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Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions

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Abstract

In this paper, we construct degenerate soliton solutions (which preserve \(\mathscr {PT}\)-symmetry/break \(\mathscr {PT}\)-symmetry) to the nonlocal Manakov system through a nonstandard bilinear procedure. Here, by degenerate we mean the solitons that are present in both the modes which propagate with same velocity. The degenerate nonlocal soliton solution is constructed after briefly indicating the form of nondegenerate one-soliton solution. To derive these soliton solutions, we simultaneously solve the nonlocal Manakov equation and a pair of coupled equations that arise from the zero curvature condition. The latter consideration yields general soliton solution which agrees with the solutions that are already reported in the literature under certain specific parametric choice. We also discuss the salient features associated with the obtained degenerate soliton solutions.

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Acknowledgements

The work of MS forms part of a research project sponsored by DST-SERB, Government of India, under the Grant No. EMR/2016/001818. The research work of ML is supported by a SERB Distinguished Fellowship and also forms part of the DAE-NBHM research project (2/48 (5)/2015/NBHM (R.P.)/R&D-II/14127).

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  1. Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, Tamil Nadu, 620 024, India

    S. Stalin, M. Senthilvelan & M. Lakshmanan

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  1. S. Stalin
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  2. M. Senthilvelan
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  3. M. Lakshmanan
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Correspondence to M. Senthilvelan.

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Appendices

Appendix

A. The constants which appear in the nondegenerate one-soliton solution (10a)–(10b)

The constants which appear in the nondegenerate one-soliton solution (10a)–(10b) have the explicit forms,

$$\begin{aligned}&\text{ e }^{\varDelta _1^{(j)}}=\frac{(-1)^j(\bar{k}_1^{(1)}-\bar{k}_1^{(2)})\alpha _1^{(1)}\alpha _1^{(2)}\beta _1^{(3-j)}}{(\bar{k}_1^{(j)}+k_1^{(3-j)})(k_1^{(3-j)}+\bar{k}_1^{(3-j)})^2}, \end{aligned}$$
(26)
$$\begin{aligned}&\text{ e }^{\gamma _1^{(j)}}=\frac{(-1)^j(k_1^{(1)}-k_1^{(2)})\alpha _1^{(3-j)}\beta _1^{(1)}\beta _1^{(2)}}{(k_1^{(j)}+\bar{k}_1^{(3-j)})(k_1^{(3-j)}+\bar{k}_1^{(3-j)})^2},\nonumber \\&\quad ~ j=1,2, \end{aligned}$$
(27)
$$\begin{aligned}&\text{ e }^{\delta _1}=\frac{-\alpha _1^{(1)}\beta _1^{(1)}}{(k_1^{(1)}+\bar{k}_1^{(1)})^2},~\text{ e }^{\delta _2}=\frac{-\alpha _1^{(2)}\beta _1^{(2)}}{(k_1^{(2)}+\bar{k}_1^{(2)})^2}, \end{aligned}$$
(28)
$$\begin{aligned}&\text{ e }^{\delta _3}=\frac{\alpha _1^{(1)}\alpha _1^{(2)}\beta _1^{(1)}\beta _1^{(2)}(k_1^{(1)}-k_1^{(2)})(\bar{k}_1^{(1)}-\bar{k}_1^{(2)})}{(k_1^{(1)}+\bar{k}_1^{(1)})^2(\bar{k}_1^{(1)}+k_1^{(2)})(k_1^{(1)}+\bar{k}_1^{(2)})(k_1^{(2)}+\bar{k}_1^{(2)})^2}.\nonumber \\ \end{aligned}$$
(29)

B. The constants which appear in the reduced form of general two-soliton solution (21a)–(21b)

The following constants appear in the two-soliton solution (21a)–(21b)

$$\begin{aligned} \text{ e }^{\varDelta _1^{(j)}}= & {} \bar{\varrho }_{12}(-1)^{3-j}k_1\beta _1^{(3-j)}\nu _1+\bar{k}_2\alpha _2^{(j)}\varGamma _{11}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{21}]/\kappa _{11}\kappa _{12}, \end{aligned}$$
$$\begin{aligned} \text{ e }^{\varDelta _2^{(j)}}= & {} \bar{\varrho }_{12}[(-1)^{3-j}k_2\beta _2^{(3-j)}\nu _1+\bar{k}_2\alpha _2^{(j)}\varGamma _{12}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{22}]/\kappa _{21}\kappa _{22},\\ \text{ e }^{\gamma _1^{(j)}}= & {} \varrho _{12}[k_2\beta _2^{(j)}\varGamma _{11}-k_1\beta _1^{(j)}\varGamma _{12}\\&+\,(-1)^{(j)}\bar{k}_1\alpha _1^{(3-j)}\nu _2]/\kappa _{11}\kappa _{21},\\ \text{ e }^{\gamma _2^{(j)}}= & {} \varrho _{12}[k_2\beta _2^{(j)}\varGamma _{21}-k_1\beta _1^{(j)}\varGamma _{22}\\&+\,(-1)^{(j)}\bar{k}_2\alpha _2^{(3-j)}\nu _2]/\kappa _{12}\kappa _{22},\\ \text{ e }^{\delta _{1}}= & {} -\frac{\varGamma _{11}}{\kappa _{11}},~\text{ e }^{\delta _{2}}=-\frac{\varGamma _{21}}{\kappa _{12}},\\ ~\text{ e }^{\delta _{3}}= & {} -\frac{\varGamma _{12}}{\kappa _{21}},~\text{ e }^{\delta _{4}}=-\frac{\varGamma _{22}}{\kappa _{22}}, \\ \text{ e }^{\delta _{5}}= & {} \varrho _{12}\bar{\varrho }_{12}(\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2\\&-\,\varrho _{3}\varGamma _{21}\varGamma _{12})/\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}. \end{aligned}$$

Correspondingly in the \(\mathscr {PT}\)-symmetry preserving two-soliton solution (23), the form of various constants can be given as follows.

$$\begin{aligned} \text{ e }^{\delta _1}= & {} \frac{2(l_1+\bar{l}_2)(l_2+\bar{l}_1)}{(l_1-l_2)(\bar{l}_1-\bar{l}_2)}\text{ e }^{i(\theta _{12}+\varphi _{12})},\\ \text{ e }^{\delta _2}= & {} \frac{-2(l_1+\bar{l}_1)(l_2+\bar{l}_2)}{(l_1-l_2)(\bar{l}_1-\bar{l}_2)}\text{ e }^{i(\theta _{22}+\varphi _{12})},\\ \text{ e }^{\delta _3}= & {} \frac{-2(l_1+\bar{l}_1)(l_2+\bar{l}_2)}{(l_1-l_2)(\bar{l}_1-\bar{l}_2)}\text{ e }^{i(\theta _{12}+\varphi _{22})},\\ \text{ e }^{\delta _4}= & {} \frac{2(l_1+\bar{l}_2)(l_2+\bar{l}_1)}{(l_1-l_2)(\bar{l}_1-\bar{l}_2)}\text{ e }^{i(\theta _{22}+\varphi _{22})},\\ \text{ e }^{\varDelta _1^{(j)}}= & {} -\frac{2(l_1+\bar{l}_2)(l_2+\bar{l}_2)}{(\bar{l}_1-\bar{l}_2)}\text{ e }^{i(\theta _{12}+\varphi _{12}+\theta _{2j})},\\ \text{ e }^{\delta _5}= & {} 4\text{ e }^{i(\theta _{12}+\theta _{22}+\varphi _{12}+\varphi _{22})},\\ \text{ e }^{\varDelta _2^{(j)}}= & {} \frac{2(l_1+\bar{l}_1)(l_2+\bar{l}_1)}{(\bar{l}_1-\bar{l}_2)}\text{ e }^{i(\theta _{12}+\varphi _{22}+\theta _{2j})}, ~j=1,2. \end{aligned}$$

C. The unfactored degenerate two-soliton solution

A general unfactored degenerate two-soliton solution can be deduced by considering the following forms of seed solution for the functions \(g_1^{(j)}(x,t)\) and \(g_1^{(j)*}(-x,t)\), for Eq. (8) that is

$$\begin{aligned}&g^{(j)}_1(x,t)=\alpha _{1}^{(j)}\text{ e }^{\bar{\xi }_1}+\alpha _{2}^{(j)}\text{ e }^{\bar{\xi }_2},~\bar{\xi _j}=i \bar{k_{j}}x+i\bar{k_{j}^{2}}t , \end{aligned}$$
(30a)
$$\begin{aligned}&g^{(j)*}_1(-x,t)=\beta _{1}^{(j)}\text{ e }^{\xi _1}+\beta _{2}^{(j)}\text{ e }^{\xi _2},\nonumber \\&\quad \xi _{j}=i k_{j}x-ik_{j}^{2}t, ~j=1,2 . \end{aligned}$$
(30b)

The above form of seed solutions truncates the series expansions (7a)–(7c) at in 7-th order in \(g^{(j)}(x,t)\) and \(g^{(j)*}(-x,t)\), at 8-th order in f(x, t) and \(f^{*}(-x,t)\) and 6-th order in \(s^{(1)}(-x,t)\) and \(s^{(2)}(-x,t)\). By solving the resultant equations that arise at each order of \(\epsilon \), we have obtained the following expressions for the unknown functions \(g^{(j)}(x,t)\), \(g^{(j)*}(-x,t)\) and f(x, t),

