Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schrödinger equations

Abstract

The integrable coupled nonlinear Schrödinger equations with four-wave mixing are investigated. We first explore the conditions for modulational instability of continuous waves of this system. Secondly, based on the generalized N-fold Darboux transformation (DT), beak-shaped higher-order rogue waves (RWs) and beak-shaped higher-order rogue wave pairs are derived for the coupled model with attractive interaction in terms of simple determinants. Moreover, we derive the simple multi-dark-dark and kink-shaped multi-dark-dark solitons for the coupled model with repulsive interaction through the generalizing DT. We explore their dynamics and classifications by different kinds of spatial–temporal distribution structures including triangular, pentagonal, ‘claw-like’ and heptagonal patterns. Finally, we perform the numerical simulations to predict that some dark solitons and RWs are stable enough to develop within a short time. The results would enrich our understanding on nonlinear excitations in many coupled nonlinear wave systems with transition coupling effects.

1. Introduction

The standard nonlinear Schrödinger (NLS) equation has been paid much attention due to its widespread applications in optics [1,2], Bose–Einstein condensates [3], hydrodynamics [4], plasma physics [5], molecular biology [6] and even finance [7,8]. To well describe other important types of nonlinear physical phenomena in a similar way, it is necessary to go beyond the standard NLS description. One prime research is to add higher-order terms and/or dissipative terms to the NLS equation to accurately model extreme wave events in some nonlinear wave systems such as micro-structured optical fibres and fibre lasers [9–11]. Another important development focuses on the investigation of coupled-wave systems, as many physical systems comprise interacting wave components of distinct modes, frequencies or polarizations. Recently, the coupled NLS (CNLS) equations have become a topic of intense research, since the components are usually more than one practically for many physical phenomena (see [8,12–22] and references therein). Generally speaking, the particle number in each component is conserved for the integrable CNLS equations. However, the CNLS equations with four-wave mixing are usually non-integrable [23,24], and thus it is not easy to study nonlinear wave dynamics analytically.

In this paper, we focus on the two-component CNLS equations with four-wave mixing (also called pair-transition effects; CNLS-p for short)

$$\begin{array}{rl}\mathrm{i}{p}_t+\frac{1}{2}{p}_{xx}+\sigma (|p{|}^2+2|q{|}^2)p+\sigma q^2{p}^{\ast }& \hspace{0.17em}=0\\ \text{and}\mathrm{i}{q}_t+\frac{1}{2}{q}_{xx}+\sigma (2|p{|}^2+|q{|}^2)q+\sigma p^2{q}^{\ast }& \hspace{0.17em}=0,\end{array}\}$$ 1.1

where p=p(x,t) and q=q(x,t) are the complex wave envelopes; σ=±1 corresponds to attractive (+) or repulsive (−) interactions, respectively; and the star represents the complex conjugation. System (1.1) is associated with a variational principle ip_t=δH/(δp*), iq_t=δH/(δq*) with the Hamiltonian

$$H={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}[\frac{1}{2}|p_x{|}^2+\frac{1}{2}|q_x{|}^2-\sigma (\frac{1}{2}|p{|}^4+|q{|}^4+2|p{|}^2|q{|}^2+q^2{p}^{\ast 2}+p^2{q}^{\ast 2})]\mathrm{d}x,$$ 1.2

The coupled model can also be used to describe the propagation of orthogonally polarized optical waves in an isotropic medium [25]. This system was shown to be Painlevé integrable and admitted both bilinear form and Lax pair [26]. The transition dynamics of some nonlinear waves of this system can also be investigated [27]. Some types of nonlinear modes of the integrable model (1.1) have been obtained in terms of the special Hirota bilinearization and the Darboux transformation (DT) [28] such as solitons, breathers and rogue waves (RWs) [29,30].

DT is a powerful method to study solitons for many integrable systems such as bright solitons, breathers and RWs [16,31]. However, the dark solitons cannot be obtained directly by the classical DT method. The dark soliton of the single NLS equation was first derived [32]. After that, the single dark soliton of the N-component NLS-type equations was given [33,34]. Recently, a formula for the multi-dark solitons of the integrable N-component NLS system was obtained through the generalizing DT [35]. For the CNLS-p equations (1.1), the single dark-dark solitons and kink-shaped dark-dark solitons have been derived [30]. To the best of our knowledge, multi-dark-dark solitons and even the kink-shaped multi-dark-dark solitons of the CNLS-p equations (1.1) have not been reported before. In our present work, motivated by the work [35], we derive multi-dark-dark solitons and kink-shaped multi-dark-dark solitons for the CNLS-p equations (1.1) by the generalizing DT method. Particularly, we analyse the collision of two dark-dark solitons and the collision of two kink-shaped dark-dark solitons.

RWs initially describing extreme wave events that emerge in the deep ocean [36,37], has recently drawn much attention to both experimental observations and theoretical predictions, in areas as diverse as hydrodynamics [38–40], capillary wave [41], optics [42–45], plasma physics [46], Bose–Einstein condensates [47] and even finance [7,8]. In fact, the higher-order RWs can be regarded as the nonlinear superposition of a certain amount of fundamental RWs, and they can be described by the complicated rational polynomials instead of the simple Peregrine soliton. Very recently, the higher-order RWs of the scalar NLS equation have been successfully observed in a water wave tank, which gives a definitive answer of the theoretical prediction of their existence [39]. Also, the complete classification of higher-order RW solutions for the scalar NLS equation has been presented [48,49]. For the CNLS-p equations (1.1), both beak-shaped RW and beak-shaped RW pair (RWp) have been given by the DT [30], but beak-shaped higher-order RWs, beak-shaped higher-order RWp and their classifications have not been derived. In this work, based on the generalized DT [50–54], we derive them and also numerically study the dynamical behaviours of some dark-dark solitons and beak-shaped RWs.

This paper is organized as follows. In §2, we analyse the modulational instability (MI) of continuous waves (CWs) of equations (1.1). We find that the MI appears only for σ=1 (i.e. attractive interactions). In §3, we present a general N-fold DT for the CNLS-p equations (1.1) based on the loop group method. In §4, the formula of general beak-shaped Nth-order RWs is derived through the generalized DT. Furthermore, we give it a complete classification by means of its dynamical distribution, and analyse the trajectories of beak-shaped Nth-order RWs. In §5, both multi-dark-dark and kink-shaped multi-dark-dark solitons of equations (1.1) with the respective interaction σ=−1 are found explicitly through the generalizing DT. As examples, we investigate the collision of two dark-dark solitons and collision of two kink-shaped dark-dark solitons, and their asymptotic properties. In §6, we present the numerical results to strengthen our prediction of triggering these dark-dark solitons and RWs within more realistic conditions, namely, in the presence of some noise. Finally, we give the conclusions and discussions.

2. Modulational instability of continuous waves

Similar to the study of Wen et al. [52,53], we study the modulational instability of CW of system (1.1). The CW solutions of system (1.1) can be given by

$$p(x,t)=c\hspace{0.17em}{\mathrm{e}}^{4\mathrm{i}\sigma c^{2}t},q(x,t)=\mu c\hspace{0.17em}{\mathrm{e}}^{4\mathrm{i}\sigma c^{2}t},{\mu }^2=1,$$ 2.1

where c is a non-zero real amplitude.

