, the dynamics of multi-hump structural solitons in some dissipative structures have been studied [48–50]. In the latest research, the above-mentioned solitons are referred to as the concept of the nondegenerate solitons [51–57]. However, nondegenerate soliton solutions of the coupled NLS Eq. (1) for the physically relevant case $\delta _{1}=\delta _{2}=1$ have not been reported yet.

In this paper, we obtain nondegenerate bright solitons in a birefringent fiber with 35 degree elliptical angle $(\delta _{1}=\delta _{2}=1)$ via the Hirota bilinear methods [39] and analyze the interaction between solitons through the asymptotic analysis. Specifically, we find that the nondegenerate solitons have a node corresponding to the excited state in an effective quantum well, while the nondegenerate solitons do not have node corresponding to the ground state. The inelastic interaction between solitons is led to the nonlinear incoherent interaction between solitons of different components and the nonlinear coherent interaction between solitons of the same component. In the process of soliton interaction in both degenerate-degenerate and nondegenerate-degenerate cases, after the interaction, the intensity is suppressed (enhanced) on the $u$ component, while it is enhanced (suppressed) on the $v$ component. In addition, we find that the interference between them exhibits periodicity.

We briefly outline the paper. In Section 2, we will construct the nondegenerate bright one-soliton solutions for system (1) in a birefringent fiber with 35 degree elliptical angle via the bilinear forms given in Refs. [37,38] and give the five types of nondegenerate bright one-soliton. In Section 3, we will give nondegenerate bright two-soliton solutions in a birefringent fiber with 35 degree elliptical angle. In addition, we will also analyze three forms of the interactions between two solitons through the asymptotic analysis. We will discuss the interaction characteristics of three types of solitons through the analytical method, and confirm that the interaction between solitons in degenerate-degenerate and nondegenerate-degenerate cases is usually inelastic, which is caused by the coherent and incoherent interaction between solitons in Section 4. Conclusions will be given in Section 5.

2. Nondegenerate bright one-soliton solutions

To construct the nondegnerate bright one-soliton solutions, we assume $g=\epsilon g_{1}+\epsilon ^{3} g_{3},~ h=\epsilon h_{1}+\epsilon ^{3} h_{3},~ f=1+\epsilon ^{2}f_{2}+\epsilon ^{4}f_{4}$, where $g_{1}=\alpha _{1}^{(1)}e^{ \eta _{1}},~h_{1}=\alpha _{1}^{(2)}e^{\xi _{1}},~g_{3}=C_{1}e^{\eta _{1}+ \xi _{1}+\xi _{1}^{*}}, ~h_{3}=C_{2}e^{\xi _{1}+\eta _{1}+\eta _{1}^{*}},~f_{2}=L_{1}e^{\eta _{1}+ \eta _{1}^{*}}+L_{2}e^{\xi _{1}+\xi _{1}^{*}},~f_{4}= L_{3}e^{\eta _{1}+\eta _{1}^{*}+\xi _{1}+\xi _{1}^{*}},~\eta _{1}=k_{1}t-\frac { \text {i}}{2}k_{1}^{2}\beta _{2}z$ and $\xi _{1}=l_{1}t-\frac {\text {i}}{2} l_{1}^{2}\beta _{2}z$, and substitute them into Eq. (2) with $\epsilon =1$, and derive the nondegnerate one-soliton solutions as