$$\begin{aligned} g^{(j)}(x,t)= & {} \alpha _{1}^{(j)}\text{ e }^{\bar{\xi }_1}+\alpha _{2}^{(j)}\text{ e }^{\bar{\xi }_2}+\text{ e }^{\xi _{1}+2\bar{\xi }_1+\varDelta _{1}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{2}+2\bar{\xi _1}+\varDelta _{2}^{(j)}}+\text{ e }^{\xi _{1}+2\bar{\xi _2}+\varDelta _{3}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{2}+2\bar{\xi _2}+\varDelta _{4}^{(j)}} +\text{ e }^{\xi _{1}+\bar{\xi _1}+\bar{\xi _2}+\varDelta _{5}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{2}+\bar{\xi _1}+\bar{\xi _2}+\varDelta _{6}^{(j)}}+\text{ e }^{2\xi _{1}+2\bar{\xi _1}+\bar{\xi _2}+\,\varDelta _{7}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+\bar{\xi _1}+2\bar{\xi _2}+\varDelta _{8}^{(j)}}+\text{ e }^{2\xi _{2}+2\bar{\xi _1}+\bar{\xi _2}+\varDelta _{9}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{2}+\bar{\xi _1}+2\bar{\xi _2}+\mu _{1}^{(j)}}+\text{ e }^{\xi _{1}+\bar{\xi _1}+\xi _2+2\bar{\xi _2}+\mu _{2}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{1}+2\bar{\xi _1}+\xi _2+\bar{\xi _2}+\mu _{3}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+2\bar{\xi _1}+\xi _{2}+2\bar{\xi _2}+\mu _{4}^{(j)}}\nonumber \\&+\,\text{ e }^{2\bar{\xi _1}+2\bar{\xi _2}+\xi _{1}+2\xi _{2}+\mu _{5}^{(j)}}\end{aligned}$$
(31a)
$$\begin{aligned} g^{(j)*}(-x,t)= & {} \beta _{1}^{(j)}\text{ e }^{\xi _{1}}+\beta _{2}^{(j)}\text{ e }^{\xi _{2}}+\text{ e }^{2\xi _{1}+\bar{\xi _1}+\gamma _{1}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+\bar{\xi _2}+\gamma _{2}^{(j)}}+\text{ e }^{2\xi _{2}+\bar{\xi _1}+\gamma _{3}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{2}+\bar{\xi _2}+\gamma _{4}^{(j)}}+\text{ e }^{\xi _{1}+\bar{\xi _1}+\xi _{2}+\gamma _{5}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{1}+\xi _{2}+\bar{\xi _2}+\gamma _{6}^{(j)}}+\text{ e }^{2\xi _{1}+2\bar{\xi _1}+\xi _{2}+\gamma _{7}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{1}+2\bar{\xi _1}+2\xi _{2}+\gamma _{8}^{(j)}}+\text{ e }^{2\xi _{1}+2\bar{\xi _2}+\xi _{2}+\gamma _{9}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{2}+2\bar{\xi _2}+\xi _{1}+\varphi _{1}^{(j)}}+\text{ e }^{2\xi _{1}+\bar{\xi _1}+\xi _{2}+\bar{\xi _2}+\varphi _{2}^{(j)}}\nonumber \\&+\,\text{ e }^{\xi _{1}+\bar{\xi _1}+2\xi _{2}+\bar{\xi _2}+\varphi _{3}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+2\bar{\xi _1}+2\xi _{2}+\bar{\xi _2}+\varphi _{4}^{(j)}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+\bar{\xi _1}+2\xi _{2}+2\bar{\xi _2}+\varphi _{5}^{(j)}}\end{aligned}$$
(31b)
$$\begin{aligned} f(x,t)= & {} 1+\text{ e }^{\xi _{1}+\bar{\xi _1}+\delta _1}+\text{ e }^{\xi _{2}+\bar{\xi _1}+\delta _2}+\text{ e }^{\xi _{1}+\bar{\xi _2}+\delta _3}\nonumber \\&+\,\text{ e }^{\xi _{2}+\bar{\xi _2}+\delta _4}+\text{ e }^{2(\xi _{1}+\bar{\xi _1})+\delta _{11}}\nonumber \\&+\,\text{ e }^{2(\xi _{2}+\bar{\xi _1})+\delta _{12}}+\text{ e }^{2(\xi _{1}+\bar{\xi _2})+\delta _{13}}\nonumber \\&+\,\text{ e }^{2(\xi _{2}+\bar{\xi _2})+\delta _{14}}+\text{ e }^{2\bar{\xi _1}+\xi _{1}+\xi _{2}+\delta _{15}}\nonumber \\&+\,\text{ e }^{2\bar{\xi _2}+\xi _{1}+\xi _{2}+\delta _{16}}+\text{ e }^{2\xi _{1}+\bar{\xi _1}+\bar{\xi _2}+\delta _{17}}\nonumber \\&+\,\text{ e }^{2\xi _{2}+\bar{\xi _1}+\bar{\xi _2}+\delta _{18}}+\text{ e }^{\xi _{1}+\bar{\xi _1}+\xi _{2}+\bar{\xi _2}+\delta _{19}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+2\bar{\xi _1}+\xi _{2}+\bar{\xi _2}+\delta _{21}}\nonumber \\&+\,\text{ e }^{2\xi _{1}+\bar{\xi _1}+\xi _{2}+2\bar{\xi _2}+\delta _{22}}\nonumber \\&+\,\text{ e }^{\xi _{1}+2\bar{\xi _1}+2\xi _{2}+\bar{\xi _2}+\delta _{23}}\nonumber \\&+\,\text{ e }^{\xi _{1}+\bar{\xi _1}+2\xi _{2}+2\bar{\xi _2}+\delta _{24}}\nonumber \\&+\,\text{ e }^{2(\xi _{1}+\bar{\xi _1}+\xi _{2}+\bar{\xi _2})+\delta _{31}}\equiv f^{*}(-x,t). \nonumber \\ \end{aligned}$$
(31c)

The explicit expression of all the constants that appear in two-soliton solution is given as