We analyse the MI by taking the perturbed form of the CW as

$$p(x,t)=[c+ϵu(x,t)]\hspace{0.17em}{\mathrm{e}}^{4\mathrm{i}\sigma c^{2}t}\mathrm{and}q(x,t)=[\mu c+ϵv(x,t)]\hspace{0.17em}{\mathrm{e}}^{4\mathrm{i}\sigma c^{2}t},$$ 2.2

where ϵ is an infinitesimal amplitude of the perturbation. Substituting equation (2.2) into system (1.1) yields a linearized system

$$\mathrm{i}{u}_t+\frac{1}{2}{u}_{xx}+2\sigma c^2[u^{\ast }+\mu (2v+v^{\ast })]=0\mathrm{and}\mathrm{i}{v}_t+\frac{1}{2}{v}_{xx}+2\sigma c^2[v^{\ast }+\mu (2u+u^{\ast })]=0,$$ 2.3

To analyse the MI by means of the complex linearized system (2.3), we rewrite u and v in the form u=u₁+iu₂, v=v₁+iv₂ with u_j, v_j, j=1,2 being real functions such that we have

$$\begin{array}{rl}{u}_{1t}+\frac{1}{2}{u}_{2xx}+2\sigma c^2(\mu v_2-u_2)& \hspace{0.17em}=0,u_{2t}-\frac{1}{2}{u}_{1xx}-2\sigma c^2(3\mu v_1+u_1)=0\\ \text{and}{v}_{1t}+\frac{1}{2}{v}_{2xx}+2\sigma c^2(\mu u_2-v_2)& \hspace{0.17em}=0,v_{2t}-\frac{1}{2}{v}_{1xx}-2\sigma c^2(3\mu u_1+v_1)=0,\end{array}\}$$ 2.4

Solutions to these real equations may be sought for in a formally complex form

$$(u_1,u_2,v_1,v_2)=(u_{01},u_{02},v_{01},v_{02})\hspace{0.17em}{\mathrm{e}}^{gt+\mathrm{i}kx},$$ 2.5

where g is the MI gain (generally speaking, this eigenvalue may be complex), k is an arbitrary real wavenumber of the small perturbation, and u_0j,v_0j, (j=1,2) are real constant amplitudes of the perturbation eigenmode. As u_0j,v_0j, (j=1,2) are all not zero, we have the condition

$$\left|\begin{array}{cccc}g& -\frac{1}{2}{k}^2-2\sigma c^2& 0& 2\sigma \mu c^2\\ \frac{1}{2}{k}^2-2\sigma c^2& g& -6\sigma \mu c^2& 0\\ 0& 2\sigma \mu c^2& g& -\frac{1}{2}{k}^2-2\sigma c^2\\ -6\sigma \mu c^2& 0& \frac{1}{2}{k}^2-2\sigma c^2& g\end{array}\right|=0,$$ 2.6

which generates the relation

$$g^2=8c^4(-1\pm \mu )\pm 4\sigma \mu c^2{k}^2-\frac{1}{4}{k}^4,$$ 2.7

For the distinct sign (±) in equation (2.7), σ=±1 and μ=±1, we can determine the conditions for the positive or negative g², that is,

• (i) for the branch for g² with sign + (−) and μ=−1 (1), we have $g^2=-{(4c^2+\frac{1}{2}\sigma k^2)}^2\le 0;$

• (ii) for the branch for g² with sign + (−), μ=1 (−1) and σ=−1, we have $g^2=4\sigma c^2{k}^2-\frac{1}{4}{k}^4\le 0$;

• (iii) for the branch for g² with sign + (−), μ=1 (−1), σ=1 and |k|≥4|c| or k=0, we have $g^2=4c^2{k}^2-\frac{1}{4}{k}^4\le 0$;

• (iv) For the branch for g² with sign + (−), μ=1 (−1), σ=1 and 0<|k|<4|c|, we have $g^2=4c^2{k}^2-\frac{1}{4}{k}^4>0$.

For Cases (i)–(iii), we do not find the MI; however, for Case (iv), that is, the interactions in system (1.1) are attractive, we find that the MI can appear, the MI gain spectrum is symmetric with respect to k and MI gain decreases with increase in the normalized frequency. For $k=\pm 2\sqrt{2}|c|$, we have g_max=4c² (figure 1).

Figure 1.

Gain spectra of the MI for distinct CW amplitudes with the positive growth rate g(k) (see Case iv). Distinct curves correspond to values of the CW amplitude c=0.5 (solid line), c=1 (dashed line), c=2 (dotted line) and c=2.5 (dotted-dashed line). (Online version in colour.)

3. N-fold Darboux transformation

The CNLS-p system (1.1) admits the following Lax pair:

$$\begin{array}{rl}{\Phi }_x& \hspace{0.17em}=U\Phi ,U(Q;\lambda )\equiv \mathrm{i}(\lambda {\sigma }_3+Q)\\ \text{and}{\Phi }_t& \hspace{0.17em}=V\Phi ,V(Q;\lambda )\equiv \mathrm{i}({\lambda }^2{\sigma }_3+\lambda Q)+\frac{1}{2}{\sigma }_3(Q_x-\mathrm{i}{Q}^2),\end{array}\}$$ 3.1

where Φ=Φ(x,t,λ) is a 4×1 column vector eigenfunction, U and V are 4×4 matrices, λ is an iso-spectral parameter, σ₃=diag(1,1,−1,−1) and the 4×4 matrix Q=Q(x,t) with the variables p(x,t) and q(x,t) is

$$Q=\left(\begin{array}{cc}0& \sigma W^{†}\\ W& 0\end{array}\right)\mathrm{and}W=\left(\begin{array}{cc}p(x,t)& q(x,t)\\ q(x,t)& p(x,t)\end{array}\right),$$

with † representing the Hermitian conjugation. The compatibility condition of system (3.1), U_t−V _x+[U,V ]=0, gives rise to equations (1.1). The Lax pair (3.1) admits two symmetric conditions. The first symmetric condition is

$$U^{†}(\lambda )=-JU({\lambda }^{\ast })J\mathrm{and}{V}^{†}(\lambda )=-JV({\lambda }^{\ast })J,$$ 3.2

where J=diag(1,1,σ,σ) is a constant diagonal matrix. The second one is

$$\mathit{\Sigma }U(\lambda )\mathit{\Sigma }=U(\lambda ),\mathit{\Sigma }V(\lambda )\mathit{\Sigma }=V(\lambda ),\mathit{\Sigma }=\left(\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}\right),{\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$ 3.3

To represent the Darboux transformation conveniently, we give the conjugate form of the Lax pair (3.1) as

$${\mathrm{\Psi }}_x=-\mathrm{\Psi }U\mathrm{and}{\mathrm{\Psi }}_t=-\mathrm{\Psi }V.$$ 3.4