The type of nondegenerate solitons mainly depends on $k_{1R}$ and $l_{1R}$. The distance between the two humps decreases as the distance between $k_{1R}$ and $l_{1R}$ increases. When $k_{1R}$ is very close to $l_{1R}$, the hump completely separates, the double-hump solitons become the two-soliton molecule. Here, the profile of nondegenerate bright solitons can be mainly divided into five different forms: soliton molecules(completely separated bimodal solitons on two components), asymmetric double-hump solitons (asymmetric bimodal solitons on two components), symmetric double-hump solitons (symmetric bimodal solitons on two components), flattop-double-hump solitons (flattop soliton on one component, symmetric bimodal soliton on another component), single-hump-double-hump solitons (unimodal solitons in one component, symmetric bimodal solitons in another component). Five different types of soliton profiles have been presented in Fig. 1, where blue and red solid lines correspond to the components $u$ and $v$, respectively. Figure 1(a1) describes the strength distribution of two-soliton molecules on two components, where the two humps are completely separated. The strength distribution of asymmetric bimodal solitons on two components is shown in Fig. 1(a2). In the effective quantum well potential, the solitons in the $u$ and $v$ components exhibit respectively the first excited state and ground state. Specifically, when $|\alpha _{1}^{(1)}|^{2}$ and $|\alpha _{1}^{(2)}|^{2}$ satisfy the following conditions,

 

Fig. 1. Intensity profiles of five typical nondegenerate one solitons via solutions (1) with $\gamma =-4,~ \beta _{0}= \beta _{1}=2,~ \beta _{2}=1,~\delta _{1}=\delta _{2}=1$. $(a1)$ soliton molecules, $(a2)$ asymmetric double-double-hump solitons, $(a3)$ symmetric double-double-hump solitons, $(a4)$ symmetric flattop-double-hump solitons, $(a5)$ symmetric double-hump-single-hump solitons. The parameters are as follows: $(a1)~k_{1}=0.5+\frac {1}{4}\text {i} ,~l_{1}=0.5001+\frac {1}{4}\text {i},~\alpha _{1}^{(1)}=\frac {1}{6}- \text {i}, ~\alpha _{1}^{(2)}=\frac {1}{2}+\text {i}$, $(a2)~k_{1}=\frac { 9}{20}+\frac {1}{4}\text {i},~l_{1}=\frac {1}{2}+\frac {1}{4}\text {i},~\alpha _{1}^{(1)}=1-2\text {i}, ~\alpha _{1}^{(2)}=2+\text {i}$, $(a3)~k_{1}=\frac {1}{5}+\frac {1}{4}\text {i},~l_{1}=\frac {1}{4}+\frac {1}{4} \text {i},~\alpha _{0}^{(1)}=\frac {3}{5} ,~\alpha _{1}^{(2)}= \frac {3}{4}$, $(a4)~k_{1}=\frac {1}{7}+\frac {1}{4}\text {i},~l_{1}=\frac {1}{5}+ \frac {1}{4}\text {i},~\alpha _{1}^{(1)}=\frac {2}{7} -\frac {\sqrt {2}}{7}\text {i},~\alpha _{1}^{(2)}=\frac {2}{5}+\frac {\sqrt {2}}{5}\text {i}$, $(a5)~k_{1}=\frac {1}{3}+\frac {1}{4}\text {i},~l_{1}= \frac {1}{2}+\frac {1}{4}\text {i}, ~\alpha _{1}^{(1)}=\frac {2}{3}-\frac { 1}{3}\text {i},~\alpha _{1}^{(2)}=\frac {1}{2}+\text {i}$.

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We find that the nondegenerate bright solitons are formed when two solitons satisfy the relative velocity of zero and certain specific coefficients. The selection of specific coefficients is consistent with the interaction between incoherent solitons of different components. The formation of double-hump solitons is related to two incoherent solitons. Therefore, we study the impact of relative velocity on the formation of nondegenerate bright solitons via studying the interaction between two solitons at different relative velocities. By adjusting the soliton velocity on each component, the incoherent interaction between two solitons can be accurately studied, as shown in Fig. 2. Here, the incoherent interaction mainly involves incoherent collision and incoherent superposition. From Fig. 2, it can be seen that the incoherent interaction increases as the relative velocity of incoherent solitons decreases. Specifically, when the relative velocity is $0$, the incoherent interaction is only affected by the incoherent superposition, which also results in bright solitons in one component being a double-hump soliton with one node, and bright solitons in the other component being a soliton (single-hump soliton, flattop soliton, double-hump soliton) without nodes.