$$\begin{aligned} \text{ e }^{\delta _{1}}= & {} -2\frac{\varGamma _{11}}{\kappa _{11}},~\text{ e }^{\delta _{2}}=-2\frac{\varGamma _{12}}{\kappa _{21}},\\&\text{ e }^{\delta _{3}}=-2\frac{\varGamma _{21}}{\kappa _{12}},~\text{ e }^{\delta _{4}}=-2\frac{\varGamma _{22}}{\kappa _{22}},\\ \varGamma _{11}= & {} \left( \alpha ^{(1)}_1\beta ^{(1)}_1+\alpha ^{(2)}_1\beta ^{(2)}_1\right) ,\\ \varGamma _{12}= & {} \left( \alpha ^{(1)}_1\beta ^{(1)}_2+\alpha ^{(2)}_1\beta ^{(2)}_2\right) ,\\ \varGamma _{21}= & {} \left( \alpha ^{(1)}_2\beta ^{(1)}_1+\alpha ^{(2)}_2\beta ^{(2)}_1\right) ,\\ \varGamma _{22}= & {} \left( \alpha ^{(1)}_2\beta ^{(1)}_2+\alpha ^{(2)}_2\beta ^{(2)}_2\right) ,\\ \kappa _{lm}= & {} (k_l+\bar{k}_m)^2 ,~l,m=1,2.\\ \text{ e }^{\varDelta _{1}^{(j)}}= & {} -\frac{\alpha _1^{(j)}\varGamma _{11}}{\kappa _{11}},~\text{ e }^{\varDelta _{2}^{(j)}}=-\frac{\alpha _1^{(j)}\varGamma _{12}}{\kappa _{21}},\\ \text{ e }^{\varDelta _{3}^{(j)}}= & {} -\frac{\alpha _2^{(j)}\varGamma _{21}}{\kappa _{12}},~\text{ e }^{\varDelta _{4}^{(j)}}=-\frac{\alpha _2^{(j)}\varGamma _{22}}{\kappa _{22}}, \\ \text{ e }^{\varDelta _{5}^{(j)}}= & {} -\bigg (\alpha _1^{(j)}\varGamma _{21}(k_1+\bar{k}_1)(k_1+2\bar{k}_1-\bar{k}_2)\\&+\,\alpha _2^{(j)}\varGamma _{11}(k_1+\bar{k}_2)(k_1+2\bar{k}_2-\bar{k}_1)\bigg )/\kappa _{11}\kappa _{12},~\\ \text{ e }^{\varDelta _{6}^{(j)}}= & {} -\bigg (\alpha _1^{(j)}\varGamma _{22}(k_2+\bar{k}_1)(k_2+2\bar{k}_1-\bar{k}_2)\\&+\,\alpha _2^{(j)}\varGamma _{12}(k_2+\bar{k}_2)(k_2+2\bar{k}_2-\bar{k}_1)\bigg )/ \kappa _{21}\kappa _{22},\\ \text{ e }^{\gamma _{1}^{(j)}}= & {} -\frac{\beta _1^{(j)}\varGamma _{11}}{\kappa _{11}},~\text{ e }^{\gamma _{2}^{(j)}}=-\frac{\beta _1^{(j)}\varGamma _{21}}{\kappa _{12}},\\ \text{ e }^{\gamma _{3}^{(j)}}= & {} -\frac{\beta _2^{(j)}\varGamma _{12}}{\kappa _{21}},~\text{ e }^{\gamma _{4}^{(j)}}=-\frac{\beta _2^{(j)}\varGamma _{22}}{\kappa _{22}},\\ \text{ e }^{\gamma _{5}^{(j)}}= & {} -\bigg (\beta _1^{(j)}\varGamma _{12}(k_1+\bar{k}_1)(\bar{k}_1+2k_1-k_2)\\&+\,\beta _2^{(j)}\varGamma _{11}(k_2+\bar{k}_1)(\bar{k}_1+2k_2-k_1)\bigg )/\kappa _{11}\kappa _{21},\\ \text{ e }^{\gamma _{6}^{(j)}}= & {} -\bigg (\beta _1^{(j)}\varGamma _{22}(k_1+\bar{k}_2)(\bar{k}_2+2k_1-k_2)\\&+\,\beta _2^{(j)}\varGamma _{21}(k_2+\bar{k}_2)(\bar{k}_2+2k_2-k_1)\bigg )/\kappa _{12}\kappa _{22}. \end{aligned}$$
$$\begin{aligned} \text{ e }^{\delta _{11}}= & {} \frac{\varGamma _{11}^2}{\kappa _{11}^2},~\text{ e }^{\delta _{12}}=\frac{\varGamma _{12}^2}{\kappa _{21}^2},~\text{ e }^{\delta _{13}}=\frac{\varGamma _{21}^2}{\kappa _{12}^2},\\ \text{ e }^{\delta _{14}}= & {} \frac{\varGamma _{22}^2}{\kappa _{22}^2},~\text{ e }^{\delta _{15}}=\frac{2\varGamma _{11}\varGamma _{12}}{\kappa _{11}\kappa _{21}},\\ \text{ e }^{\delta _{16}}= & {} \frac{2\varGamma _{21}\varGamma _{22}}{\kappa _{12}\kappa _{22}},~\text{ e }^{\delta _{17}}=\frac{2\varGamma _{11}\varGamma _{21}}{\kappa _{11}\kappa _{12}},~\text{ e }^{\delta _{18}}=\frac{2\varGamma _{12}\varGamma _{22}}{\kappa _{21}\kappa _{22}},\\ \text{ e }^{\delta _{19}}= & {} \frac{2(\kappa _{21}\kappa _{12})^{\frac{1}{2}}\varLambda _3+2(\kappa _{11}\kappa _{22})^{\frac{1}{2}}\varLambda _4+4\varLambda _5}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}},\\ \varLambda _3= & {} (k_1(2\bar{k}_1+k_2-\bar{k}_2)+2k_2\bar{k}_2+\bar{k}_1(\bar{k}_2-k_2))\\&(\alpha _1^{(1)}\beta _1^{(1)}\alpha _2^{(2)}\beta _2^{(2)}+\alpha _1^{(1)}\beta _1^{(2)}\alpha _2^{(1)}\beta _2^{(1)}),\\ \varLambda _4= & {} (-k_2\bar{k}_2+\bar{k}_1(2k_2+\bar{k}_2)+k_1(2\bar{k}_2-\bar{k}_1+k_2))\\&(\alpha _1^{(2)}\beta _1^{(1)}\alpha _2^{(1)}\beta _2^{(2)}+\alpha _1^{(1)}\beta _1^{(2)}\alpha _2^{(2)}\beta _2^{(1)}),\\ \varLambda _5= & {} (k_2^2\bar{k}_2^2+k_2\bar{k}_1\bar{k}_2(\bar{k}_2-k_2)\\&+\,\bar{k}_1^2(k_2^2+k_2\bar{k}_2+\bar{k}_2^2)+k_1^2(\bar{k}_1^2+k_2^2\\&+\,\bar{k}_1(k_2-\bar{k}_2)+k_2\bar{k}_2\\&+\,\bar{k}_2^2)+k_1[k_2\bar{k}_2(k_2-\bar{k}_2)+\bar{k}_1^2(\bar{k}_2-k_2)\\&+\,\bar{k}_1(k_2^2+5k_2\bar{k}_2+\bar{k}_2^2)])\\&(\alpha _1^{(2)}\beta _1^{(2)}\alpha _2^{(2)}\beta _2^{(2)}+\alpha _1^{(1)}\beta _1^{(1)}\alpha _2^{(1)}\beta _2^{(1)}).\\ \text{ e }^{\varDelta _{7}^{(j)}}= & {} \bar{\varrho }_{12}\varGamma _{11}\bigg ((-1)^{j}k_1\beta _1^{(3-j)}\nu _1\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{11}+\bar{k}_1\alpha _1^{(j)}\varGamma _{21}\bigg )/\kappa _{11}^2\kappa _{12},\\ \text{ e }^{\varDelta _{8}^{(j)}}= & {} \bar{\varrho }_{12}\varGamma _{21}\bigg ((-1)^{j}k_1\beta _1^{(3-j)}\nu _1\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{11}+\bar{k}_1\alpha _1^{(j)}\varGamma _{21}\bigg )/\kappa _{11}\kappa _{12}^2,\\ \text{ e }^{\varDelta _{9}^{(j)}}= & {} \bar{\varrho }_{12}\varGamma _{12}\bigg ((-1)^{j}k_2\beta _2^{(3-j)}\nu _1\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{12}+\bar{k}_1\alpha _1^{(j)}\varGamma _{22}\bigg )/\kappa _{21}^2\kappa _{22},\\ \text{ e }^{\mu _{1}^{(j)}}= & {} \bar{\varrho }_{12}\varGamma _{22}\bigg ((-1)^{j}k_2\beta _2^{(3-j)}\nu _1\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{12}+\bar{k}_1\alpha _1^{(j)}\varGamma _{22}\bigg )/\kappa _{21}\kappa _{22}^2,\\ \end{aligned}$$
$$\begin{aligned} \text{ e }^{\gamma _{7}^{(j)}}= & {} \varrho _{12}\varGamma _{11}\bigg (-k_2\beta _2^{(j)}\varGamma _{11}\\&+\,k_1\beta _1^{(j)}\varGamma _{12}+(-1)^{(3-j)}\bar{k}_1\alpha _1^{(3-j)}\nu _2\bigg )/\kappa _{11}^2\kappa _{21},\\ \text{ e }^{\gamma _{8}^{(j)}}= & {} \varrho _{12}\varGamma _{12}\bigg (-k_2\beta _2^{(j)}\varGamma _{11}\\&+\,k_1\beta _1^{(j)}\varGamma _{12}+(-1)^{(3-j)}\bar{k}_1\alpha _1^{(3-j)}\nu _2\bigg )/\kappa _{11}\kappa _{21}^2,\\ \text{ e }^{\gamma _{9}^{(j)}}= & {} \varrho _{12}\varGamma _{21}\bigg (-k_2\beta _2^{(j)}\varGamma _{21}\\&+\,k_1\beta _1^{(j)}\varGamma _{22}+(-1)^{(3-j)}\bar{k}_2\alpha _2^{(3-j)}\nu _2\bigg )/\kappa _{12}^2\kappa _{22},\\ \text{ e }^{\varphi _{1}^{(j)}}= & {} \varrho _{12}\varGamma _{22}\bigg (-k_2\beta _2^{(j)}\varGamma _{21}\\&+\,k_1\beta _1^{(j)}\varGamma _{22}+(-1)^{(3-j)}\bar{k}_2\alpha _2^{(3-j)}\nu _2\bigg )/\kappa _{12}\kappa _{22}^2,\\ \text{ e }^{\mu _2^{(j)}}= & {} \frac{\bar{\varrho }_{12}\varLambda _6}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}},~\text{ e }^{\mu _3^{(j)}}=\frac{\bar{\varrho }_{12}\varLambda _7}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}},\\ \text{ e }^{\varphi _2^{(j)}}= & {} \frac{\varrho _{12}\varLambda _8}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}},~ \text{ e }^{\varphi _3^{(j)}}=\frac{\varrho _{12}\varLambda _9}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}},\\ \varLambda _6= & {} \left( -2k_2^2\bar{k}_2\alpha _2^{(j)}\varGamma _{11}\varGamma _{22}+2\bar{k}_1^3\alpha _1^{(j)}\varGamma _{21}\varGamma _{22}\right. \\&+\,k_1^2\left[ -2\bar{k}_2\alpha _2^{(j)}\varGamma _{21}\varGamma _{12}+\bar{k}_1(\alpha _2^{(j)}\varGamma _{11}\right. \\&+\,\alpha _1^{(j)}\varGamma _{21})\varGamma _{22}+k_2\nu _1(-\alpha _2^{(j)}\nu _2\\&\left. +\beta _2^{(3-j)}(-1)^j\varGamma _{21})\right] +\bar{k}_1k_2[k_2\varGamma _{21}(\alpha _1^{(j)}\varGamma _{22}+\alpha _2^{(j)}\\&\varGamma _{12})+\bar{k}_2\alpha _2^{(j)}(\alpha _1^{(1)}(-2\beta _1^{(1)}\varGamma _{22}-\alpha _2^{(2)}\nu _2)\\&+\,\alpha _1^{(2)}(-2\beta _1^{(2)}\varGamma _{22}+\alpha _2^{(1)}\nu _2))]\\&+\,\bar{k}_1^2[k_2\varGamma _{21}(2\alpha _1^{(j)}\varGamma _{22}+(-1)^j\beta _2^{(3-j)}\nu _1)\\&-\,\bar{k}_2\alpha _2^{(j)}(\alpha _1^{(1)}(\beta _2^{(1)}\varGamma _{21}+\beta _1^{(1)}\varGamma _{22})+\alpha _1^{(2)}\\&(\beta _1^{(2)}\varGamma _{22}+\beta _2^{(2)}\varGamma _{21}))]+k_1\left[ \bar{k}_1^2\varGamma _{22}(2\alpha _1^{(j)}\varGamma _{21}\right. \\&+\,(-1)^j\beta _1^{(3-j)}\nu _1)+k_2(k_2\nu _1(-1)^j((-1)^j\\&\alpha _2^{(j)}\nu _2+\beta _1^{(3-j)}\varGamma _{22})+\bar{k}_2\alpha _2^{(j)}(\varGamma _{11}\varGamma _{22}+\varGamma _{21}\varGamma _{12}))\\&+\,\bar{k}_1\left[ \bar{k}_2\alpha _2^{(j)}\big (\alpha _1^{(1)}(-2\beta _2^{(1)}\varGamma _{21}\right. \\&+\,\alpha _2^{(2)}\nu _2)+\alpha _1^{(2)}(-\alpha _2^{(1)}\nu _2-2\beta _2^{(2)}\varGamma _{21})\big )\\&+\,k_2\big (-3\alpha _1^{(3-j)}\alpha _2^{(j)}(\beta _2^{(3-j)}\varGamma _{21}+\beta _1^{(3-j)}\varGamma _{22})\\&+\,\alpha _1^{(j)}\big (\alpha _2^{(3-j)}\beta _1^{(3-j)}\varGamma _{22}+\alpha _2^{(3-j)}\beta _2^{(3-j)}\varGamma _{21}\\&\left. \left. \left. +2\alpha _2^{(j)2}\beta _1^{(3-j)}\beta _2^{(3-j)}-2\alpha _2^{(j)2}\beta _1^{(j)}\beta _2^{(j)}\big )\big )\right] \right] \right) , \end{aligned}$$
$$\begin{aligned} \varLambda _7= & {} \bigg (-k_2\varGamma _{11}[2\bar{k}_2^2\alpha _2^{(j)}\varGamma _{12}+k_2\bar{k}_2(2\alpha _2^{(j)}\varGamma _{12}\\&+\,(-1)^{(3-j)}\beta _2^{(3-j)}\nu _1)+k_2^2(\alpha _1^{(j)}\varGamma _{22}+\alpha _2^{(j)}\\&\varGamma _{12})]+k_1^2[-\bar{k}_2(\alpha _1^{(j)}\varGamma _{21}+\alpha _2^{(j)}\varGamma _{11})\varGamma _{12}\\&+\,2\bar{k}_1\alpha _1^{(j)}\varGamma _{11}\varGamma _{22}+(-1)^jk_2\nu _1((-1)^{(3-j)}\alpha _1^{(j)}\\&\nu _2+\beta _2^{(3-j)}\varGamma _{11})]+\bar{k}_1\alpha _1^{(j)}[2k_2^2\varGamma _{21}\varGamma _{12}\\&+\,\bar{k}_2^2\big (\alpha _1^{(1)}(\beta _2^{(1)}\varGamma _{21}+\beta _1^{(1)}\varGamma _{22})+\alpha _1^{(2)}(\beta _1^{(2)}\varGamma _{22}\\&+\,\beta _2^{(2)}\varGamma _{21})\big )+k_2\bar{k}_2\big (\alpha _1^{(1)}(2\beta _2^{(1)}\varGamma _{21}-\alpha _2^{(2)}\nu _2)\\&+\,\alpha _1^{(2)}(2\beta _2^{(2)}\varGamma _{21}+\alpha _2^{(1)}\nu _2)\big )]-k_1\\&[\bar{k}_2^2(2\alpha _2^{(j)}\varGamma _{11}+(-1)^{(3-j)}\beta _1^{(3-j)}\nu _1)\varGamma _{12}\\&+\,(-1)^{(3-j)}k_2^2\nu _1((-1)^j\alpha _1^{(j)}\nu _2+\beta _1^{(3-j)}\varGamma _{12})\\&+\,k_2\bar{k}_2\big ((-1)^{(3-j)}\beta _1^{(3-j)}\varGamma _{12}\nu _1\\&+\,(-1)^{(3-j)}\beta _2^{(3-j)}\varGamma _{11}\nu _1-2\alpha _1^{(j)}\alpha _2^{(3-j)}(\beta _1^{(3-j)}\\&\varGamma _{12}+\beta _2^{(3-j)}\varGamma _{11})+2\alpha _1^{(3-j)2}\alpha _2^{(j)}\beta _1^{(3-j)}\beta _{2}^{(3-j)}\\&-\,2\alpha _1^{(j)2}\alpha _2^{(j)}\beta _1^{(j)}\beta _2^{(j)}\big )+\bar{k}_1\alpha _1^{(j)}\big (\bar{k}_2\\&(-2\varGamma _{11}\varGamma _{22}+\nu _1\nu _2)+k_2(\varGamma _{11}\varGamma _{22}+\varGamma _{12}\varGamma _{21})\big )]\bigg ),\\ \varLambda _8= & {} \bigg (-\bar{k}_1^2[\bar{k}_2(\beta _1^{(j)}\nu _1+(-1)^j\alpha _2^{(3-j)}\varGamma _{11})\nu _2\\&+\,k_2\varGamma _{21}(\beta _2^{(j)}\varGamma _{11}+\beta _1^{(j)}\varGamma _{12})]-k_2\varGamma _{11}\\&[2k_2^2\varGamma _{21}\beta _2^{(j)}+\bar{k}_2^2(\beta _2^{(j)}\varGamma _{21}+\beta _1^{(j)}\varGamma _{22})\\&+\,k_2\bar{k}_2(2\beta _2^{(j)}\varGamma _{21}+(-1)^j\alpha _2^{(3-j)}\nu _2)]\\&+\,k_1\beta _1^{(j)}[2\bar{k}_2^2\varGamma _{12}\varGamma _{21}+2\bar{k}_1^2\varGamma _{11}\varGamma _{22}\\&+\,k_2\bar{k}_2(2\varGamma _{21}\varGamma _{12}+\nu _1\nu _2)+k_2^2(\varGamma _{12}\varGamma _{21}+\varGamma _{11}\varGamma _{22})\\&+\,\bar{k}_1\big (-\bar{k}_2(\varGamma _{12}\varGamma _{21}+\varGamma _{11}\varGamma _{22})\\&+\,k_2(2\varGamma _{11}\varGamma _{22}-\nu _2\nu _2)\big )]+\bar{k}_1[(-1)^{(3-j)}\bar{k}_2^2\nu _2\\&((-1)^{(3-j)}\beta _1^{(j)}\nu _1+\alpha _1^{(3-j)}\varGamma _{21})-k_2^2\varGamma _{21}(2\beta _2^{(j)}\\ \end{aligned}$$
$$\begin{aligned}&\varGamma _{11}+(-1)^{(j)}\alpha _1^{(3-j)}\nu _2)+k_2\bar{k}_2\\&\big (\alpha _1^{(j)}\beta _1^{(j)}(2\beta _1^{(j)}\varGamma _{22}+(-1)^{(3-j)}\alpha _2^{(3-j)}\nu _2)\\&+\alpha _1^{(3-j)}((-1)^{(3-j)}\varGamma _{21}\nu _2+2\beta _1^{(j)}\beta _2^{(3-j)}\\&\varGamma _{21}+3(-1)^{(3-j)}\alpha _2^{(3-j)}\beta _1^{(3-j)}\nu _2)\big )]\bigg ),\\ \varLambda _9= & {} \bigg (-2k_2\bar{k}_2^2\varGamma _{11}\varGamma _{22}\beta _2^{(j)}+2k_1^3\beta _1^{(j)}\varGamma _{12}\varGamma _{22}\\&+\,\bar{k}_1^2(-1)^j[(-1)^{(3-j)}2k_2\varGamma _{21}\beta _2^{(j)}+(-1)^{(3-j)}\\&\bar{k}_2\nu _2(\nu _1\beta _2^{(j)}+(-1)^{(j)}\alpha _2^{(3-j)}\varGamma _{12})]\\&+\,k_1^2[\bar{k}_1\varGamma _{22}(2\beta _1^{(j)}\varGamma _{12}+(-1)^{(3-j)}\alpha _1^{(3-j)}\nu _2)\\&+\,\bar{k}_2\varGamma _{12}(2\beta _1^{(j)}\varGamma _{22}+(-1)^{(3-j)}\alpha _2^{(3-j)}\nu _2)\\&-\,k_2\beta _2^{(j)}(\varGamma _{11}\varGamma _{22}+\varGamma _{12}\varGamma _{21})]+\bar{k}_1\bar{k}_2[(-1)^{(3-j)}\bar{k}_2\\&\nu _2(\alpha _1^{(3-j)}\varGamma _{22}+(-1)^{(3-j)}\beta _2^{(j)}\nu _1)\\&+\,k_2\beta _2^{(j)}\big (\alpha _1^{(1)}(\beta _2^{(1)}\varGamma _{21}+\beta _1^{(1)}\varGamma _{22})+\alpha _1^{(2)}\\&(\beta _1^{(2)}\varGamma _{22}+\beta _2^{(2)}\varGamma _{21})\big )]\\&+\,k_1[\bar{k}_1^2\varGamma _{22}(\beta _2^{(j)}\varGamma _{11}+\beta _1^{(j)}\varGamma _{12})\\&+\,\bar{k}_2\big (\bar{k}_2\varGamma _{12}(\beta _1^{(j)}\varGamma _{22}+\beta _2^{(j)}\varGamma _{21})+k_2\beta _2^{(j)}\\&(-2\varGamma _{11}\varGamma _{22}+\nu _1\nu _2)\big )\\&+\,\bar{k}_1\big (k_2\beta _2^{(j)}[\alpha _1^{(1)}(-2\beta _2^{(1)}\varGamma _{21}+\alpha _2^{(2)}\nu _2)\\&+\,(-2\beta _2^{(2)}\varGamma _{21}-\alpha _2^{(1)}\nu _2)]\\&+\,\bar{k}_2[\alpha _1^{(j)}\beta _2^{(j)}(-2\beta _2^{(j)}\varGamma _{21}+(-1)^{(3-j)}\alpha _2^{(3-j)}\nu _2)\\&+\,\alpha _1^{(3-j)}(-2\beta _1^{(j)}\beta _2^{(j)}\varGamma _{22}+(-1)^{(3-j)}\nu _2\\&\varGamma _{22}+3(-1)^{(3-j)}\alpha _2^{(3-j)}\beta _2^{(j)}\nu _2)]\big )]\bigg ),\\ \nu _1= & {} \alpha _1^{(2)}\alpha _2^{(1)}-\alpha _1^{(1)}\alpha _2^{(2)},~\nu _2=\beta _1^{(1)}\beta _2^{(2)}-\beta _1^{(2)}\beta _2^{(1)},\\ \varrho _{12}= & {} (k_1-k_2),~\bar{\varrho }_{12}=(\bar{k}_1-\bar{k}_2), \\ \text{ e }^{\delta _{21}}= & {} -2\varrho _{12}\bar{\varrho }_{12}\varGamma _{11}\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2\\&-\,\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )/\kappa _{11}^2\kappa _{12}\kappa _{21}\kappa _{22},\\ \text{ e }^{\delta _{22}}= & {} -2\varrho _{12}\bar{\varrho }_{12}\varGamma _{21}\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2\\&-\,\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )/\kappa _{11}\kappa _{12}^2\kappa _{21}\kappa _{22}, \end{aligned}$$
$$\begin{aligned} \text{ e }^{\delta _{23}}= & {} -\,2\varrho _{12}\bar{\varrho }_{12}\varGamma _{12}\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2\\&-\,\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )/\kappa _{11}\kappa _{12}\kappa _{21}^2\kappa _{22},\\ \text{ e }^{\delta _{24}}= & {} -\,2\varrho _{12}\bar{\varrho }_{12}\varGamma _{22}\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2\\&-\,\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )/\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}^2,\\ \varrho _{1}= & {} (k_2\bar{k}_2+k_1\bar{k}_1),~\varrho _{2}=(k_1k_2+\bar{k}_1\bar{k}_2),\\ \varrho _{3}= & {} (k_1\bar{k}_2+k_2\bar{k}_1).\\ \text{ e }^{\mu _{4}^{(j)}}= & {} -\varrho _{12}\bar{\varrho }_{12}^2\bigg ((-1)^{j}k_1\beta _1^{(3-j)}\nu _1\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{11}+\bar{k}_1\alpha _1^{(j)}\varGamma _{21}\bigg )\\&\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2-\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )\bigg /\kappa \kappa _{11}\kappa _{12} ,\\ \text{ e }^{\mu _{5}^{(j)}}= & {} -\varrho _{12}\bar{\varrho }_{12}^2\bigg ((-1)^{j}k_2\beta _2^{(3-j)}\nu _1\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{12}+\bar{k}_1\alpha _1^{(j)}\varGamma _{22}\bigg )\\&\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2-\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )\bigg /\kappa \kappa _{21}\kappa _{22} \\ \text{ e }^{\varphi _{4}^{(j)}}= & {} -\varrho _{12}^2\bar{\varrho }_{12}\bigg (-k_2\beta _2^{(j)}\varGamma _{11}\\&+\,k_1\beta _1^{(j)}\varGamma _{12}+(-1)^{(3-j)}\bar{k}_1\alpha _1^{(3-j)}\nu _2\bigg )\\&\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2-\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )/\kappa \kappa _{11}\kappa _{21},\\ \text{ e }^{\varphi _{5}^{(j)}}= & {} -\varrho _{12}^2\bar{\varrho }_{12}\bigg (-k_2\beta _2^{(j)}\varGamma _{21}\\&+\,k_1\beta _1^{(j)}\varGamma _{22}+(-1)^{(3-j)}\bar{k}_2\alpha _2^{(3-j)}\nu _2\bigg )\\&\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2-\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )/\kappa \kappa _{12}\kappa _{22},\\ \text{ e }^{\delta _{31}}= & {} \varrho _{12}^2\bar{\varrho }_{12}^2\bigg (\varrho _{1}\varGamma _{11}\varGamma _{22}-\varrho _{2}\nu _1\nu _2\\&-\,\varrho _{3}\varGamma _{21}\varGamma _{12}\bigg )^2\bigg /\kappa _{11}^2\kappa _{12}^2\kappa _{21}^2\kappa _{22}^2,\\ \kappa= & {} \kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}. \end{aligned}$$

