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Dynamics of breathers and rogue waves in scalar and multicomponent nonlinear systems

  • Weiying Wang , 1 ,
  • Xiubin Wang , 2
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  • 1School of Economics, Harbin University of Commerce, Harbin 150028, China
  • 2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received date: 2021-07-30

  Revised date: 2021-11-20

  Accepted date: 2022-02-07

  Online published: 2022-05-20

Copyright

© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this paper, we propose a new method, the variable separation technique, for obtaining a breather and rogue wave solution to the nonlinear evolution equation. Integrable systems of the derivative nonlinear Schrödinger type are used as three examples to illustrate the effectiveness of the presented method. We then obtain a family of rational solutions. This family of solutions includes the Akhmediev breather, the Kuznetsov-Ma breather, versatile rogue waves, and various interactions of localized waves. Moreover, the main characteristics of these solutions are discussed and some graphics are presented. More importantly, our results show that more abundant and novel localized waves may exist in the multicomponent coupled equations than in the uncoupled ones.

Cite this article

Weiying Wang , Xiubin Wang . Dynamics of breathers and rogue waves in scalar and multicomponent nonlinear systems[J]. Communications in Theoretical Physics, 2022 , 74(4) : 045001 . DOI: 10.1088/1572-9494/ac5238

1. Introduction

As we all know, many nonlinear systems of physical interest support solitons, which are localized waves that arise from a balance between nonlinearity and dispersion and can propagate steadily for a long time. It has recently been found that another class of rational solutions, i.e., breathers, is also of great importance. In addition, because of their localization properties, breathers have been recognized as models of rogue waves, which have recently become a popular research topic [13]. They also often appear in many fields, such as optics, Bose–Einstein condensates, plasma physics, hydrodynamics, photonics, finance, etc [410].
The standard nonlinear Schrödinger equation (NLSE) is completely integrable [11], and many kinds of exact solution have been found. In particular, the Peregrine soliton [12], the Akhmediev breather (AB) [1, 13], and the Kuznetsov-Ma (KM) breathers [14, 15] have been associated with rogue waves as a potential outcome of the modulational instability (MI) of a plane wave. In particular, a recent work has examined the relationship between extraordinary MI in optics and hydrodynamics and the generation of large-amplitude periodic wave trains [16]. Earlier, in 1993, some scholars applied the powerful analytic method developed in the abovementioned paper by Akhmediev et al to the normal-dispersion regime [1720]. Because the Peregrine soliton is localized in both time and space, it is recognized as a rogue wave prototype and reveals the main features of rogue waves [21]. More importantly, a Peregrine soliton of the NLSE has been experimentally observed in water wave tanks [22] and nonlinear fiber optics [23, 24].
To show the effectiveness of the variable separation technique in this paper, we focus on the following derivative nonlinear Schrödinger-type equations:
$\begin{eqnarray}{\rm{i}}{{ \mathcal U }}_{t}+{{ \mathcal U }}_{{xx}}+{\rm{i}}{\left({ \mathcal U }{{ \mathcal U }}^{\dagger }{ \mathcal U }\right)}_{x}=0,\end{eqnarray}$
where
$\begin{eqnarray*}\begin{array}{l}{ \mathcal U }=q(x,t),\,\,{ \mathcal U }=\left({q}_{1}(x,t),{q}_{2}(x,t)\right),\\ {\rm{and}}\,\,{ \mathcal U }=\left(\begin{array}{cc}{q}_{1} & {q}_{0}\\ {q}_{0} & {q}_{-1}\end{array}\right).\end{array}\end{eqnarray*}$
Here, ${{ \mathcal U }}^{\dagger }$ is the Hermitian conjugate of ${ \mathcal U }$, and the subscript x, t denotes partial differentiation throughput.
Many studies of localized waves in nonlinear science have been carried out recently [2545]. In addition, there have been some developments in the field of the variable separation technique and soliton structures as well as in its application. For example, [39] presents a system with controllable parameters that describes the evolution of polarization modes in nonlinear fibers. In [40], Dai et al investigated scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials. In addition, [41, 42] studied wick-type stochastic multi-solitons, soliton molecules, and fractional soliton solutions of the NLSE. In [43], vector breathers for the coupled fourth-order NLSE were investigated. In [44], explicit soliton–cnoidal wave interaction solutions for the (2+1)-dimensional negative-order breaking soliton equation were discussed in detail. These results help us to study the soliton structures associated with nonlinear differential equations in the field of mathematical physics. It is also known that the variable separation technique is a powerful method for the derivation of soliton solutions. This concept, to the best of the authors’ knowledge, has never been reported before. The chief purpose of this work is to employ a variable separation technique to investigate breathers and rogue wave solutions of the derivative NLSEs. In addition, the dynamic behaviors of the localized wave solutions are also considered by selecting suitable parameters.