 

Fig. 2. At different relative velocities $(l_{1I}-k_{1I})\beta _{2}$, the interaction between two solitons. The parameters are as follows: $(a1)$ and $(a2)\;l_{1I}=-k_{1I}=0.1$, $(b1)$ and $(b2)$ $l_{1I}=-k_{1I}=0.02$, $(c1)$ and $(c2)$ $l_{1I}=k_{1I}=0$. The other parameters are $\alpha _{1}^{(1)}=1-2\text {i},~\alpha _{1}^{(2)}=2+\text {i},~\gamma =-8,~\beta _{0}=\beta _{1}=2,~\beta _{2}=1, ~\delta _{1}=\delta _{2}=1$.

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3. Nondegenerate two-soliton solutions and asymptotic analysis

In order to obtain the nondegnerate two-soliton solutions, we substitute $g=\epsilon g_{1}+\epsilon ^{3}g_{3}+\epsilon ^{5}g_{5}+\epsilon ^{7}g_{7},~h=\epsilon h_{1}+\epsilon ^{3}h_{3}+\epsilon ^{5}h_{5}+\epsilon ^{7}h_{7},~f=1+ \epsilon ^{2}f_{2}+\epsilon ^{4}f_{4} +\epsilon ^{6}f_{6}+\epsilon ^{8}f_{8}$ into Eq. (2) with $\epsilon =1$, and derive the nondegnerate two-soliton solutions of system (1) as

The coefficients in solution (6) satisfy $l_{j}=k_{j}$ in this situation. We consider $k_{2I}>k_{1I}$, $\beta _{2}>0$ and $k_{1R},k_{2R}>0$, and the asymptotic analysis of degenerate-degenerate solitons is as follows

(1) Before the interaction ($z\rightarrow -\infty$)

Soliton $S^{1-}$ $(\eta _{1}+\eta _{1}^{*}\sim 0,~\eta _{2}+\eta _{2}^{*} \rightarrow -\infty )$:

Soliton $S^{2-}$ $(\eta _{2}+\eta _{2}^{*}\sim 0,~\eta _{1}+\eta _{1}^{*} \rightarrow +\infty )$:

(2) After the interaction ($z\rightarrow +\infty$)

Soliton $S^{1+}$ $(\eta _{1}+\eta _{1}^{*}\sim 0,~\eta _{2}+\eta _{2}^{*} \rightarrow +\infty )$:

Soliton $S^{2+}$ $(\eta _{2}+\eta _{2}^{*}\sim 0,~\eta _{1}+\eta _{1}^{*} \rightarrow -\infty )$:

The asymptotic analysis of the nondegenerate-degenerate and the nondegenerate-nondegenerate case are shown in Appendix B and C, respectively.

4. Interaction characteristics of the solitons

We will discuss the interaction characteristics of the solitons through analysis and graphical methods. Next, we analyze the interactions between the following three types of solitons (degenerate-degenerate solitons, nondegenerate-degenerate solitons, nondegenerate-nondegenerate solitons).

4.1 Degenerate-degenerate (D-D)

Before the interaction, the amplitudes are $(A_{1}^{1-},A_{1}^{2-})=\left ( \frac {\alpha _{11}}{2\sqrt {m_{1}}},\frac {m_{2}}{2\sqrt {m_{1}m_{4}}}\right )$ for the component $u$, and the amplitudes of component $v$ are $(A_{2}^{1-},A_{2}^{2-})=\left (\frac {\alpha _{12}}{2 \sqrt {m_{1}}},\frac {m_{3}}{2\sqrt {m_{1}m_{4}}}\right )$. After the interaction, the amplitudes of component $u$ are $(A_{1}^{1+},A_{1}^{2+})= \left (\frac {n_{1}}{2\sqrt {n_{2}m_{4}}},\frac {\alpha _{21}}{2\sqrt {n_{2}}} \right )$, and the amplitudes of component $v$ are $(A_{2}^{1+},A_{2}^{2+})=\left (\frac {n_{3}}{2 \sqrt {n_{2}m_{4}}},\frac {\alpha _{22}}{2\sqrt {n_{2}}}\right )$. By directly calculating the expressions for the amplitudes of the degenerate solitons mentioned above, we obtain that