We arrive at the degenerate two-soliton solution by substituting the expression given in (31a)–(31c) in Eq. (4). The auxiliary functions are found to be

$$\begin{aligned}&s^{(1)}(-x,t)=s^{(2)}(-x,t)=\varGamma _{11}\text{ e }^{\xi _1+\bar{\xi }_1}+\varGamma _{21}\text{ e }^{\xi _1+\bar{\xi }_2}\\&\quad +\,\varGamma _{12}\text{ e }^{\bar{\xi }_1+\xi _2}+\varGamma _{22}\text{ e }^{\xi _2+\bar{\xi }_2}\\&\quad +\,\text{ e }^{\xi _1+2\bar{\xi }_1+\xi _2+\phi _1}+e^{2\xi _1+\bar{\xi }_1+\bar{\xi }_2+\phi _2}\\&\quad +\,\text{ e }^{\xi _1+\xi _2+2\bar{\xi }_2+\phi _3}+\text{ e }^{2\xi _2+\bar{\xi }_1+\bar{\xi }_2+\phi _4}\\&\quad +\,\text{ e }^{\xi _1+\bar{\xi }_1+\xi _2+\bar{\xi }_2+\phi _5}+\text{ e }^{2\xi _1+2\bar{\xi }_1+\xi _2+\bar{\xi }_2+\phi _{11}}\\&\quad +\,\text{ e }^{\xi _1+2\bar{\xi }_1+2\xi _2+\bar{\xi }_2+\phi _{12}}+\text{ e }^{2\xi _1+\bar{\xi }_1+\xi _2+2\bar{\xi }_2+\phi _{13}}\\&\quad +\,\text{ e }^{\xi _1+\bar{\xi }_1+2\xi _2+2\bar{\xi }_2+\phi _{14}} \end{aligned}$$