By the first symmetric condition (3.2), we can find that if Φ(x,t) is a solution for linear system (3.1), then Φ^†(x,t)J is a solution for the conjugate system (3.4) with λ=λ*. Based on the loop group method [55], the following Darboux matrix:

$$T[1]=I+\frac{({\lambda }_1^{\ast }-{\lambda }_1)y_1(x,t)y_1^{†}(x,t)J}{(\lambda -{\lambda }_1^{\ast })y_1^{†}(x,t)J{y}_1(x,t)},$$ 3.5

can transform system (3.1) with (U,V) into a new system with (U[1],V [1]), which can also admit the first kind of symmetric condition U[1]^†(λ)=−JU[1](λ*)J, V [1]^†(λ)=−JV [1](λ*)J, where U[1]=(T[1]_x+T[1]U)T[1]⁻¹ and V [1]=(T[1]_t+T[1]V)T[1]⁻¹. We see that Φ₁(x,t)≡Φ(x,t,λ₁) is a special solution to system (3.1) with λ=λ₁ and y₁(x,t)≡y(x,t,λ₁)=v(x,t,λ₁)Φ₁(x,t), where v(x,t,λ) is a function which has no root on $(x,t,\lambda )\in \mathbb{R}\times \mathbb{R}\times \mathbb{C}$. To make the new system keep the second kind of symmetric condition ΣU[1](λ)Σ=U[1](λ), ΣV [1](λ)Σ=V [1](λ), we take a special solution Φ₁ such that Φ₁=±ΣΦ₁. Therefore, through the loop group method [55], we can present the DT for CNLS-p equations (1.1) as follows.

Theorem 3.1 —

The Darboux matrix (3.5) can transform the Lax pair (3.1) into a new one

$$\Phi {[1]}_x=U(Q[1];\lambda )\Phi [1]\mathrm{and}\Phi {[1]}_t=V(Q[1];\lambda )\Phi [1],$$

where Φ[1]=T[1]Φ, Φ₁ is a special solution for system (3.1) with λ=λ₁, which keeps Φ₁=±ΣΦ₁. The Darboux transformation between the potential functions Q and Q[1] is

$$Q[1]=Q+({\lambda }_1^{\ast }-{\lambda }_1)[\frac{y_1{y}_1^{†}J}{y_1^{†}J{y}_1},{\sigma }_3].$$ 3.6

Iterating the above-mentioned process, we can establish the general N-fold DT in terms of simple determinants as follows.

Theorem 3.2 —

For the given N distinct solutions Φ_i of system (3.1) with the initial solution (p,q) and λ=λ_i, which keeps Φ_i=±ΣΦ_i, (i=1,2,…,N), we have the N-fold Darboux matrix

$$T[N]=I-Y{M}^{-1}{(\lambda I-S)}^{-1}{Y}^{†}J,$$ 3.7

and the N-fold DT between the new potential functions (p[N],q[N]) and old ones (p,q)

$$p[N]=p+\frac{2\hspace{0.17em}\det \left(\begin{array}{cc}M& Y_1^{†}\\ Y_3& 0\end{array}\right)}{\det \hspace{0.17em}M}\mathrm{and}q[N]=q+\frac{2\hspace{0.17em}\det \left(\begin{array}{cc}M& Y_1^{†}\\ Y_4& 0\end{array}\right)}{\det M},$$ 3.8

where $S=\mathrm{diag}({\lambda }_1^{\ast },{\lambda }_2^{\ast },\dots ,{\lambda }_N^{\ast }),$ Y =(y₁,y₂,…,y_N) with y_i=v_iΦ_i, M=(M_ij)_N×N with $M_{ij}=y_i^{†}J{y}_j/({\lambda }_j-{\lambda }_i^{\ast }),$ and Y _i is the ith row of Y .

To derive new potential functions through the above-mentioned DT, we first need to find the fundamental solution Φ(x,t,λ) of Lax pair (3.1) starting from the known (seed) solutions. Now, we start with the seed solutions $p=q=\frac{1}{2}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}\sigma t}$ (see equation (2.1) with c=1/2, μ=1). Substituting the seed solutions into the Lax pair (3.1) yields the fundamental solution:

$$\Phi (x,t,\lambda )=CHKL\hspace{0.17em}{\mathrm{e}}^{-(\mathrm{i}\sigma /2)t},H=\left(\begin{array}{cccc}1& 0& 1& 1\\ -1& 0& 1& 1\\ 0& 1& {\displaystyle \frac{\mathrm{i}}{\mu +\mathrm{i}\lambda }}& {\displaystyle \frac{\mathrm{i}}{-\mu +\mathrm{i}\lambda }}\\ 0& -1& {\displaystyle \frac{\mathrm{i}}{\mu +\mathrm{i}\lambda }}& {\displaystyle \frac{\mathrm{i}}{-\mu +\mathrm{i}\lambda }}\end{array}\right),$$ 3.9

where C=diag(1,1,e^iσt,e^iσt), L=diag(l₁,l₂,l₃,l₄), K=diag(e^g(iλ),e^g(−iλ),e^g(μ),e^g(−μ)) with g(z)=z[x+((z²+3λ²+σ)/2λ)t] being a function of variable z, $\mu =\sqrt{-{\lambda }^2-\sigma }$. Let v(x,t,λ)=e^(iσ/2)t. Then, we can obtain y(x,t,λ)=CHKL.

4. Higher-order rogue wave solutions

(a). Beak-type rogue wave solutions

In this section, we consider system (1.1) with the attractive interaction σ=1, where the MI possibly appears. We derive the beak-shaped RW solutions by the plane-wave solutions and obtain different kinds of spatial–temporal distribution structures for beak-shaped RW solutions such as triangular, pentagonal, ‘claw-like’ and heptagonal structures.

Let

$$y_1=C\left(\begin{array}{c}-k_2\hspace{0.17em}{\mathrm{e}}^{\hat{g}(\mu )}+k_1\hspace{0.17em}{\mathrm{e}}^{-\hat{g}(\mu )}\\ -k_2\hspace{0.17em}{\mathrm{e}}^{\hat{g}(\mu )}+k_1\hspace{0.17em}{\mathrm{e}}^{-\hat{g}(\mu )}\\ \mathrm{i}{k}_1\hspace{0.17em}{\mathrm{e}}^{\hat{g}(\mu )}-\mathrm{i}{k}_2\hspace{0.17em}{\mathrm{e}}^{-\hat{g}(\mu )}\\ \mathrm{i}{k}_1\hspace{0.17em}{\mathrm{e}}^{\hat{g}(\mu )}-\mathrm{i}{k}_2\hspace{0.17em}{\mathrm{e}}^{-\hat{g}(\mu )}\end{array}\right),k_{\pm }=\frac{1}{\sqrt{h^2-1}\sqrt{h\pm \sqrt{h^2-1}}},k_1=k_{-},k_2=k_{+},$$ 4.1

where h=−iλ, $\mu =\sqrt{h^2-1}$, $\hat{g}(\mu )=\mu (x+(1/2\lambda )({\mu }^2+3{\lambda }^2+1)t+F)$ with F being an arbitrary complex number.