From expression (11), it can be seen that there is an energy redistribution phenomenon between degenerate solitons of different components. And we also obtain that two degenerate solitons $S^{1}$ and $S^{2}$ are inelastic interactions on components $u$ and $v$ when $\frac {\alpha _{21}}{\alpha _{11}}\neq \frac {\alpha _{22}}{ \alpha _{12}}$ or $\frac {|\alpha _{12}|^{2}}{|\alpha _{11}|^{2}}\neq -\frac { \delta _{1}}{\delta _{2}}$. Specifically, two degenerate bright solitons $S^1$ and $S^2$ in both $u$ and $v$ components are elastic interactions, when the following conditions are satisfied

Whether the interaction is inelastic or elastic, we find that the total intensities of solitons before the interaction are equal to that after the soliton interaction, namely $\sum _{i,j=1}^{2}|A_{i}^{j-}|^{2}= \sum _{i,j=1}^{2}|A_{i}^{j+}|^{2}$. The velocities of solitons before and after the interaction remain unchanged as $v_{1}=-\beta _{2}k_{1I}$ and $v_{2}=-\beta _{2}k_{2I}$, respectively, where $v_1$ and $v_2$ represent the velocities of solitons $S^1$ and $S^2$, respectively. After the interaction between two solitons, the solitons undergo phase shift. The phase shifts of solitons $S^{1}$ and $S^{2}$ in components $u$ and $v$ are $\frac {1}{2}\ln \frac {m_{4}}{n_{2}m_{1}}$ and $\frac {1}{2}\ln \frac {m_{1}n_{2}}{m_{4}}$, respectively. The velocities of solitons $S^1$ and $S^2$ in components $u$ and $v$ remain unchanged before and after the interaction. From the expressions of velocity and amplitude, we observe that the velocity and amplitude of the solitons also increase as $|\beta _{2}|$ increases, and the amplitude of solitons decreases as $|\gamma |$ increases. The influence of $\delta _{1}$ and $\delta _{2}$ on the amplitude of solitons is relatively complex, so specific discussions are needed for specific situations. For example, for soliton $S_{1}^{1-}$ in component $u$, when $\frac { |\alpha _{11}|^{2}}{4k_{1R}^{2}}>\frac {|\alpha _{12}|^{2}}{4l_{1R}^{2}}$, $\frac {\delta _{1}}{\delta _{2}}$ increases, the amplitude of solitons increases, while when $\frac {|\alpha _{11}|^{2}}{4k_{1R}^{2}}<\frac { |\alpha _{12}|^{2}}{4l_{1R}^{2}}$, $\frac {\delta _{1}}{\delta _{2}}$ increases, the amplitude of solitons decreases. $\beta _{0}$ and $\beta _{1}$ only affect the phase shift and frequency of solitons.

Due to the redistribution of the strengths of degenerate solitons among components, two degenerate solitons undergo shape changing interactions. The weak inelastic interactions are presented in Fig. 3, where the strength of soliton $S^1$ in the component $u$ is enhanced while the corresponding intensity of soliton $S^1$ in the component $v$ is suppressed, resulting in the redistribution of intensity. In order to maintain energy conservation between components, the strength of soliton $S^2$ is suppressed in the component $u$ and enhanced in the component $v$. Figures 3(a) and (b) show the evolution diagrams of the interaction between two degenerate solitons on components $u$ and $v$, respectively. Figures 3(c) and (d) depict the strength distributions of before and after the interaction of two degenerate solitons on components $u$ and $v$, respectively, where the blue solid line represents before interaction $(z=-10)$ and the red solid line represents after interaction $(z=10)$.