where the constants are obtained as

$$\begin{aligned} \text{ e }^{\phi _{1}}= & {} \frac{-\varrho _{12}^2\varGamma _{11}\varGamma _{12}}{\kappa _{11}\kappa _{21}},~\text{ e }^{\phi _{2}}=\frac{-\bar{\varrho }_{12}^2\varGamma _{11}\varGamma _{21}}{\kappa _{11}\kappa _{12}},\\ \text{ e }^{\phi _{3}}= & {} \frac{-\varrho _{12}^2\varGamma _{21}\varGamma _{22}}{\kappa _{12}\kappa _{22}},~\text{ e }^{\phi _{4}}=\frac{-\bar{\varrho }_{12}^2\varGamma _{12}\varGamma _{22}}{\kappa _{21}\kappa _{22}},\\ \text{ e }^{\phi _5}= & {} \frac{\varGamma _{11}\varGamma _{22}(\kappa _{12}\kappa _{21})^{1/2}\varLambda _1+\varGamma _{12}\varGamma _{21}(\kappa _{11}\kappa _{22})^{1/2}\varLambda _2}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}}, \end{aligned}$$
$$\begin{aligned} \varLambda _1= & {} (2\kappa _{11}(\kappa _{21}\kappa _{12})^{\frac{1}{2}}-\kappa _{11}\kappa _{12}^{\frac{1}{2}}(2\bar{k}_1+k_2-\bar{k}_2)\\&-\,\kappa _{11}\kappa _{21}^{\frac{1}{2}}(2k_1-k_2+\bar{k}_2)\\&+\,\kappa _{21}^{\frac{1}{2}}\kappa _{22}(-k_1+\bar{k}_1-2\bar{k}_2)\\&+\,(k_1-\bar{k}_1-2k_2)\kappa _{12}+2(\kappa _{12}\kappa _{21})^{\frac{1}{2}}\kappa _{22}),\\ \varLambda _2= & {} \left( \kappa _{11}^{\frac{1}{2}}(k_1-2k_2-\bar{k}_2)\kappa _{21}\right. \\&+\,\kappa _{11}^{\frac{1}{2}}\kappa _{12}(\bar{k}_1-k_2-2\bar{k}_2)\\&+\,2(\kappa _{11}\kappa _{22})^{\frac{1}{2}}\kappa _{21}-\kappa _{21}\kappa _{22}^{\frac{1}{2}}(k_1+2\bar{k}_1-\bar{k}_2)\\&\left. +\,2\kappa _{12}\kappa _{22}\kappa _{11}^{\frac{1}{2}}-(2k_1+\bar{k}_1-k_2)\kappa _{12}\kappa _{22}^{\frac{1}{2}}\right) ,\\ \text{ e }^{\phi _{11}}= & {} \frac{\varrho _{12}\bar{\varrho }_{12}\varGamma _{11}\psi }{\kappa _{11}^2\kappa _{21}\kappa _{12}},~\text{ e }^{\phi _{12}}=\frac{\varrho _{12}\bar{\varrho }_{12}\varGamma _{12}\psi }{\kappa _{11}\kappa _{21}^2\kappa _{22}},\\ \text{ e }^{\phi _{13}}= & {} \frac{\varrho _{12}\bar{\varrho }_{12}\varGamma _{21}\psi }{\kappa _{11}\kappa _{12}^2\kappa _{22}},~\text{ e }^{\phi _{14}}=\frac{\varrho _{12}\bar{\varrho }_{12}\varGamma _{22}\psi }{\kappa _{21}\kappa _{12}\kappa _{22}^2},\\ \psi= & {} \bigg (k_2\bar{k}_2\varGamma _{11}\varGamma _{22}+k_1(-\bar{k}_2\varGamma _{21}\varGamma _{12}\\&+\,\bar{k}_1\varGamma _{11}\varGamma _{22}-k_2\nu _1\nu _2)-\bar{k}_1(k_2\varGamma _{21}\varGamma _{12}+\bar{k}_2\nu _1\nu _2)\bigg ), \end{aligned}$$

D. The constants which appear in the degenerate three-soliton solution (24a)–(24c)

The constants which appear in factorized degenerate three-soliton solution (24a)–(24c) have the explicit forms,