To obtain a higher-order solution for equations (1.1), we take $h=1+{\epsilon }^2,F=\sum _{i=0}^{N-1}{s}_i{\epsilon }^{2\mathrm{i}}$, where s_i is an arbitrary complex constant [50,52–54]. It follows from equation (4.1) that C⁻¹y₁(x,t,ε) can be of the form $C^{-1}{y}_1(x,t,\epsilon )=\sum _{i=0}^{+\mathrm{\infty }}{f}_i(x,t){\epsilon }^{2\mathrm{i}}.$

Theorem 4.1 —

The general eye-shaped N-fold RW solutions for system (1.1) are

$$p[N]=\frac{1}{2}\frac{\det (A-4\hspace{0.17em}{\mathit{\Xi }}_1^{†}{\mathit{\Xi }}_3)}{\det \hspace{0.17em}A}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t},q[N]=\frac{1}{2}\frac{\det (A-4\hspace{0.17em}{\mathit{\Xi }}_1^{†}{\mathit{\Xi }}_4)}{\det \hspace{0.17em}A}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t},$$ 4.2

where Ξ(x,t)=(f₀,f₁,…,f_N−1), Ξ_j is the jth row of Ξ, (j=1,2,3,4), A=(A_ij)_N×N with

$$A_{mj}=-\frac{\mathrm{i}}{2}\sum _{k=0}^{m+j-2}\sum _{l=\max (0,k-m+1)}^{\min (j-1,k)}{(-\frac{1}{2})}^k{C}_k^l{f}_{m-1-k+l}^{†}{f}_{j-1-l},$$

Based on this theorem, we can obtain the eye-shaped N-order RW solutions. However, as the speciality of y₁, the RW solution p[N]=q[N]. General beak-shaped higher-order RW solutions are

$$p_b[N]=\frac{1}{2}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t}+p_{\mathrm{e}}[N]\hspace{0.17em}{\mathrm{e}}^{-2\mathrm{i}\theta }\mathrm{and}{q}_b[N]=\frac{1}{2}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t}-p_{\mathrm{e}}[N]\hspace{0.17em}{\mathrm{e}}^{-2\mathrm{i}\theta },$$ 4.3

where $\frac{1}{2}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t}$ is the plane-wave solution for equation (1.1). We see that p_e[N], q_e[N] are eye-shaped N-order RW solutions for equation (1.1) and p_e[N]=q_e[N] according to theorem 4.1. In this section, we take θ=1. We call the Nth-order RW with F=0 a standard one.

 Case 1. N=1,F=0. The eye-shaped first-order RW (fundamental RW) solution is

$$p_{\mathrm{e}}[1]=-\frac{1}{2}\frac{4(x^2+t^2-2\mathrm{i}t)-3}{4(x^2+t^2)+1}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t},$$ 4.4

which is symmetrical about the x- and t-axes, and the limit of |p_e[1]|² is 1/4 as $|x|,|t|\to \mathrm{\infty }$. Thus, the beak-shaped first-order RW (fundamental RW) solutions are of the form

$$p_b[1]=\frac{4(i+1)(x^2+t^2)+8t+1-3\mathrm{i}}{2(4x^2+4t^2+1)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t}\mathrm{and}{q}_b[1]=-\frac{4(i-1)(x^2+t^2)+8t-1-3\mathrm{i}}{2(4x^2+4t^2+1)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}t},$$ 4.5

which are symmetrical about t-axis, and the limits of both |p_e[1]|² and |q_e[1]|² are both $\frac{1}{2}$ as $|x|,|t|\to \mathrm{\infty }$ (figure 2a,b for p_b[1]. Note that the structures of the RWs q_b[1] and p_b[1] are opposite). It is seen that the pattern is quite different from one of the eye-shaped RW. The beak-shaped RW has one hump and two valleys on the temporal–spatial distribution, which admits a distinct spatial–temporal distribution, in contrast to the well-known eye-shaped, four-petalled and anti-eye-shaped structure. The structure of the RW p_b[1] is inverse for the q_b[1]. The hump in one component corresponds to two valleys in the other component, which clearly demonstrate the transition dynamics between the two components. The one hump and two valleys structure for a fundamental RW can be proved exactly through calculating the extreme points.

Figure 2.

(a,b) Beak-shaped fundamental RW with s₀=0. (c,d) Beak-shaped second-order RW. (c) Standard one with s₀=s₁=0. (d) Triangle structure with s₀=0,s₁=20. (Online version in colour.)

For the beak-shaped first-order RW shown in figure 2a,b nearby t=0, the wave has one hump and two valleys. At a large t, the peak values of the hump and valley approach the background. But the wave keeps the structure before or after the moment t=0. Therefore, we can define the trajectory by the motion of its hump and valleys, which can be described by the motions of the hump and valley's centre locations. The motion of its hump's centre is defined

$$X_{\mathrm{h}}=0(t>-\frac{3}{2}),$$ 4.6

and the motions of the two valleys’ centres are

$$X_{\mathrm{v}}=\pm \frac{1}{2}\frac{\sqrt{(1-2t)(8t^3+12t^2+2t+3)}}{2t-1}(-\frac{3}{2}<t<\frac{1}{2})\mathrm{and}{X}_{\mathrm{v}}=0(t<-\frac{3}{2}).$$ 4.7

Then, the beak-shaped first-order RW's trajectory is shown in figure 3a, in which the red curve represents its valleys and the blue curve denotes its hump. It can be seen that the valleys are symmetric about the x-axis and the hump is a straight line which is a part of t-axis. Furthermore, we can define the width of the RW as the distance between the two valleys’ centres, which corresponds to the distance between the two red lines in figure 3a. Its evolution is of the form

$$X_{\mathrm{d}}=\frac{\sqrt{(1-2t)(8t^3+12t^2+2t+3)}}{2t-1}.$$ 4.8

It is obvious that the width is increasing from 0 to $\mathrm{\infty }$ as t changes from −1.5 to 0.5. Substituting equations (4.6) and (4.7) into the density expression of the first-order RW, we can obtain explicit expression for the evolution of the RW's hump and valleys. The expression for the evolution of the RW's hump is

$$A_{\mathrm{h}}(t)=\frac{1}{2}\hspace{0.17em}\frac{4\hspace{0.17em}{t}^2+8\hspace{0.17em}t+5}{4\hspace{0.17em}{t}^2+1}(t>-\frac{3}{2}),$$ 4.9

and the expression for the evolution of the RW's valley is

$$A_{\mathrm{v}}(t)=\frac{1}{4}\hspace{0.17em}\frac{4\hspace{0.17em}{t}^2+4\hspace{0.17em}t+1}{4\hspace{0.17em}{t}^2+1}(-\frac{3}{2}<t<\frac{1}{2})\mathrm{and}{A}_{\mathrm{v}}(t)=\frac{1}{2}\hspace{0.17em}\frac{4\hspace{0.17em}{t}^2+8\hspace{0.17em}t+5}{4\hspace{0.17em}{t}^2+1}(t<-\frac{3}{2}).$$ 4.10

Furthermore, we can obtain that the highest value of the RW is $(3+2\sqrt{2})/2$ at $(x,t)=(0,(\sqrt{2}-1)/2)$, and the steepest value of the RW is 0 at $(x,t)=(\pm \sqrt{2}/2,-1/2)$.

Figure 3.

The trajectories for the humps and valleys of the beak-shaped standard RWs. (a) The first-order RW shown in figure 2a, (b) the second-order RW shown in figure 2c, (c) the third-order RW shown in figure 4a and (d) the fourth-order RWs shown in figure 5a. (Online version in colour.)