 

Fig. 3. Two degenerate solitons are weakly inelastic interactions. (a) and (b) represent the dynamic evolution of degenerate soliton interactions. (c) and (d) are the intensity plots of the $u$ and $v$ components before interaction ($z=-10$, solid blue line) and after interaction ($z=10$, solid red line), respectively. It can be seen that the contour of degenerate solitons has changed after the interaction. The parameters are as follows: $k_{1}=1-\frac {1}{2}\text {i},~k_{2}=\frac {3}{2}+\text {i},~\alpha _{11}=1-\frac {1}{2}\text {i},~\alpha _{12}= 1+\text {i},~\alpha _{21}=\frac {\sqrt {2}}{2}+\frac {1}{2}\text {i},~\alpha _{22}= \frac {1}{4}+\text {i},~\gamma =-4, ~\beta _{0}=\beta _{1}=2,~\beta _{2}=1,~\delta _{1}=\delta _{2}=1$.

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4.2 Nondegenerate-degenerate (ND-D)

Before the interaction, the intensities of the degenerate soliton $S^{2}$ are $|\frac {\chi _{11}}{2\sqrt {L_{11}(\delta _{11}+d_{11})}}|^{2}$ and $|\frac { \theta _{11}}{2\sqrt {L_{11}(\delta _{11}+d_{11})}}|^{2}$ on components $u$ and $v$, respectively. After the interaction, the intensities of the degenerate soliton $S^{2}$ are $|\frac {\alpha _{21}}{2\sqrt {\mu _{22}+\nu _{22}} }|^{2}$ and $|\frac {\alpha _{22}}{2\sqrt {\mu _{22}+\nu _{22}}}|^{2}$ on components $u$ and $v$, respectively. The phase shift and velocity for the degenerate solion $S^{2}$ are $\frac {1}{2}[\ln (\delta _{11}+d_{11})-\ln L_{11}-\ln (\mu _{22}+\nu _{22})]$ and $-\beta _{2}k_{2I}$, respectively. Through the above research, we obtain that the intensities and phase shift of the degenerate soliton $S^2$ change before and after the interaction, while the velocity remains unchanged. Similarly, from expressions. (18) and (20), we find that the nondegenerate solion $S^{1}$ also undergo changes before and after the interaction. The intensity of the nondegenerate and degenerate solitons before and after interaction still satisfies $\sum _{i,j=1}^{2}|A_{i}^{j-}|^{2}=\sum _{i,j=1}^{2}|A_{i}^{j+}|^{2}$. Studying the interaction between the nondegenerate and degenerate solitons, we find that their interference has periodicity. The periodic functions are $\sin [ (k_{1I}-k_{2I})t+\frac {\beta _{2}}{2} (k_{1I}^{2}-k_{2I}^{2}-k_{1R}^{2}+k_{2R}^{2})z]$ and $\sin [(k_{1I}-k_{2I})t+ \frac {\beta _{2}}{2}(k_{1I}^{2}-k_{2I}^{2}-l_{1R}^{2}+k_{2R}^{2})z]$, and their corresponding cosine functions. This means that their interference exhibits three periodic oscillation behaviors, with the temporal period of $T=\frac {2\pi }{|k_{1I}-k_{2I}|}$ and the spatial periods of $D_{1}=\frac {2\pi }{|\frac {\beta _{2}}{2}||k_{1I}^{2}-k_{2I}^{2}-k_{1R}^{2}+k_{2R}^{2}|}$ and $D_{2}=\frac {2\pi }{|\frac {\beta _{2}}{2} ||k_{1I}^{2}-k_{2I}^{2}-l_{1R}^{2}+k_{2R}^{2}|}$. We note that the temporal period is dependent on the velocity of solitons, namely $T=\frac { 2\pi |\beta _{2}|}{|v_{1}-v_{2}|}$. Based on the quantitative expression of interference period, the relative velocity, relative phase, and nonlinear interaction intensity of solitons can be measured. By adjusting the characteristic variables of velocity and amplitude, highly visible interference and tunneling modes can be obtained, as shown in Figs. 4, 5, 8 and 10. Design atomic soliton interferometers in optical fibers and Bose Einstein condensates using nonlinear interference characteristics [57–59].