2. A variable separation technique

In this section, we introduce a variable separation technique for constructing rogue wave solutions to nonlinear wave equations. A general integrable NLSE has the following Lax pair:
$\begin{eqnarray}\left\{\begin{array}{ll}{{\rm{\Psi }}}_{x}={ \mathcal U }{\rm{\Psi }}=\left({\rm{i}}{\lambda }^{{s}_{1}}\widetilde{\sigma }+{ \mathcal Q }\right){\rm{\Psi }}, & {s}_{1}=1,2,\ldots \\ {{\rm{\Psi }}}_{t}={ \mathcal V }{\rm{\Psi }}=\left({\rm{i}}{\lambda }^{{s}_{2}}\widetilde{\sigma }+\widetilde{{Q}}\right){\rm{\Psi }}, & {s}_{2}=1,2,\ldots ,\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray*}\widetilde{\sigma }=\left(\begin{array}{cc}{{ \mathcal I }}_{m} & 0\\ 0 & -{{ \mathcal I }}_{n}\end{array}\right).\end{eqnarray*}$
To determine nonlinear wave solutions explicitly, we take the following six steps:
Step 1
We consider the following plane wave solution as a seed solution for the general integrable NLSE:
$\begin{eqnarray}\begin{array}{l}{q}_{1,[0]}={a}_{1}{{\rm{e}}}^{{\rm{i}}\widetilde{\varphi }(x,t)},\,\,{q}_{2,[0]}={a}_{2}{{\rm{e}}}^{{\rm{i}}\widetilde{\varphi }(x,t)}, \ldots ,\,\,{q}_{n,[0]}={a}_{n}{{\rm{e}}}^{{\rm{i}}\widetilde{\varphi }(x,t)},\end{array}\end{eqnarray}$
where a1, a2,…,an are free parameters.
Step 2
We then seek a family of the solutions of the Lax system (2) in the following form:
$\begin{eqnarray}{\rm{\Psi }}={\rm{\Lambda }}{ \mathcal R }{ \mathcal E }{ \mathcal Z },\end{eqnarray}$
and
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal R } & = & \exp ({\rm{i}}{\rm{\Theta }}x),\,\,{ \mathcal E }=\exp ({\rm{i}}{\rm{\Omega }}t),\\ {\rm{\Lambda }} & = & {\rm{diag}}\left(\mathop{\underbrace{1,1,\ldots 1}}\limits_{m},\mathop{\underbrace{{{\rm{e}}}^{\pm {\rm{i}}\widetilde{\varphi }},{{\rm{e}}}^{\pm {\rm{i}}\widetilde{\varphi }},\ldots ,{{\rm{e}}}^{\pm {\rm{i}}\widetilde{\varphi }}}}\limits_{n}\right),\end{array}\end{eqnarray}$
where ${ \mathcal Z }$ is an arbitrary complex vector. In addition, it is assumed that
$\begin{eqnarray}\left[{\rm{\Theta }},{\rm{\Omega }}\right]={\rm{\Theta }}{\rm{\Omega }}-{\rm{\Omega }}{\rm{\Theta }}=0.\end{eqnarray}$
Step 3
Substituting (4) into (2) yields
$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Lambda }}}_{x}+{\rm{i}}{\rm{\Lambda }}{\rm{\Theta }}={ \mathcal U }{\rm{\Lambda }},\\ {{\rm{\Lambda }}}_{t}+{\rm{i}}{\rm{\Lambda }}{\rm{\Omega }}={ \mathcal V }{\rm{\Lambda }}.\end{array}\right.\end{eqnarray}$
By combining conditions (6) and (7), we obtain Θ and Ω.
Step 4
Using the mathematical software Maple, we rewrite the exponential matrix ${ \mathcal R },{ \mathcal E }$ in (5) as a matrix function whose elements can be expressed by trigonometric functions and exponential functions.
Step 5
Taking the solution (3) as the seed solution in the Darboux transformation (DT), we can obtain a periodic solution of the general integrable NLSE composed of trigonometric functions and exponential functions.
Step 6
Let the period of the periodic wave go to infinity in the breather solution; the function $\exp ({\rm{i}}{\rm{\Theta }}x+{\rm{i}}{\rm{\Omega }}t)$ then becomes a combination of exponential and polynomial functions of x and t. Thus, we get the first-order rogue wave.