The trajectory of the beak-shaped first-order RW can be derived exactly. However, for the higher-order beak-shaped RWs, we cannot obtain the exact expressions for their trajectories, because the higher-order algebraic equations emerge for these extreme points. When x or t $\to \mathrm{\infty }$, we can readily prove that the trajectories of the higher-order RWs are asymptotic to the first-order RW. But we cannot obtain the trajectories of higher-order RWs in the neighbourhood (x,t)=(0,0) in a simple way. Here, we use the numerical method to exhibit the trajectories of the higher-order RWs.

 Case 2. N=2, F=s₀+s₁ε². There are two free parameters in the general beak-shaped second-order RW solutions. By varying them, we can derive different kinds of excitation patterns. We find that the beak-shaped second-order RW can demonstrate the coexistence of three beak-shaped RWs, which can arrange with two different kinds of distribution structures. (i) When s₁=0, these three fundamental RWs merge together (figure 2c), which is symmetrical about t-axis. (ii) When s₁≠0, we find that these three fundamental RWs will separate gradually by increasing the value |s₁| and constitute a triangle structure (figure 2d). We numerically display the locations of humps and valleys (figure 3b) of the standard second-order RWs (figure 2c).

 Case 3. N=3, F=s₀+s₁ ε²+s₂ ε⁴. The general beak-shaped third-order RW possesses three free parameters s_1,2,3, which can be used to obtain different kinds of excitation patterns by varying them. We find that the beak-shaped third-order RW consists of six fundamental RWs, which can mainly arrange with four types of distribution structures. (i) When s₁=0,s₂=0, we find that these six fundamental RWs merge together completely and constitute an identical distribution structure with the standard beak-shaped second-order RW (figure 4a). (ii) When s₁≠0,s₂=0, the six fundamental RWs will separate gradually by increasing the value |s₁| and arrange with a triangle structure (figure 4b). (iii) When s₁=0,s₂≠0, the six fundamental RWs will separate gradually by increasing the value |s₂| and constitute a pentagonal structure (figure 4c). (iv) When s₁≠0,s₂≠0, by choosing certain s₁,s₂, three among the six fundamental RWs merge together and fuse into a beak-shaped standard second-order RW (shown in figure 2c). These six fundamental RWs constitute a claw-like structure (figure 4d). Similarly, we use the numerical method to display the locations of humps and valleys (figure 3c) of the beak-shaped standard third-order RW.

Figure 4.

Beak-shaped third-order RW. (a) Standard one with s_0,1,2=0. (b) Triangular structure with s_0,2=0,s₁=20. (c) Pentagon structure with s_0,1=0,s₂= 250, (d) claw-like structure with s₀=0,s₁=31,s₂=500. (Online version in colour.)

 Case 4. N=4, F=s₀+s₁ ε²+s₂ ε⁴+s₃ ε⁶. The beak-shaped fourth-order RW possesses four parameters s_0,1,3,4, which can be used to derive different kinds of excitation patterns. We find that beak-shaped fourth-order RW consists of ten fundamental RWs, which mainly constitute eight types of distribution patterns. (i) When s_j=0,j=0,1,2,3, these ten fundamental RWs merge together and fuse into an identical pattern with beak-shaped standard fourth-order RW (figure 5a(i)). (ii) When s₀=0,s₁≠0,s₂=0,s₃=0,s₄=0, these 10 fundamental RWs will separate gradually by increasing the value |s₂| and arrange with a triangular structure (figure 5b(i)). There are four fundamental RWs in each side of the triangle and one fundamental RW in the inner. (iii) When s₀=0,s₁=0,s₂≠0,s₃=0, these 10 fundamental RWs exhibit a pentagram structure (figure 5c(i)). (iv) When s₀=0,s₁=0,s₂=0,s₃≠0, by increasing the value |s₃|, a heptagonal structure will arise (figure 5d(i)). Three fundamental RWs fuse into a beak-shaped standard second-order RW located in the inner and seven fundamental RWs and around it are the other seven fundamental RWs. (v) When s₀=0,s₁≠0,s₂=0,s₃≠0, by choosing certain s₁,s₃, we can obtain a so-called double column structure (figure 5a(ii)), which is constituted by two standard second-order RWs and four fundamental RWs arranging in a quadrangle; (vi) When s₀=0,s₁=0,s₂≠0,s₃≠0, by choosing certain s₂,s₃, a claw-line-I structure can be obtained (figure 5b(ii)), which consists of a claw-like structure and three fundamental RWs arranging in a line. This claw-like structure contains four claws and a standard second-order RW. (vii) When s₀=0,s₁=8,s₂=108,s₃=0, we can obtain a claw-line-II structure (figure 5c(ii)), which consists of a claw-like structure and two fundamental RWs arranging in a line. But the number of claw is five. (viii) When s₁≠0,s₂≠0,s₃≠0, by choosing certain s₁,s₂,s₃, we can obtain a claw-like structure (figure 5d(ii)), which is constituted by a standard third-order RW and four fundamental RWs arranging its claws. Similarly, we use the numerical method to display the locations of humps and valleys (figure 3d) of the standard third-order RW.

Figure 5.

Beak-shaped fourth-order RW. (a(i)) Standard one with s_0,1,2,3=0. (b(i)) Triangle structure with s_0,2,3=0,s₁=60. (c(i)) Pentagram structure with s_0,1= 0,s₂=500,s₃=0. (d(i)) Heptagonal structure with s_0,1,2=0,s₃=2000. (a(ii)) Double column structure with s_0,2=0,s₁=9,s₃=450. (b(ii)) Claw-line-I structure with s_0,1= 0,s₂=104,s₃=960. (c(ii)) Claw-line-II structure with s_0,3=0,s₁=8,s₂=108. (d(ii)) Claw-like structure with s₀=0,s₁=12,s₂=101,s₃=1016. (Online version in colour.)

(b). Beak-type rogue wave pair solutions

Theorem 4.2 —

If equation (1.1) has the two families of solutions p=q=g₁(x,t) and p=q=g₂(x,t), then p=g₁(x−x₁,t−t₁)+g₂(x−x₂,t−t₂) e^−2iθ, q=g₁(x− x₁,t−t₁)−g₂(x−x₂,t−t₂) e^−2iθ are also the solutions of equation (1.1), where x_1,2, t_1,2 are any real constants, and θ∈[0,π/2].

In this section, we derive the beak-shaped RWp solutions by beak-shaped RW solutions and obtain different kinds of spatial–temporal distribution structures such as triangular, pentagonal and claw-like pattern. General beak-shaped higher-order RWp solutions are obtained in the following formula:

$$\begin{array}{rl}p[N]& \hspace{0.17em}=p_{\mathrm{e}}[N](x-x_1,t-t_1)+p_{\mathrm{e}}[N](x-x_2,t-t_2)\hspace{0.17em}{\mathrm{e}}^{-2\mathrm{i}\theta }\\ \text{and}q[N]& \hspace{0.17em}=p_{\mathrm{e}}[N](x-x_1,t-t_1)-p_{\mathrm{e}}[N](x-x_2,t-t_2)\hspace{0.17em}{\mathrm{e}}^{-2\mathrm{i}\theta },\end{array}\}$$ 4.11

where p_e[N](x,t) is the N-order eye-shaped RW solution by theorem 4.1, and x₁, t₁ are real constants. When the distance between two RWs is far enough, they behave as two RWs superposition on the plane-wave background. When the distance of two RWs is close, their interaction will induce new behaviours. In this section, we take x_1,2=0, t₁=−t₂=1, $\theta =\arccos \left(3/5\right)$.