 

Fig. 4. The weak inelastic interaction between a degenerate and nondegenerate soliton. (a) and (b) represent the density plots of the interaction between a degenerate soliton and a nondegenerate soliton. (c) and (d) are the intensity plots of the $u$ and $v$ components before interaction ($z=-30$, solid blue line) and after interaction ($z=30$, solid red line), respectively. The parameters are as follows: $k_{1}=\frac {1}{4}- \text {i},l_{1}=\frac {1}{3}-\text {i},~k_{2}=\frac {1}{3}+\frac {3}{2}\text {i},~ \alpha _{11} =\frac {\sqrt {3}}{4}-\frac {2}{4}\text {i},~\alpha _{12}=\frac {2}{3}-\frac {\sqrt {3}}{3}\text {i},~\alpha _{21}=1+\text {i},~\alpha _{22}=1 +\frac {1}{2}\text {i},~\gamma =-4, ~\beta _{0}=\beta _{1}=2,~\beta _{2}=1,~\delta _{1}=\delta _{2}=1$.

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Fig. 5. (a) and (b) represent respectively the weak inelastic interaction between a double-hump and degenerate solitons on components $u$ and $v$. The parameters are equal to Fig. 4, except that $k_{2}= \frac {1}{4}+\text {i}$.

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Figures 4(a) and (b) show the density diagrams of the weak inelastic interaction between a double-hump soliton and a degenerate soliton on components $u$ and $v$, respectively. We can observe that a double-hump bright soliton have one node on component $u$, while a bimodal bright soliton have no node on components $v$. Figures 4(c) and (d) represent the profile plots before and after the interaction on the corresponding components. From Figs. 4(c) and (d), it can be seen that the interaction between bimodal soliton and degenerate soliton is weakly inelastic. We observe that before the interaction, double-hump solitons are symmetric, and after the interaction, there are asymmetric double-hump solitons. We can clearly see the interference fringes in both time and space in Figs. 4(a) and (b). Specifically, when $k_{2I}=-k_{1I},~k_{2R}=k_{1R}$, only the temporal interference pattern [57] is visible, as presented in Figs. 5(a) and (b). Figure 6 displays the three-dimensional evolution diagram of the strong inelastic interaction between bound state and degenerate soliton. We can observe that the nondegenerate soliton is an asymmetric bimodal soliton with one node before the interaction in Fig. 6(a), while it is an asymmetric bimodal soliton without nodes in Fig. 6(b), and after the interaction, they are bound solitons in both the $u$ and $v$ components. We note that the soliton wave numbers $k_j$ and $l_j$ affect the phase of all solitons. Similar to the strong inelastic interaction shown in Fig. 6, it can also be obtained by adjusting the effective potential [60,61].

 

Fig. 6. The strong inelastic interaction between a degenerate and bound state soliton. The parameters are the same as Fig. 4, except for $k_{1}=\frac {1}{3}-\frac {1}{8}\text {i},~l_{1}=\frac {1}{2}-\frac {1}{8}\text {i} ,~k_{2}=\frac {1}{3}+\frac {1}{4}\text {i}$.

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Figure 7 illustrates the strong inelastic interactions between a nondegenerate soliton $S^1$ and a degenerate soliton $S^2$ on components $u$ and $v$, respectively. And it can be obtained that the single-hump soliton $S^1$ forms an asymmetric double-hump soliton after interaction.

 

Fig. 7. (a) and (b) represent respectively the strong inelastic interaction between a nondegenerate and degenerate solitons on components $u$ and $v$. The parameters are as follows: $k_{1}=1-\text {i},~l_{1}=\frac {3}{2}-\frac {1}{2}\text {i},~k_{2}=1+\text {i},~\alpha _{11}=1+\text {i},~\alpha _{12}=\frac {1}{2}- \frac {1}{2}\text {i},~\alpha _{21}=\frac {1}{6}+\frac {1}{6}\text {i},~\alpha _{22}=\frac {1}{5}+\frac {1}{5}\text {i},\gamma =-4,~\beta _{0} =\beta _{1}=2,~\beta _{2}=1,\delta _{1}=\delta _{2}=1$.