$$\begin{aligned} \text{ e }^{\varDelta _1^{(j)}}= & {} \bar{\varrho }_{12}[(-1)^{3-j}k_1\beta _1^{(3-j)}\nu _1^{(1)}+\bar{k}_2\alpha _2^{(j)}\varGamma _{11}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{21}]/\kappa _{11}\kappa _{12},\\ \text{ e }^{\varDelta _2^{(j)}}= & {} \bar{\varrho }_{13}[(-1)^{3-j}k_1\beta _1^{(3-j)}\nu _2^{(1)}+\bar{k}_3\alpha _3^{(j)}\varGamma _{11}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{31}]/\kappa _{11}\kappa _{13},\\ \text{ e }^{\varDelta _3^{(j)}}= & {} \bar{\varrho }_{23}[(-1)^{3-j}k_1\beta _1^{(3-j)}\nu _3^{(1)}+\bar{k}_3\alpha _3^{(j)}\varGamma _{21}\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{31}]/\kappa _{12}\kappa _{13},\\ \text{ e }^{\varDelta _4^{(j)}}= & {} \bar{\varrho }_{12}[(-1)^{3-j}k_2\beta _2^{(3-j)}\nu _1^{(1)}+\bar{k}_2\alpha _2^{(j)}\varGamma _{12}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{22}]/\kappa _{21}\kappa _{22},\\ \text{ e }^{\varDelta _5^{(j)}}= & {} \bar{\varrho }_{13}[(-1)^{3-j}k_2\beta _2^{(3-j)}\nu _2^{(1)}+\bar{k}_3\alpha _3^{(j)}\varGamma _{12}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{32}]/\kappa _{21}\kappa _{23},\\ \text{ e }^{\varDelta _6^{(j)}}= & {} \bar{\varrho }_{23}[(-1)^{3-j}k_2\beta _2^{(3-j)}\nu _3^{(1)}+\bar{k}_3\alpha _3^{(j)}\varGamma _{22}\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{32}]/\kappa _{22}\kappa _{23},\\ \text{ e }^{\varDelta _7^{(j)}}= & {} \bar{\varrho }_{12}[(-1)^{3-j}k_3\beta _3^{(3-j)}\nu _1^{(1)}+\bar{k}_2\alpha _2^{(j)}\varGamma _{13}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{23}]/\kappa _{31}\kappa _{32},\\ \end{aligned}$$
$$\begin{aligned} \text{ e }^{\varDelta _8^{(j)}}= & {} \bar{\varrho }_{13}[(-1)^{3-j}k_3\beta _3^{(3-j)}\nu _2^{(1)}+\bar{k}_3\alpha _3^{(j)}\varGamma _{13}\\&-\,\bar{k}_1\alpha _1^{(j)}\varGamma _{33}]/\kappa _{31}\kappa _{33},\\ \text{ e }^{\varDelta _9^{(j)}}= & {} \bar{\varrho }_{23}[(-1)^{3-j}k_3\beta _3^{(3-j)}\nu _3^{(1)}+\bar{k}_3\alpha _3^{(j)}\varGamma _{23}\\&-\,\bar{k}_2\alpha _2^{(j)}\varGamma _{33}]/\kappa _{32}\kappa _{33},\\ \text{ e }^{\gamma _1^{(j)}}= & {} \varrho _{12}[k_2\beta _2^{(j)}\varGamma _{11}-k_1\beta _1^{(j)}\varGamma _{12}\\&+\,(-1)^{(j)}\bar{k}_1\alpha _1^{(3-j)}\nu _1^{(2)}]/\kappa _{11}\kappa _{21},\\ \text{ e }^{\gamma _2^{(j)}}= & {} \varrho _{13}[k_3\beta _3^{(j)}\varGamma _{11}-k_1\beta _1^{(j)}\varGamma _{13}\\&+\,(-1)^{(j)}\bar{k}_1\alpha _1^{(3-j)}\nu _2^{(2)}]/\kappa _{11}\kappa _{31},\\ \text{ e }^{\gamma _3^{(j)}}= & {} \varrho _{23}[k_3\beta _3^{(j)}\varGamma _{12}-k_2\beta _2^{(j)}\varGamma _{13}\\&+\,(-1)^{(j)}\bar{k}_1\alpha _1^{(3-j)}\nu _3^{(2)}]/\kappa _{21}\kappa _{31},\\ \text{ e }^{\gamma _4^{(j)}}= & {} \varrho _{12}[k_2\beta _2^{(j)}\varGamma _{21}-k_1\beta _1^{(j)}\varGamma _{22}\\&+\,(-1)^{(j)}\bar{k}_2\alpha _2^{(3-j)}\nu _1^{(2)}]/\kappa _{12}\kappa _{22},\\ \text{ e }^{\gamma _5^{(j)}}= & {} \varrho _{13}[k_3\beta _3^{(j)}\varGamma _{21}-k_1\beta _1^{(j)}\varGamma _{23}\\&+\,(-1)^{(j)}\bar{k}_2\alpha _2^{(3-j)}\nu _2^{(2)}]/\kappa _{12}\kappa _{32},\\ \text{ e }^{\gamma _6^{(j)}}= & {} \varrho _{23}[k_3\beta _3^{(j)}\varGamma _{22}-k_2\beta _2^{(j)}\varGamma _{23}\\&+\,(-1)^{(j)}\bar{k}_2\alpha _2^{(3-j)}\nu _3^{(2)}]/\kappa _{22}\kappa _{32},\\ \text{ e }^{\gamma _7^{(j)}}= & {} \varrho _{12}[k_2\beta _2^{(j)}\varGamma _{31}-k_1\beta _1^{(j)}\varGamma _{32}\\&+\,(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\nu _1^{(2)}]/\kappa _{13}\kappa _{23}, \end{aligned}$$
$$\begin{aligned} \text{ e }^{\gamma _8^{(j)}}= & {} \varrho _{13}[k_3\beta _3^{(j)}\varGamma _{31}-k_1\beta _1^{(j)}\varGamma _{33}\\&+\,(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\nu _2^{(2)}]/\kappa _{13}\kappa _{33},\\ \text{ e }^{\gamma _9^{(j)}}= & {} \varrho _{23}[k_3\beta _3^{(j)}\varGamma _{32}-k_2\beta _2^{(j)}\varGamma _{33}\\&+\,(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\nu _3^{(2)}]/\kappa _{23}\kappa _{33},\\ \text{ e }^{\delta _{1}}= & {} -\frac{\varGamma _{11}}{\kappa _{11}},~\text{ e }^{\delta _{2}}=-\frac{\varGamma _{21}}{\kappa _{12}},~\text{ e }^{\delta _{3}}=-\frac{\varGamma _{31}}{\kappa _{13}},\\ \text{ e }^{\delta _{4}}= & {} -\frac{\varGamma _{12}}{\kappa _{21}},~\text{ e }^{\delta _{5}}=-\frac{\varGamma _{13}}{\kappa _{31}},\\ \text{ e }^{\delta _{6}}= & {} -\frac{\varGamma _{22}}{\kappa _{22}},~\text{ e }^{\delta _{7}}=-\frac{\varGamma _{32}}{\kappa _{23}}, \\ \text{ e }^{\delta _{8}}= & {} -\frac{\varGamma _{23}}{\kappa _{32}},~\text{ e }^{\delta _{9}}=-\frac{\varGamma _{33}}{\kappa _{33}},\\ \text{ e }^{\delta _{11}}= & {} \varrho _{12}\bar{\varrho }_{12}(\varrho _{1}\varGamma _{11}\varGamma _{22}\\&-\,\varrho _{2}\nu _1^{(1)}\nu _2^{(2)}-\varrho _{3}\varGamma _{21}\varGamma _{12})/\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22},\\ \text{ e }^{\delta _{12}}= & {} \varrho _{12}\bar{\varrho }_{13}(\varrho _{4}\varGamma _{11}\varGamma _{32}\\&-\,\varrho _{6}\nu _2^{(1)}\nu _1^{(2)}-\varrho _{5}\varGamma _{31}\varGamma _{12})/\kappa _{11}\kappa _{21}\kappa _{13}\kappa _{23},\\ \text{ e }^{\delta _{13}}= & {} \varrho _{12}\bar{\varrho }_{23}(\varrho _{9}\varGamma _{1}\varGamma _{32}\\&-\,\varrho _{7}\nu _3^{(1)}\nu _1^{(2)}-\varrho _{8}\varGamma _{31}\varGamma _{22})/\kappa _{12}\kappa _{22}\kappa _{13}\kappa _{23},\\ \text{ e }^{\delta _{14}}= & {} \varrho _{13}\bar{\varrho }_{12}(\varrho _{10}\varGamma _{11}\varGamma _{23}\\&-\,\varrho _{12}\nu _1^{(1)}\nu _2^{(2)}-\varrho _{11}\varGamma _{21}\varGamma _{13})/\kappa _{11}\kappa _{12}\kappa _{31}\kappa _{32},\\ \text{ e }^{\delta _{15}}= & {} \varrho _{23}\bar{\varrho }_{12}(\varrho _{13}\varGamma _{12}\varGamma _{23}\\&-\,\varrho _{15}\nu _1^{(1)}\nu _3^{(2)}-\varrho _{14}\varGamma _{22}\varGamma _{13})/\kappa _{21}\kappa _{22}\kappa _{31}\kappa _{32},\\ \text{ e }^{\delta _{16}}= & {} \varrho _{13}\bar{\varrho }_{13}(\varrho _{16}\varGamma _{11}\varGamma _{33}\\&-\,\varrho _{18}\nu _2^{(1)}\nu _2^{(2)}-\varrho _{17}\varGamma _{31}\varGamma _{13})/\kappa _{11}\kappa _{31}\kappa _{13}\kappa _{33},\\ \text{ e }^{\delta _{17}}= & {} \varrho _{23}\bar{\varrho }_{13}(\varrho _{19}\varGamma _{12}\varGamma _{33}\\&-\,\varrho _{21}\nu _2^{(1)}\nu _3^{(2)}-\varrho _{20}\varGamma _{32}\varGamma _{13})/\kappa _{21}\kappa _{31}\kappa _{23}\kappa _{33},\\ \text{ e }^{\delta _{18}}= & {} \varrho _{13}\bar{\varrho }_{23}(\varrho _{22}\varGamma _{21}\varGamma _{33}\\&-\,\varrho _{24}\nu _3^{(1)}\nu _2^{(2)}-\varrho _{23}\varGamma _{31}\varGamma _{23})/\kappa _{12}\kappa _{32}\kappa _{13}\kappa _{33},\\ \text{ e }^{\delta _{19}}= & {} \varrho _{23}\bar{\varrho }_{23}(\varrho _{25}\varGamma _{22}\varGamma _{33}\\&-\,\varrho _{27}\nu _3^{(1)}\nu _3^{(2)}-\varrho _{26}\varGamma _{32}\varGamma _{23})/\kappa _{22}\kappa _{32}\kappa _{23}\kappa _{33},\\ \end{aligned}$$