 Case 1. N=1. The dynamics of the fundamental RWp is shown in figure 6. It is seen that there are two humps in one component, and correspondingly there are four valleys in the other component. We numerically display the locations of humps and valleys (figure 7a(i)(ii)).

Figure 6.

Beak-shaped first-order RWp with s₀=0. (Online version in colour.)

Figure 7.

The trajectory of beak-shaped standard RWp. (a) The first-order RW shown in figure 2a, (b) the second-order RW shown in figure 2c, (c) the third-order RW shown in figure 4a and (d) the fourth-order RW shown in figure 5a. (Online version in colour.)

 Case 2. N=2, F=s₀+s₁ ε². There are two free parameters in the general beak-shaped second-order RWp solution, which can be used to distinct types of excitation patterns. We find that beak-shaped second-order RWp consists of three fundamental RWp, which can arrange with different kinds of patterns by varying the two free parameters. There are mainly two kinds of beak-shaped RWp solutions: (i) when s₁=0, we give standard beak-shaped second-order RWp (figure 8a,b), which symmetrical about x- and t-axes; (ii) when s₁≠0, by increasing the value |s₁|, we can obtain a triangle spatial symmetry structure (figure 8c,d). We numerically display the locations of humps and valleys (figure 7b(i)(ii)) of the standard one.

Figure 8.

Beak-shaped second-order RWp. (a,b) Standard one with s₀=s₁=0. (c,d) triangular structure with s₀=0,s₁=20. (Online version in colour.)

 Case 3. N=3, F=s₀+s₁ε²+s₂ε⁴. There are three free parameters in the general beak-shaped third-order RWp. We find that beak-shaped third-order RWp consists of six fundamental RWp solutions, which can arrange with four types of distribution structures. (i) When s₁=0,s₂=0,s₃=0, the six fundamental RWps merge together (figure 9a(i),b(i)). (ii) When s₁=0,s₂≠0,s₃=0, by increasing the value |s₂|, these six fundamental RWps will separate gradually and constitute a triangular structure (figure 9c(i),d(i)). (iii) When s₁=0,s₂=0,s₃≠0, these six fundamental RWps will separate and arrange with a pentagonal structure by increasing the value |s₃| (figure 9a(ii),b(ii)). (iv) When s₁=0,s₂≠0,s₃≠0, by choosing certain s₁,s₂, claw-like structure can be derived (figure 9c(ii),d(ii)), which consists of three claws and a beak-shaped standard second-order RWp. We numerically display the locations of humps and valleys (figure 7c(i)(ii)) of the beak-shaped standard third-order one.

Figure 9.

Beak-shaped three-order RWp. (a(i),b(i)) Standard one with s_0,1,2=0. (c(i),d(i)) Triangular structure with s_0,2=0,s₁=20. (a(ii),b(ii)) Pentagonal structure with s_0,1=0,s₂=250. (c(ii),d(ii)) ‘Claw-like’ structure with s₀=0,s₁=31,s₂=500. (Online version in colour.)

5. Multi-dark-dark solitons and asymptotic properties

In this section, we consider the CNLS-p system (1.1) with the repulsive interaction σ=−1, where the MI does not appear. We will derive the multi-dark-dark soliton and the multi-kink-dark-dark soliton by the generalizing DT [35]. As applications, we investigate the dynamics of the single dark-dark, single kink-dark-dark, two dark-dark and two kink-dark-dark soltions.

To derive the one-dark-dark solitons for the CNLS-p system (1.1), we restrict λ₁∈(−1,1) such that ${\mu }_1=\sqrt{1-{\lambda }_1^2}\in \mathbb{R}$. By using this relation 1/(μ₁+iλ₁)=μ₁−iλ₁, we can obtain $1-1/|{\mu }_1+\mathrm{i}{\lambda }_1{|}^2=2\mathrm{i}({\lambda }_1-{\lambda }_1^{\ast })/({\mu }_1^{\ast }+{\mu }_1)$.

Taking

$$y_1=C\left(\begin{array}{c}{\mathrm{e}}^{A_1}+{\alpha }_1({\lambda }_1-{\lambda }_1^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_1}\\ {\mathrm{e}}^{A_1}+{\alpha }_1({\lambda }_1-{\lambda }_1^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_1}\\ {\displaystyle \frac{\mathrm{i}}{{\mu }_1+\mathrm{i}{\lambda }_1}}\hspace{0.17em}{\mathrm{e}}^{A_1}+{\displaystyle \frac{\mathrm{i}}{-{\mu }_1+\mathrm{i}{\lambda }_1}}{\alpha }_1({\lambda }_1-{\lambda }_1^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_1}\\ {\displaystyle \frac{\mathrm{i}}{{\mu }_1+\mathrm{i}{\lambda }_1}}\hspace{0.17em}{\mathrm{e}}^{A_1}+{\displaystyle \frac{\mathrm{i}}{-{\mu }_1+\mathrm{i}{\lambda }_1}}{\alpha }_1({\lambda }_1-{\lambda }_1^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_1}\end{array}\right),$$ 5.1

where α₁ is an arbitrary constant, A₁=g(μ₁)=μ₁(x+λ₁t). Letting β₁=2μ₁Im[(1−1/(μ₁+iλ₁)* (−μ₁+iλ₁))α₁], we can obtain $y_1^{†}J{y}_1/({\lambda }_1-{\lambda }_1^{\ast })=(4\mathrm{i}/({\mu }_1+{\mu }_1^{\ast }))({\mathrm{e}}^{2A_1}+{\beta }_1)$. To avoid the singular dark solitons, we need to restrict β₁>0 (β₁=0 corresponds to a trivial solution). Based on theorem 3.1, we have the one-dark-dark solitons of the CNLS-p system (1.1).

 Case 1. The single dark-dark solitons for the CNLS-p system (1.1) are

$$p[1]=q[1]=\frac{1}{2}[1-\frac{{\mu }_1}{{\mu }_1+\mathrm{i}{\lambda }_1}-\frac{{\mu }_1}{{\mu }_1+\mathrm{i}{\lambda }_1}\tanh (A_1-\frac{1}{2}\ln \beta )]{\mathrm{e}}^{-\mathrm{i}t}.$$ 5.2

When $x\to -\mathrm{\infty }$, we have p[1]→e^−it/2. When $x\to +\mathrm{\infty }$, we have p[1]→(−(μ₁+iλ₁)/(μ₁+iλ₁))(e^−it/2). Moreover, |p[1]|→1/2 as $x\to \pm \mathrm{\infty }$. In addition, as x varies from $-\mathrm{\infty }$ to $+\mathrm{\infty }$, the phase of the p[1] component acquires a shift in the amount of φ₁ with e^iφ₁=−(μ₁+iλ₁)/(μ₁+iλ₁). The density function |p[1]| of the soliton moves at a velocity −λ₁. The centre of the dark soliton p[1] is along the line ${\mu }_1(x+{\lambda }_{1}t)-\ln {\beta }_1/2{\chi }_{1I}=0$. The intensity |p[1]|² at the centre is ${\lambda }_1^2/4=(\cos {\phi }_1+1)/4$. The centre intensity is lower than the background intensity $\frac{1}{4}$. Thus, the soliton p[1] is a dark soliton. Noting that the intensity dip at the centre of the p[1] component is controlled by its phase shift φ₁, the phase shift dictates how ‘dark’ the centre is. To illustrate the dark soliton, we take the parameters as λ₁=1/3,β₁=1 to exhibit the dark soliton (figure 10a).