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4.3 Nondegenerate-nondegenerate (ND-ND)

We demonstrate that the intensities and velocities of two nondegenerate solitons remain unchanged after interaction through the asymptotic expressions (21)–(24). The velocities of soliton $S^1$ and $S^2$ are $-k_{1I}\beta _{2}$ and $-k_{2I}\beta _{2}$, respectively. To confirm that the strengths of solitons are equal before and after the collision, we calculate the transition intensities

We first choose the center of the soliton as the reference frame, and the effective intrinsic energy of the soliton is defined as $E^{*}=\frac {\text {d} \phi _{j}}{\text {d} z}$, where $\phi _{j}$ represents the phase of the soliton [57]. Therefore, the effective intrinsic energies of soliton $S^1$ in components $u$ and $v$ are $E_{1}^{1*}=\frac {\beta _{2}}{2} (k_{1I}^{2}-k_{1R}^{2})$ and $E_{2}^{1*}=\frac {\beta _{2}}{2} (k_{1I}^{2}-l_{1R}^{2})$, respectively, and the effective intrinsic energies of soliton $S^2$ in components $u$ and $v$ are $E_{1}^{2*}=\frac {\beta _{2}}{2 }(k_{2I}^{2}-k_{2R}^{2})$ and $E_{2}^{2*}=\frac {\beta _{2}}{2} (k_{2I}^{2}-l_{2R}^{2})$, respectively. Finally, we obtain $D_{1}=\frac {2\pi }{|E_{1}^{1*}-E_{1}^{2*}|}$ and $D_{2}=\frac {2\pi }{|E_{2}^{1*}-E_{2}^{2*}|}$, from which we can conclude that the interference and tunneling period of nondegenerate solitons in the time direction is determined by the effective intrinsic energy difference between the two solitons.

Figures 8(a) and (b) respectively show the elastic interaction between two symmetric double-hump solitons on the component $u$ and $v$. Figures 8(c) and (d) show the profile plots of solitons before and after the interaction on two components. We note that there is no redistribution of strength between components $u$ and $v$. In ND-ND cases, the interaction between solitons is usually elastic. However, we note that the structure of nondegenerate solitons change when the phase change, as shown in Fig. 9. Figure 9(a) displays the interaction between two bimodal solitons, and (b) represents the interaction between a bimodal solitons and a single-hump soliton. We note that soliton $S^1$ is the symmetric soliton before the interaction, and the asymmetric soliton after the interaction, while soliton $S^2$ is the asymmetric soliton before the interaction, and the symmetric soliton after the interaction. In addition, the tunneling dynamics of two nondegenerate bright solitons are observed, as shown in Figs. 10(a) and (b) represent the density evolution of the components $u$ and $v$, respectively. The tunneling dynamics of nondegenerate bright solitons are significantly more complex than those of degenerate bright solitons. Periodic tunneling dynamics can be considered as a nonlinear Josephson type oscillation [62].

 

Fig. 8. The elastic interaction between two nondegenerate solitons. (a) and (b) represent the dynamic evolution of the interaction between two bimodal solitons. (c) and (d) are the intensity plots of the $u$ and $v$ components before interaction ($z=-35$, solid blue line) and after interaction ($z=35$, solid red line), respectively. The parameters are as follows: $k_{1}=\frac {1}{4}-\text {i},l_{1}=\frac {1}{3}-\text {i},~k_{2}=\frac { 1}{5}+\frac {1}{2}\text {i}, ~l_{2}=\frac {1}{4}+\frac {1}{2}\text {i},~\alpha _{11}=\frac {\sqrt {3}}{4}-\frac {1}{2}\text {i},~\alpha _{12}= \frac {\sqrt {3}}{3}-\frac {2}{3}\text {i},~\alpha _{21}= \frac {3}{5},~\alpha _{22}=\frac {3}{4},~\gamma =-4, ~\beta _{0}=\beta _{1}=2,~\beta _{2}=1,~\delta _{1}= \delta _{2}=1$.