$$\begin{aligned} \text{ e }^{\chi _1^{(j)}}= & {} \frac{\varrho _{12}\bar{\varrho }_{12}\bar{\varrho }_{13}\bar{\varrho }_{23}\varLambda _1^{(j)}}{\kappa _{11}\kappa _{12}\kappa _{21}\kappa _{22}\kappa _{13}\kappa _{23}},~\text{ e }^{\chi _2^{(j)}}=\frac{\varrho _{13}\bar{\varrho }_{12}\bar{\varrho }_{13}\bar{\varrho }_{23}\varLambda _2^{(j)}}{\kappa _{11}\kappa _{12}\kappa _{31}\kappa _{32}\kappa _{13}\kappa _{33}},\\ \text{ e }^{\chi _3^{(j)}}= & {} \frac{\varrho _{23}\bar{\varrho }_{12}\bar{\varrho }_{13}\bar{\varrho }_{23}\varLambda _3^{(j)}}{\kappa _{21}\kappa _{22}\kappa _{31}\kappa _{32}\kappa _{23}\kappa _{33}},~\text{ e }^{\varphi _1^{(j)}}=\frac{\bar{\varrho }_{12}\varrho _{12}\varrho _{13}\varrho _{23}\varLambda _4^{(j)}}{\kappa _{11}\kappa _{21}\kappa _{12}\kappa _{22}\kappa _{31}\kappa _{32}},\\ \text{ e }^{\varphi _2^{(j)}}= & {} \frac{\bar{\varrho }_{13}\varrho _{12}\varrho _{13}\varrho _{23}\varLambda _5^{(j)}}{\kappa _{11}\kappa _{21}\kappa _{31}\kappa _{13}\kappa _{23}\kappa _{33}},~\text{ e }^{\varphi _3^{(j)}}=\frac{\bar{\varrho }_{23}\varrho _{12}\varrho _{13}\varrho _{23}\varLambda _6^{(j)}}{\kappa _{12}\kappa _{22}\kappa _{32}\kappa _{13}\kappa _{23}\kappa _{33}},\\ \varLambda _1^{(j)}= & {} (-1)^{(3-j)}k_2\bar{k}_2\bar{k}_3\varGamma _{11}\big [(-1)^{j}k_2\beta _2^{(3-j)}\nu _3^{(1)}\\&-\,\bar{k}_3\alpha _3^{(j)}\varGamma _{22}+\bar{k}_2\alpha _2^{(j)}\varGamma _{32}\big ]\\&+\,\bar{k}_1\big [k_2^2\beta _2^{(3-j)}\big (\bar{k}_3\nu _2^{(1)}\varGamma _{21}-\bar{k}_2\nu _1^{(1)}\varGamma _{11}\big )\\&+\,(-1)^jk_2\varGamma _{12}\big (-\bar{k}_3^2\alpha _3^{(j)}\varGamma _{21}+\bar{k}_2^2\alpha _2^{(j)}\varGamma _{31}\big )\\&+\,(-1)^j\bar{k}_2\bar{k}_3\nu _1^{(2)}\big (-\bar{k}_3\nu _1^{(1)}\alpha _3^{(j)}+\bar{k}_2\alpha _2^{(j)}\nu _2^{(1)}\big )\big ]\\&+\,k_1^2\beta _1^{(3-j)}\big [\bar{k}_2\bar{k}_3\nu _3^{(1)}\varGamma _{12}+\bar{k}_1\big (-k_2\nu _3^{(1)} \varGamma _{12}\\&-\,\bar{k}_3\nu _2^{(1)}\varGamma _{22}+\bar{k}_2\nu _1^{(1)}\varGamma _{32} \big )+k_2\big (\bar{k}_2\nu _2^{(1)}\varGamma _{22}\\&-\,\bar{k}_3\nu _1^{(1)}\varGamma _{32}\big )\big ]+\bar{k}_1^2\alpha _1^{(j)}\big [(-1)^{(3-j)}\bar{k}_2\bar{k}_3\nu _3^{(1)}\\&\nu _1^{(2)}+(-1)^{(3-j)}k_2\big (\bar{k}_2\varGamma _{31}\varGamma _{22}-\bar{k}_3\varGamma _{21}\varGamma _{32}\big )\big ]\\&+\,k_1\big [\bar{k}_2\varGamma _{21}\big (-k_2^2\nu _2^{(1)}\beta _2^{(3-j)}+(-1)^{(3-j)}\bar{k}_3^2\alpha _3^{(j)}\varGamma _{12}\big )\\&+\,\bar{k}_1\varGamma _{11}\big (k_2^2\nu _3^{(1)}\beta _2^{(3-j)}+(-1)^{(j)}\bar{k}_3^2\alpha _3^{(j)}\varGamma _{22}\\&+\,(-1)^{(3-j)}\bar{k}_2^2\alpha _2^{(j)}\varGamma _{32}\big )+(-1)^{(j)}\bar{k}_2^2\alpha _2^{(j)}\big (\bar{k}_3\varGamma _{31}\varGamma _{12}\\&+\,k_2\nu _2^{(1)}\nu _1^{(2)}\big )+(-1)^{(3-j)}\bar{k}_1^2\alpha _1^{(j)}\big (\bar{k}_3\varGamma _{31}\varGamma _{22}\\&+\,\bar{k}_2\varGamma _{21}\varGamma _{32}+k_2\nu _3^{(1)}\nu _1^{(2)}\big )\\&+\,k_2\bar{k}_3\nu _1^{(1)}\big (k_2\varGamma _{31}\beta _2^{3-(j)}+(-1)^{(3-j)}\bar{k}_3\alpha _3^{(j)}\nu _1^{(2)}\big )\big ], \end{aligned}$$
$$\begin{aligned} \varLambda _2^{(j)}= & {} -\,\bar{k}_2k_3\bar{k}_3\varGamma _{11}\big [(-1)^{(j)}k_3\nu _3^{(1)}\beta _3^{(3-j)}\\&-\,\bar{k}_3\alpha _3^{(j)}\varGamma _{23}+\bar{k}_2\alpha _2^{(j)}\varGamma _{33}\big ]\\&+\,k_1^2\beta _1^{(3-j)}\big [(-1)^{(3-j)}k_3\bar{k}_3\nu _1^{(1)}\varGamma _{33}\\&+\,(-1)^{(j)}\bar{k}_2\big (\bar{k}_3\nu _3^{(1)}\varGamma _{13}+k_3\\&\nu _2^{(1)}\varGamma _{23}\big )+(-1)^{(3-j)}\bar{k}_1\big (k_3\nu _3^{(1)}\varGamma _{13}\\&+\,\bar{k}_3\nu _2^{(1)}\varGamma _{23}+(-1)^{(j)}\bar{k}_2\nu _1^{(1)}\\&\varGamma _{33}\big )\big ]+\bar{k}_1\big [(-1)^{(j)}k_3\bar{k}_3\varGamma _{21}\big (k_3\nu _2^{(1)}\beta _3^{(3-j)}\\&+\,(-1)^{(3-j)}\bar{k}_3\alpha _3^{(j)}\varGamma _{13}\big )+\bar{k}_2^2\alpha _2^{(j)}\big (k_3\varGamma _{13}\varGamma _{31}\\&+\,\bar{k}_3\nu _2^{(1)}\nu _2^{(2)}\big )+(-1)^{(3-j)}\big (k_3^2\varGamma _{31}\beta _3^{(3-j)}\\&+\,(-1)^{(3-j)}\alpha _3^{(j)}\nu _2^{(2)}\big )\big ]\nonumber \\&+\,\bar{k}_1^2\alpha _1^{(j)}\big [k_3\bar{k}_3\varGamma _{21}\varGamma _{33}-\bar{k}_2\big (k_3\varGamma _{31}\varGamma _{23}+\bar{k}_3\nu _3^{(1)}\nu _2^{(2)}\big )\big ]\\&+\,k_1\big [-\bar{k}_2\varGamma _{21}\big ((-1)^{(j)}k_3^2\nu _2^{(1)}\beta _3^{(3-j)}+\bar{k}_3^2\alpha _3^{(j)}\varGamma _{13}\big )\\&+\,\bar{k}_1\varGamma _{11}\big ((-1)^{(j)}k_3^2\nu _3^{(1)}\beta _3^{(3-j)}+\bar{k}_3^2\alpha _3^{(j)}\varGamma _{23}\\&-\,\bar{k}_2^2\alpha _2^{(j)}\varGamma _{33}\big )+(-1)^{(j)}k_3\\&\bar{k}_3\nu _1^{(1)}\big (k_3\varGamma _{31}\beta _3^{(3-j)}+(-1)^{(3-j)}\bar{k}_3\alpha _3^{(j)}\nu _2^{(2)}\big )\\&+\,\bar{k}_2\alpha _2^{(j)} \big (\bar{k}_3\varGamma _{31}\varGamma _{13}+k_3\nu _2^{(1)}\nu _1^{(2)}\big )\\&+\,\bar{k}_1^2\alpha _1^{(j)}\big (-\bar{k}_3\varGamma _{31}\varGamma _{23}+\bar{k}_2\varGamma _{21}\varGamma _{33}\\&-\,k_3\nu _3^{(1)}\nu _2^{(2)}\big )\big ],\\ \varLambda _3^{(j)}= & {} -\,\bar{k}_2k_3\bar{k}_3\varGamma _{12}\big [(-1)^{(j)}k_3\nu _3^{(1)}\beta _3^{(3-j)}\\&-\,\bar{k}_3\alpha _3^{(j)}\varGamma _{23}+\bar{k}_2\alpha _2^{(j)}\varGamma _{33}\big ]\\&+\,k_2^2\beta _2^{(3-j)}\big [(-1)^{(3-j)}k_3\bar{k}_3\nu _1^{(1)}\varGamma _{33}\\&+\,\bar{k}_2\big ((-1)^{(j)}\bar{k}_3\nu _3^{(1)}\varGamma _{13}+(-1)^{(j)}k_3\nu _2^{(1)}\varGamma _{23}\big )\big ]\\&+\,\bar{k}_1\big [(-1)^{(j)}k_3\bar{k}_3\varGamma _{22}\big (k_3\nu _2^{(1)}\beta _3^{(3-j)}+(-1)^{(3-j)}\\&\bar{k}_3\alpha _3^{(j)}\varGamma _{13}\big )+k_2\varGamma _{12}\big ((-1)^{(j)}k_3^2\nu _3^{(1)}\beta _3^{(3-j)}\\&+\,\bar{k}_3^2\alpha _3^{(j)}\varGamma _{23}-\bar{k}_2^2\alpha _2^{(j)}\varGamma _{33}\big )\\ \end{aligned}$$
$$\begin{aligned}&+\,k_2^2\beta _2^{(3-j)}\big ((-1)^{(3-j)}k_3\nu _3^{(1)}\varGamma _{13}\\&+\,(-1)^{(3-j)}\bar{k}_3\nu _2^{(1)}\varGamma _{23}+(-1)^{(j)}\\&\bar{k}_2\nu _1^{(1)}\varGamma _{33}\big )+(-1)^{(3-j)}\bar{k}_2\nu _1^{(1)}\big (k_3^2\varGamma _{32}\beta _3^{(3-j)}\\&+\,(-1)^{(j)}\bar{k}_3^2\alpha _3^{(j)}\nu _3^{(2)}\big )\\&+\,\bar{k}_2^2\alpha _2^{(j)}\big (k_3\varGamma _{32}\varGamma _{13}+\bar{k}_3\nu _2^{(1)}\nu _3^{(2)}\big )\big ]\\&+\,\bar{k}_1^2\alpha _1^{(j)}\big [k_3\bar{k}_3\varGamma _{22}\varGamma _{33}+k_2\big (-\bar{k}_3\\&\varGamma _{32}\varGamma _{23}+\bar{k}_2\varGamma _{22}\varGamma _{33}-k_3\nu _3^{(1)}\nu _3^{(2)}\big )\\&-\,\bar{k}_2\big (k_3\varGamma _{32}\varGamma _{23}+\bar{k}_3\nu _3^{(1)}\nu _3^{(2)}\big )\big ]\\&+\,k_2\big [-\bar{k}_2\varGamma _{22}\big ((-1)^{(j)}k_3^2\nu _2^{(1)}\beta _3^{(3-j)}\\&+\,\bar{k}_3^2\alpha _3^{(j)}\varGamma _{13}\big )+(-1)^{(j)}k_3\bar{k}_3\nu _1^{(1)}\\&\big (k_3\varGamma _{32}\beta _3^{(3-j)}+\bar{k}_3\alpha _3^{(j)}\nu _3^{(2)}\big )\\&+\,\bar{k}_2^2\alpha _2^{(j)}\big (\bar{k}_3\varGamma _{32}\varGamma _{13}+k_3\nu _2^{(1)}\nu _3^{(2)}\big )\big ],\\ \varLambda _4^{(j)}= & {} -\,k_2\bar{k}_2k_3\varGamma _{11}\big [-k_3\varGamma _{22}\beta _3^{(j)}\\&+\,k_2\beta _2^{(j)}\varGamma _{23}+(-1)^{(3-j)}\alpha _2^{(3-j)}\bar{k}_2\nu _3^{(2)}\big ]\\&+\,\bar{k}_1\big [(-1)^{(3-j)}\bar{k}_2k_3\nu _1^{(2)}\big ((-1)^{(j)}k_3\nu _1^{(1)}\beta _3^{(j)}\\&+\,\bar{k}_2\alpha _2^{(j)}\varGamma _{13}\big )+k_2^2\\&\beta _2^{(j)}\big (k_3\varGamma _{21}\varGamma _{13}+\bar{k}_2\nu _1^{(1)}\nu _1^{(2)}\big )\\&-\,k_2\varGamma _{12}\big (k_3^2\varGamma _{21}\beta _3^{(j)}+(-1)^{(3-j)}\\&\bar{k}_2^2\alpha _2^{(3-j)}\nu _2^{(2)}\big )\big ]+k_1^2\beta _1^{(j)}\big [\bar{k}_2k_3\varGamma _{12}\varGamma _{23}\\&+\,\bar{k}_1\big (-k_3\varGamma _{22}\varGamma _{13}+k_2\varGamma _{12}\varGamma _{23}\\&-\,\bar{k}_2\nu _1^{(1)}\nu _3^{(2)}\big )-k_2\big (\bar{k}_2\varGamma _{22}\varGamma _{13}\\&+\,k_3\nu _1^{(1)}\nu _3^{(2)}\big )\big ]+\bar{k}_1^2\alpha _1^{(3-j)}\big [(-1)^{(j)}\bar{k}_2k_3\\&\nu _1^{(2}\varGamma _{23}+(-1)^{(3-j)}k_2\big (\bar{k}_2\varGamma _{22}\nu _2^{(2)}\\&+\,k_3\varGamma _{21}\nu _3^{(2)}\big )\big ]+k_1\big [(-1)^{(j)}k_2\nu _1^{(2)}\\&\big ((-1)^{(3-j)}k_3^2\nu _1^{(1)}\beta _3^{(j)}+\bar{k}_2^2\alpha _2^{(3-j)}\varGamma _{13}\big )\\&-\,\bar{k}_2k_3\varGamma _{12}\big (k_3\varGamma _{21}\beta _3^{(j)}+(-1)^{(j)}\\&\bar{k}_2\alpha _2^{(j)}\nu _2^{(2)}\big )+k_2^2\beta _2^{(j)}\big (\bar{k}_2\varGamma _{21}\varGamma _{13}\\&+\,k_3\nu _1^{(1)}\nu _2^{(2)}\big )+\bar{k}_1^2\alpha _1^{(3-j)}\big ((-1)^{(3-j)}\\&k_2\nu _1^{(2)}\varGamma _{23}+(-1)^{(j)}k_3\varGamma _{22}\nu _3^{(2)}\\&+\,(-1)^{(j)}\bar{k}_2\varGamma _{21}\nu _3^{(2)}\big )+\bar{k}_1\varGamma _{11}\big (k_3^2\varGamma _{22}\\&\beta _3^{(j)}-k_2^2\beta _2^{(j)}\varGamma _{23}+(-1)^{(3-j)}\bar{k}_2^2\alpha _2^{(3-j)}\nu _3^{(2)}\big )\big ], \end{aligned}$$