Figure 10.

(a) Dark soliton, (b,c) kink-dark solitons. Parameters are λ₁=1/3,β₁= 1,θ=π/6. (Online version in colour.)

 Case 2. The single kink-dark-dark solitons for the CNLS-p system (1.1) are

$$\begin{array}{rl}& p[1]=p_{+},q[1]=p_{-},p_{\pm }=\frac{1}{2}\\ & [1\pm (1-\frac{{\mu }_1}{{\mu }_1+\mathrm{i}{\lambda }_1}-\frac{{\mu }_1}{{\mu }_1+\mathrm{i}{\lambda }_1}\tanh (A_1-\frac{1}{2}\ln \beta )){\mathrm{e}}^{-2\mathrm{i}\theta }]{\mathrm{e}}^{-\mathrm{i}t},\end{array}$$ 5.3

where θ∈[0,π/2].

For $x\to -\mathrm{\infty }$, we have p[1]→(1+e^−2iθ)(e^−it/2) and q[1]→(1−e^−2iθ)(e^−it/2). When $x\to +\mathrm{\infty }$, we have p[1]→(1+e^{iφ₁−2iθ})(e^−it/2)=(1+e^{iφ₁−2iθ})(e^−it/2) and q[1]→(1−e^{iφ₁−2iθ})(e^−it/2)=(1−e^{iφ₁−2iθ})(e^−it/2). Moreover, $|p[1]|\to \frac{1}{2}|1+{\mathrm{e}}^{-2\mathrm{i}\theta }|$ as $x\to -\mathrm{\infty }$ and $|p[1]|\to \frac{1}{2}|1+{\mathrm{e}}^{\mathrm{i}{\phi }_1-2\mathrm{i}\theta }|$ as $x\to +\mathrm{\infty }$. Similarly, we have $|q[1]|\to \frac{1}{2}|1-{\mathrm{e}}^{-2\mathrm{i}\theta }|$ as $x\to -\mathrm{\infty }$ and $|q[1]|\to \frac{1}{2}|1-{\mathrm{e}}^{\mathrm{i}{\phi }_1-2\mathrm{i}\theta }|$ as $x\to +\mathrm{\infty }$. Therefore, when θ≠(φ₁−2πk)/4, these components p[1]and q[1] approach different constant amplitudes as $x\to \pm \mathrm{\infty }$, that is, these components p[1] and q[1] form the kink-dark (or grey) solitons, which differs from the usual dark (grey) solitons. It is easy to see that the density functions |p[1]| and |q[1]| of the kink solitons move at velocity −λ₁. The centre of the kink-dark soliton p[1] is along the line ${\mu }_1(x+{\lambda }_{1}t)-\ln {\beta }_1/2{\chi }_{1I}=0$. It is easy to verify that the densities |p[1]| and |q[1]| at the centre are lower than their two different background densities. Thus, the kink solitons p[1] and q[1] are kink-dark solitons. We choose λ₁=1/3,β₁=1,θ=π/6 to illustrate these kink-dark solitons (figure 10b,c).

 Case 3. We still consider the same seed solutions $p=q=\frac{1}{2}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}t}$ to solve the Lax pair (3.1) at λ=λ_j. We take

$$y_j=C\left(\begin{array}{c}{\mathrm{e}}^{A_j}+{\alpha }_j({\lambda }_j-{\lambda }_j^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_j}\\ {\mathrm{e}}^{A_j}+{\alpha }_j({\lambda }_j-{\lambda }_j^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_j}\\ {\displaystyle \frac{\mathrm{i}}{{\mu }_j+\mathrm{i}{\lambda }_j}}\hspace{0.17em}{\mathrm{e}}^{A_j}+{\displaystyle \frac{\mathrm{i}}{-{\mu }_j+\mathrm{i}{\lambda }_j}}{\alpha }_j({\lambda }_j-{\lambda }_j^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_j}\\ {\displaystyle \frac{\mathrm{i}}{{\mu }_j+\mathrm{i}{\lambda }_j}}\hspace{0.17em}{\mathrm{e}}^{A_j}+{\displaystyle \frac{\mathrm{i}}{-{\mu }_j+\mathrm{i}{\lambda }_j}}{\alpha }_j({\lambda }_j-{\lambda }_j^{\ast })\hspace{0.17em}{\mathrm{e}}^{-A_j}\end{array}\right),$$ 5.4

where α_j are constants, A_j=g(μ_j)=μ_j(x+λ_jt), λ_j∈(−1,1) (j=1,2,…,N) N distinct real parameters such that we can obtain the N-dark-dark solitons for the CNLS-p system (1.1)

$$p[N]=q[N]=\frac{2\hspace{0.17em}\det \left(\begin{array}{cc}\hat{M}& {\hat{Y}}_1^{†}\\ {\hat{Y}}_3& 0\end{array}\right)}{\det \hspace{0.17em}\hat{M}}+\frac{1}{2}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}t},$$ 5.5

in terms of simple determinants and the relation

$$1-\frac{1}{{({\mu }_i+\mathrm{i}{\lambda }_i)}^{\ast }({\mu }_j+i{\lambda }_j)}=\frac{2\mathrm{i}({\lambda }_j-{\lambda }_i^{\ast })}{{({\mu }_i+\mathrm{i}{\lambda }_i)}^{\ast }+({\mu }_j+i{\lambda }_j)},(1\le i,j\le N),$$

where $\hat{M}={({\hat{M}}_{ij})}_{N\times N}$, ${\hat{M}}_{ij}=(4\mathrm{i}/{({\mu }_i+\mathrm{i}{\lambda }_i)}^{\ast }+({\mu }_j+\mathrm{i}{\lambda }_j))({\mathrm{e}}^{A_i-A_j}+{\delta }_{ij}{\beta }_j)$, β_j=2μ_jIm[(1−1/(μ_j+iλ_j)*(−μ_j+iλ_j))α_j]>0, δ_ij is the Kronecker delta function, $\hat{Y}={({\hat{Y}}_{ij})}_{4\times N}$, ${\hat{Y}}_{1j}={\hat{Y}}_{2j}={\mathrm{e}}^{A_j}$, and ${\hat{Y}}_{3j}={\hat{Y}}_{4j}=(\mathrm{i}/{\mu }_j+\mathrm{i}{\lambda }_j)\hspace{0.17em}{\mathrm{e}}^{A_j}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}t}$.

We know that p[N]→pe^iφ_±N as $x\to \pm \mathrm{\infty }$, where φ_±N is a phase constant. Thus, the solution p[N] approaches a constant amplitude |p| at large distances.