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Fig. 9. The weak inelastic interaction between two nondegenerate solitons. The parameters are as follows: $k_{1}=\frac {1}{4}-\frac {1}{2}\text {i},l_{1}= \frac {1}{3}-\frac {1}{2}\text {i},~k_{2}=\frac {1}{4}+\frac {1}{4}\text {i}, ~l_{2}=\frac {1}{2}+\frac {1}{4}\text {i},~\alpha _{11}=\frac {\sqrt {3}}{4}-\frac {1}{2}\text {i},~\alpha _{12}= \frac {\sqrt {3} }{3}-\frac {2}{3}\text {i},~\alpha _{21}=\frac {\sqrt {3}}{4},~ \alpha _{22}=\frac {\sqrt {3}}{2}, ~\gamma =-4,~\beta _{0}=\beta _{1}=2,~\beta _{2}=1,~\delta _{1}= \delta _{2}=1$.

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Fig. 10. The tunneling dynamics between two nondegenerate bright solitons. The parameters are as follows: $k_{1}=2.8,~l_{1}=3,~k_{2}=3.1, ~l_{2}=3.5,~ \alpha _{11}=\alpha _{12}= \alpha _{21}=\alpha _{22}=\frac {3}{5},~\gamma =-4, ~\beta _{0}=\beta _{1}=2,~\beta _{2}=1,~\delta _{1}=\delta _{2}=1$.

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5. Conclusions

In this paper, we have studied the optical nondegenerate solitons in a birefringent fibers with 35 degree elliptical angle. The nondegenerate one- and two-soliton solutions have been derived via given the bilinear forms. We have demonstrated that the formation of nondegenerate solitons is led to the incoherent superposition of solitons with different components. Based on the different structures of nondegenerate solitons, we classify nondegenerate one-soliton on the $u$ and $v$ components into the following five types: soliton molecules, asymmetric double-hump solitons, symmetric double-hump solitons, flattop-double-hump solitons, single-hump-double-hump solitons. In an effective quantum well, we have found that nondegenerate solitons have nodes corresponding to the first excited state, while nondegenerate solitons without nodes correspond to the ground state.

Two degenerate soliton are inelastic interactions, and during the process, the strengths of the solitons on the two components are redistributed. Specifically, the interaction between two solitons is elastic under the condition $\frac {\alpha _{21}}{\alpha _{11}}=\frac {\alpha _{22}}{ \alpha _{12}}$ and $\frac {|\alpha _{12}|^{2}}{|\alpha _{11}|^{2}}=-\frac { \delta _{1}}{\delta _{2}}$. In addition, we have also analyzed the influence of the Kerr nonlinear intensity coefficient $\gamma$ and the second-order group velocity dispersion $\beta _{2}$ in this system on solitons: the amplitude and velocity of solitons are proportional to $|\beta _{2}|$, while the amplitude of solitons is inversely proportional to $\gamma$.

A nondegenerate and a degenerate soliton are inelastic interactions, and similarly, the strengthes of the solitons are redistributed between the two components. After interaction, nondegenerate solitons can propagate in the shape of the bound state solitons or in the shape of the double-hump solitons, by adjusting the wave numbers $k_1$ and $l_1$. In addition, we have noticed that the interference between nondegenerate and degenerate solitons is periodic, and by adjusting the wave numbers $k_{1},~k_{2}$ and $l_1$, the visibility of the interference fringes can be changed.

Two nondegenerate solitons are elastic interactions, but the phase of the soliton can be adjusted to make it inelastic. Similarly, the interference between two nondegenerate solitons also exhibits periodicity, which is influenced by the wave numbers $k_{1},~k_{2},~l_{1}$ and $l_{2}$. Specifically, when $k_{1I}=k_{2I}$, the tunneling phenomenon between two nondegenerate states can be observed.

Whether the interaction modes, total intensities of the solitons before the interaction are equal to that after the soliton interaction. The inelastic interaction between solitons is led to the incoherent interaction between solitons of different components and the coherent interaction between solitons of the same component.