$$\begin{aligned} \varLambda _5^{(j)}= & {} -\,k_2\bar{k}_3k_3\varGamma _{11}\big [-k_3\varGamma _{32}\beta _3^{(j)}+k_2^{(j)}\beta _2^{(j)}\varGamma _{33}\\&-\bar{k}_3\alpha _3^{(3-j)}\nu _3^{(2)}\big ]+\bar{k}_1\big [-k_3\bar{k}_3\\&\nu _1^{(2)}\big (k_3\beta _3^{(j)}\nu _2^{(1)}+(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\varGamma _{13}\big )\\&+\,k_2^2\beta _2^{(j)}\big (k_3\varGamma _{31}\varGamma _{13}+\bar{k}_3\nu _2^{(1)}\nu _2^{(2)}\big )\\&-\,k_2\varGamma _{12}\big (k_3^2\varGamma _{31}\beta _3^{(j)}+(-1)^{(3-j)}\bar{k}_3^2\alpha _3^{(3-j)}\nu _2^{(2)}\big )\big ]\\&+\,k_1^2\beta _1^{(j)}\big [k_3\bar{k}_3\varGamma _{12}\varGamma _{33}\\&-\,k_2\big (\bar{k}_3\varGamma _{32}\varGamma _{13}+k_3\nu _2^{(1)}\nu _3^{(2)}\big )+\bar{k}\big (-k_3\varGamma _{32}\varGamma _{13}\\&+\,k_2\varGamma _{12}\varGamma _{33}-\bar{k}_3\nu _2^{(1)}\nu _3^{(2)}\big )\big ]\\&+\,\bar{k}_1^2\alpha _1^{(3-j)}\big [(-1)^{(j)}k_3\bar{k}_3\nu _1^{(2)}\varGamma _{33}\\&+\,k_2\big ((-1)^{(3-j)}\bar{k}_3\varGamma _{32}\nu _2^{(2)}+(-1)^{(j)}k_3\varGamma _{31}\\&\nu _3^{(2)}\big )\big ]+k_1\big [(-1)^{(j)}k_2\nu _1^{(2)}\big ((-1)^{(3-j)}k_3^2\nu _2^{(1)}\beta _3^{(j)}\\&+\,\bar{k}_3^2\alpha _3^{(3-j)}\varGamma _{13}\big )-k_3\bar{k}_3\varGamma _{12}\\&\big (k_3\varGamma _{31}\beta _3^{(j)}+(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\nu _2^{(2)}\big )\\&+\,k_2^2\beta _2^{(j)}\big (\bar{k}_3\varGamma _{31}\varGamma _{13}+k_3\nu _2^{(1)}\nu _2^{(2)}\big )\\&+\,\bar{k}_1^2\alpha _1^{(3-j)}\big ((-1)^{(3-j)}k_2\nu _1^{(2)}\varGamma _{33}\\&+\,(-1)^{(j)}k_3\varGamma _{32}\nu _2^{(2)}+(-1)^{(j)}\bar{k}_3\varGamma _{31}\nu _3^{(2)}\big )\\&+\,\bar{k}_1\varGamma _{11}\big (k_3^2\varGamma _{32}\beta _3^{(j)}-k_2^2\beta _2^{(j)}\varGamma _{33}\\&+\,(-1)^{(3-j)}\bar{k}_3^2\alpha _3^{(3-j)}\nu _3^{(2)}\big )\big ],\\ \varLambda _6^{(j)}= & {} (-1)^{(j)}\bar{k}_2\bar{k}_3k_3\nu _1^{(2)}\big [(-1)^{(3-j)}k_3\nu _3^{(1)}\beta _3^{(j)}\\&-\,\bar{k}_3\alpha _3^{(3-j)}\varGamma _{23}+\bar{k}_2\alpha _2^{(3-j)}\varGamma _{33}\big ]\\&+\,k_2^2\beta _2^{(j)}\big [-k_3\bar{k}_3\varGamma _{21}\varGamma _{33}+\bar{k}_2\big (k_3\varGamma _{31}\varGamma _{23}\\&+\,\bar{k}_3\nu _3^{(1)}\nu _2^{(2)}\big )\big ]+k_1^2\beta _1^{(j)}\big [k_3\bar{k}_3\varGamma _{22}\\&\varGamma _{33}+k_2\big (-\bar{k}_3\varGamma _{32}\varGamma _{23}+\bar{k}_2\varGamma _{22}\varGamma _{33}-\,k_3\nu _3^{(1)}\nu _3^{(2)}\big )\\&-\,\bar{k}_2\big (k_3\varGamma _{32}\varGamma _{23}+\bar{k}_3\nu _3^{(1)}\\&\nu _3^{(2)}\big )\big ]+k_1\big [-k_2\nu _1^{(2)}\big (k_3^2\nu _3^{(1)}\beta _3^{(j)}\\&+\,(-1)^{(3-j)}\bar{k}_3^2\alpha _3^{(3-j)}\varGamma _{23}+(-1)^{(j)}\\ \end{aligned}$$
$$\begin{aligned}&\bar{k}_2^2\alpha _2^{(3-j)}\varGamma _{33}\big )-k_3\bar{k}_3\varGamma _{22}\big (k_3\varGamma _{31}\beta _3^{(3-j)}\\&+\,(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\nu _2^{(2)}\big )+k_2^2\beta _2^{(j)}\\&\big (\bar{k}_3\varGamma _{31}\varGamma _{23}-\bar{k}_2\varGamma _{21}\varGamma _{33}+k_3\nu _3^{(1)}\nu _2^{(2)}\big )\\&+\,\bar{k}_2^2\alpha _2^{(3-j)}\big ((-1)^{(j)}k_3\varGamma _{32}\nu _2^{(2)}\\&+\,(-1)^{(j)}\bar{k}_3\varGamma _{31}\nu _3^{(2)}\big )+\bar{k}_2\varGamma _{21}\big (k_3^2\varGamma _{32}\beta _3^{(j)}\\&+\,(-1)^{(3-j)}\bar{k}_3^2\alpha _3^{(3-j)}\nu _3^{(2)}\big )\big ]\\&+\,k_2\big [-\bar{k}_2\varGamma _{22}\big (k_3^2\varGamma _{31}\beta _3^{(j)}\\&+\,(-1)^{(3-j)}\bar{k}_3^2\alpha _3^{(3-j)}\nu _2^{(2)}\big )+k_3\bar{k}_3\varGamma _{21}\big (k_3\varGamma _{32}\\&\beta _3^{(j)}+(-1)^{(j)}\bar{k}_3\alpha _3^{(3-j)}\nu _3^{(2)}\big )\\&+\,\bar{k}_2^2\alpha _2^{(3-j)}\big ((-1)^{(3-j)}\bar{k}_3\varGamma _{32}\nu _2^{(2)}\\&+\,(-1)^{(3-j)}k_3\varGamma _{31}\nu _3^{(2)}\big )\big ],\\ \text{ e }^{\delta _{20}}= & {} \frac{-\varLambda _{71}}{(k_1+\bar{k}_1+k_2+\bar{k}_2+k_3+\bar{k}_3)^2},\\ \varLambda _{71}= & {} (-k_1-\bar{k}_1+k_2+\bar{k}_2+k_3+\bar{k}_3)^2\text{ e }^{\delta _{19}+\delta _1}\\&+\,(-k_1+\bar{k}_1+k_2-\bar{k}_2+k_3+\bar{k}_3)^2\\&\text{ e }^{\delta _{17}+\delta _2}+(-k_1+\bar{k}_1+k_2+\bar{k}_2+k_3\\&-\,\bar{k}_3)^2\text{ e }^{\delta _3+\delta _{15}}+\text{ e }^{\delta _4+\delta _{18}}\\&(k_1-\bar{k}_1-k_2+\bar{k}_2+k_3+\bar{k}_3)^2+\text{ e }^{\delta _5+\delta _{13}}\\&(k_1-\bar{k}_1+k_2+\bar{k}_2-k_3+\bar{k}_3)^2\\&+\,(k_1+\bar{k}_1-k_2-\bar{k}_2+k_3+\bar{k}_3)^2\text{ e }^{\delta _6+\delta _{16}}\\&+\,(k_1+\bar{k}_1-k_2+\bar{k}_2+k_3-\bar{k}_3)^2\\&\text{ e }^{\delta _7+\delta _{14}}+(k_1+\bar{k}_1+k_2-\bar{k}_2-k_3\\&+\,\bar{k}_3)^2\text{ e }^{\delta _8+\delta _{12}}+\text{ e }^{\delta _9+\delta _{11}}\\&(k_1+\bar{k}_1+k_2+\bar{k}_2-k_3-\bar{k}_3)^2+\text{ e }^{\gamma _9^{(1)}+\varDelta _1^{(1)}}\\&+\,\text{ e }^{\gamma _9^{(2)}+\varDelta _1^{(2)}}+\text{ e }^{\gamma _6^{(1)}+\varDelta _2^{(1)}}\\&+\,\text{ e }^{\gamma _6^{(2)}+\varDelta _2^{(2)}}+\text{ e }^{\gamma _3^{(1)}+\varDelta _3^{(1)}}+\text{ e }^{\gamma _3^{(2)}+\varDelta _3^{(2)}}\\&+\,\text{ e }^{\gamma _8^{(1)}+\varDelta _4^{(1)}}+\text{ e }^{\gamma _8^{(2)}+\varDelta _4^{(2)}}\\&+\,\text{ e }^{\gamma _5^{(1)}+\varDelta _5^{(1)}}+\text{ e }^{\gamma _5^{(2)}+\varDelta _5^{(2)}}++\text{ e }^{\gamma _2^{(1)}+\varDelta _6^{(1)}}\\&+\,\text{ e }^{\gamma _2^{(2)}+\varDelta _6^{(2)}}++\text{ e }^{\gamma _7^{(1)}+\varDelta _7^{(1)}}\\&+\,\text{ e }^{\gamma _7^{(2)}+\varDelta _7^{(2)}}+\text{ e }^{\gamma _4^{(1)}+\varDelta _8^{(1)}}+\text{ e }^{\gamma _4^{(2)}+\varDelta _8^{(2)}}\\&+\,e^{\gamma _1^{(1)}+\varDelta _9^{(1)}}+\text{ e }^{\gamma _1^{(2)}+\varDelta _9^{(2)}}\\&+\,\sum _{j=1}^{2}\alpha _3^{(j)}\text{ e }^{\varphi _1^{(j)}}+\sum _{j=1}^{2}\alpha _2^{(j)}\text{ e }^{\varphi _2^{(j)}}\\&+\,\sum _{j=1}^{2}\alpha _1^{(j)}\text{ e }^{\varphi _3^{(j)}}+\sum _{j=1}^{2}\beta _3^{(j)}\text{ e }^{\chi _1^{(j)}}\\&+\,\sum _{j=1}^{2}\beta _2^{(j)}\text{ e }^{\chi _2^{(j)}}+\sum _{j=1}^{2}\beta _1^{(j)}\text{ e }^{\chi _3^{(j)}},\\ \end{aligned}$$
$$\begin{aligned} \varGamma _{nm}= & {} (\alpha _n^{(1)}\beta _{m}^{(1)}\!+\!\alpha _n^{(2)}\beta _{m}^{(2)}),\kappa _{nm}\!=\!(k_n\!+\! \bar{k}_m)^2,~n\!=\!m\!=\!1,2,3,\\ \nu _1^{(1)}= & {} \alpha _1^{(2)}\alpha _2^{(1)}-\alpha _1^{(1)}\alpha _2^{(2)},~\nu _2^{(1)}=\alpha _1^{(2)}\alpha _3^{(1)}-\alpha _1^{(1)}\alpha _3^{(2)},\\ \nu _3^{(1)}= & {} \alpha _2^{(2)}\alpha _3^{(1)}-\alpha _2^{(1)}\alpha _3^{(2)},~\nu _1^{(2)}=\beta _1^{(1)}\beta _2^{(2)}-\beta _1^{(2)}\beta _2^{(1)},\\ \nu _2^{(2)}= & {} \beta _1^{(1)}\beta _3^{(2)}-\beta _1^{(2)}\beta _3^{(1)},~\nu _3^{(2)}=\beta _2^{(1)}\beta _3^{(2)}-\beta _2^{(2)}\beta _2^{(1)},\\ \varrho _1= & {} k_1\bar{k}_1+k_2\bar{k}_2,~\varrho _2=k_1k_2+\bar{k_1}\bar{k}_2,\\ \varrho _3= & {} k_1\bar{k}_2+k_2\bar{k}_1,~\varrho _4=k_2\bar{k}_3+k_1\bar{k}_1,\\ \varrho _5= & {} k_1\bar{k}_3+k_2\bar{k}_1,~\varrho _6=k_1k_2+\bar{k}_1\bar{k}_3,\\ \varrho _7= & {} k_1k_2+\bar{k}_2\bar{k}_3,~\varrho _8=k_2\bar{k}_2+k_1\bar{k}_3,\\ \varrho _9= & {} k_2\bar{k}_3+k_1\bar{k}_2,~\varrho _{10}=k_3\bar{k}_2+k_1\bar{k}_1,\\ \varrho _{11}= & {} k_3\bar{k}_1+k_1\bar{k}_2,~\varrho _{12}=k_3k_1+\bar{k}_2\bar{k}_1,\\ \varrho _{13}= & {} k_3\bar{k}_2+k_2\bar{k}_1,~\varrho _{14}=k_3\bar{k}_1+k_2\bar{k}_2,\\ \varrho _{15}= & {} \bar{k}_1\bar{k}_2+k_2k_3,~\varrho _{16}=k_3\bar{k}_3+k_1\bar{k}_1,\\ \varrho _{17}= & {} k_1\bar{k}_3+k_3\bar{k}_1,~\varrho _{18}=k_3k_1+\bar{k}_1\bar{k}_1,\\ \varrho _{19}= & {} k_3\bar{k}_3+k_2\bar{k}_1,~\varrho _{20}=k_2\bar{k}_3+k_3\bar{k}_1,\\ \varrho _{21}= & {} k_2k_3+\bar{k}_3\bar{k}_1,~\varrho _{22}=k_3\bar{k}_3+k_1\bar{k}_2,\\ \varrho _{23}= & {} k_1\bar{k}_3+k_3\bar{k}_2,~\varrho _{24}=k_3k_1+\bar{k}_2\bar{k}_3,\\ \varrho _{25}= & {} k_3\bar{k}_3+k_2\bar{k}_2,~\varrho _{26}=k_2\bar{k}_3+k_3\bar{k}_2,\\ \varrho _{27}= & {} k_2k_3+\bar{k}_2\bar{k}_3,\\ \varrho _{12}= & {} (k_1-k_2),~\varrho _{13}=(k_1-k_3),\\ \varrho _{23}= & {} (k_2-k_3),~\bar{\varrho }_{12}=(\bar{k}_1-\bar{k}_2),\\ \bar{\varrho }_{13}= & {} (\bar{k}_1-\bar{k}_3),~\bar{\varrho }_{23}=(\bar{k}_2-\bar{k}_3). \end{aligned}$$

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Stalin, S., Senthilvelan, M. & Lakshmanan, M. Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions. Nonlinear Dyn 95, 343–360 (2019). https://doi.org/10.1007/s11071-018-4567-5

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