We take parameters N=2, λ₁=−1/4,λ₂=−1/3,β_1,2=1 to demonstrate the collision of two-dark-dark solitons (figure 11a(i)). We find that two dark solitons for the component p[2] pass through each other without any change of shape and velocity or darkness after interaction. Thus, there is no energy exchange between the p[2] and q[2] components after interaction. The primary cause for the complete energy transmission in both components in dark-dark solitons collisions is that the intensity of each dark-dark soliton is completely characterized by the background p,q and the soliton parameters μ_j's. These background parameters are the same for both colliding solitons, and clearly do not change before and after collision. The soliton parameters μ_j's are constants of motion throughout collision. As a consequence, the intensity of each dark-dark soliton (in both p[2] and q[2] components) cannot change before and after collision. The positions of two-dark-dark solitons do shift after collision though (figure 11a(ii)). This position shift is always towards the soliton's moving direction, which is the same as collision of bright solitons.

Figure 11.

(a(i)(ii)) Collision of two dark-dark solitons. (b(i)(ii),c(i)(ii)) Collision of two kink-dark-dark solitons. Parameters are λ₁=−1/4,λ₂=−1/3,β_1,2=1,θ=π/6. (Online version in colour.)

 Case 4. The kink-shaped multi-dark-dark solitons for the CNLS-p system (1.1) are

$$\begin{array}{rl}p[N]& \hspace{0.17em}={\displaystyle \frac{1+{\mathrm{e}}^{-2\mathrm{i}\theta }}{2}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}t}+\frac{2\hspace{0.17em}\det \left(\begin{array}{cc}\hat{M}& {\hat{Y}}_1^{†}\\ {\hat{Y}}_3& 0\end{array}\right){\mathrm{e}}^{-2\mathrm{i}\theta }}{\det \hspace{0.17em}\hat{M}}}\\ \text{and}q[N]& \hspace{0.17em}={\displaystyle \frac{1-{\mathrm{e}}^{-2\mathrm{i}\theta }}{2}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}t}-\frac{2\hspace{0.17em}\det \left(\begin{array}{cc}\hat{M}& {\hat{Y}}_1^{†}\\ {\hat{Y}}_3& 0\end{array}\right){\mathrm{e}}^{-2\mathrm{i}\theta }}{\det \hspace{0.17em}\hat{M}}.}\end{array}\}$$ 5.6

It can be verified that the kink-shaped N-dark-dark solitons p[N] and q[N] approach two distinct constant amplitudes as $x\to \pm \mathrm{\infty }$. In what follows, we investigate the collision of kink-type two-dark-dark solitons in CNLS-p (1.1). To demonstrate the collision of kink-shaped two-dark-dark solitons, we take parameters N=2, λ₁=−1/4,λ₂=−1/3,β_1,2=1,θ=π/6. The corresponding kink-shaped two-dark-dark solitons p[2] and q[2] are shown in figure 11b(i)(ii),c(i)(ii). We can see that after collision, the two kink-dark solitons pass through each other with these changes of shape, darkness and velocity in p[2] and q[2] components.

6. Numerical simulations for the dynamics of solutions

In this section, we perform the numerical simulations to explore the dynamical behaviours of some above-mentioned multi-dark solitons and multi-RWs and pairs of equations (1.1) by using the finite difference method [56].

 Figure 12a(i) exhibits the numerical results of the unperturbed kink-dark soliton |q|², using the analytical solution (5.3) at t=−3 as an initial condition and identical parameters as in figure 10b,c. Figure 12c(i) shows the numerical results of the unperturbed two-kink-dark soliton |q|², using the analytical solution (5.6) at t=−6 as an initial condition and identical parameters as in figure 11b(i),c(i),b(ii),c(ii). Figure 12a(ii) shows the numerical results of the unperturbed kink-shaped standard 2-order RW |p|², using the analytical solution at t=−2 as an initial condition and identical parameters as in figure 2c. Figure 12c(ii) shows the numerical results of the unperturbed kink-shaped 2-order RW |p|², using the analytical solution at t=−3 as an initial condition and the parameters F=10ε² (similar to one in figure 2d). It is clearly seen that our numerical solutions exactly reproduce the analytical solutions at least until t=3 (a(i)), t=6 (c(i)), t=2 (a(ii)) and t=3 (c(ii)), respectively. This manifests the accuracy of our numerical scheme and anticipates the stable evolutions of these solutions without a noise.

Figure 12.

Numerical simulations: (a(i),b(i)) one-kink-dark-dark soliton |q|² with parameters λ₁=1/3,β₁=1,θ=π/6. (c(i),d(i)) two-kink-dark-dark soliton |q|² with parameters λ₁=−1/4,λ₂=−1/3,β_1,2=1,θ=π/6. (a(ii),b(ii)) beak-shaped standard second-order RW with s₀=s₁=0. (c(ii),d(ii)) Triangular structure of a beak-shaped second-order RW with s₀=0,s₁=10. (Online version in colour.)

We further numerically inspect the stability of these solutions by perturbing the above initial conditions. Particularly, we multiply both the p and q components at t=−3 (b(i)), t=−6 (d(i)), t=−2 (b(ii)), t=−3 (d(ii)), respectively, by 1+0.02Random [−1,1]. Numerical results are provided in figure 12b(i),d(i),b(ii),d(ii), showing that they can evolve as before. In other words, these solutions are robust against small noise.

7. Conclusion and discussion

In conclusion, we have investigated the integrable CNLS-p equations (1.1), which can be used to describe transition dynamics of a one-dimensional two-component Bose–Einstein condensate system [25,57–60]. Based on the loop group method, we construct the general N-fold DT for CNLS-p equations (1.1) with both attractive and repulsive interactions. It is well known that the dark solitons are hard to be obtained through the classical DT, while we derived the general N-dark-dark solitons of this system through the generalizing DT, showing that when these solitons collide with each other, energies in all components of the solitons completely transmit through. We also derive the general kink-shaped multi-dark-dark solitons and analyse the collision of two kink-dark solitons, showing that the two kink-dark solitons pass through each other with these changes of shape, darkness and velocity in p[2] and q[2] components. By using the Taylor series expansion coefficients of a special solution to the Lax pair (1.1) with a seed solution and a fixed spectral parameter, the generalizing DT is constructed. As applications, we derive general beak-shaped higher-order RW and RWp solutions and give them a classification. Finally, we perform the numerical simulations to show that these dark solitons and RWs are stable enough to develop.

Our results can be regarded as the generalization of the work in [35] to the integrable CNLS equations with four-wave mixing. Continuing the generalized DT process, the more complicated localized wave solutions can also be generated, which may possess more abundant striking dynamics. The method to derive dark solitons can also be extended to other nonlinear wave equations, which are useful to study the dynamics of nonlinear localized waves analytically. The results in this paper further reveal the intriguing dynamic distributions of localized waves in the CNLS equations with four-wave mixing, and we hope our results can be verified in real experiments in the future.

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Authors' contributions

All the authors have contributed equally to this work.

Competing interests

We declare we have no competing interests.

Funding

This work was partially supported by the NSFC under grant no. 11571346 and the Youth Innovation Promotion Association CAS.