3. Example 1

In this section, as an example, we consider the famous derivative derivative NLSE. As a fundamental and important nonlinear physical model, the derivative nonlinear Schrödinger equation (DNLSE)
$\begin{eqnarray}{\rm{i}}{q}_{t}+{q}_{{xx}}+{\rm{i}}{\left({q}^{2}\bar{q}\right)}_{x}=0\end{eqnarray}$
has several physical applications, such as weak nonlinear electromagnetic waves in ferromagnetic [46], antiferromagnetic [47], and dielectric [48] systems subjected to external magnetic fields. It accurately describes the propagation of small-amplitude Alfvén waves in a low-β plasma [49, 50] and the evolution of large-amplitude magnetohydrodynamic waves in a high-β plasma [51, 52]; and also excellently models the transmission of ultra-short optical pulses in single-mode optical fibers [53, 54].

3.1. Darboux transformation

Equation (8) is completely integrable. Its Lax pair is
$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Psi }}}_{x}=U{\rm{\Psi }}=-{\rm{i}}{\lambda }^{2}{\sigma }_{1}+\lambda Q,\\ {{\rm{\Psi }}}_{t}=V{\rm{\Psi }}=-2{\rm{i}}{\lambda }^{4}{\sigma }_{1}+2{\lambda }^{3}Q-{\rm{i}}{\lambda }^{2}{Q}^{2}{\sigma }_{1}+\lambda {Q}^{3}-{\rm{i}}\lambda {Q}_{x}{\sigma }_{1},\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray*}{\sigma }_{1}=\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right),\,\,Q=\left(\begin{array}{cc}0 & q\\ -\bar{q} & 0\end{array}\right).\end{eqnarray*}$
It is straightforward to check that system (8) can be easily generated from the compatibility condition UtVx + [U,V] = 0.

Using ${\rm{\Psi }}(x,t,{\lambda }_{1})=\left(\begin{array}{c}\phi \\ \psi \end{array}\right)$ to denote a special vector solution of system (9) with $\lambda ={\lambda }_{1}$, the Lax pair admits the following Darboux transformation

$\begin{eqnarray*}{\rm{\Psi }}[1]=T{\rm{\Psi }},\,\,T\,=\,I+\displaystyle \frac{{\lambda }^{2}{{\rm{\Gamma }}}_{2}[1]+\lambda {{\rm{\Gamma }}}_{1}[1]}{| {\lambda }_{1}{| }^{2}},\end{eqnarray*}$
where
$\begin{eqnarray*}\begin{array}{l}{{\rm{\Gamma }}}_{2}=\left(\begin{array}{cc}\displaystyle \frac{{{\rm{\Psi }}}^{\dagger }{\delta }^{\dagger }{\rm{\Psi }}}{{{\rm{\Psi }}}^{\dagger }\delta {\rm{\Psi }}} & 0\\ 0 & \displaystyle \frac{{{\rm{\Psi }}}^{\dagger }\delta {\rm{\Psi }}}{{{\rm{\Psi }}}^{\dagger }{\delta }^{\dagger }{\rm{\Psi }}}\end{array}\right),\\ {{\rm{\Gamma }}}_{1}=\left(\begin{array}{cc}0 & \displaystyle \frac{\left({\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}\right){\rm{\Psi }}{{\rm{\Psi }}}^{\dagger }}{{{\rm{\Psi }}}^{\dagger }\delta {\rm{\Psi }}}\\ \displaystyle \frac{\left({\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}\right){\rm{\Psi }}{{\rm{\Psi }}}^{\dagger }}{{{\rm{\Psi }}}^{\dagger }{\delta }^{\dagger }{\rm{\Psi }}} & 0\end{array}\right),\end{array}\end{eqnarray*}$
Here, $\delta ={\rm{diag}}({\lambda }_{1},{\bar{\lambda }}_{1})$ and † represents Hermite conjugation. The above system (9) can now be converted into a new system
$\begin{eqnarray*}{\rm{\Psi }}{[1]}_{x}=U{[1]{\rm{\Psi }}[1],\,\,{\rm{\Psi }}[1]}_{t}=V[1]{\rm{\Psi }}[1],\end{eqnarray*}$
and the transformation between potential functions yields
$\begin{eqnarray}{q}_{[1]}={q}_{[0]}-\displaystyle \frac{{\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}}{| {\lambda }_{1}{| }^{2}}{\left(\displaystyle \frac{\phi \bar{\psi }}{{\lambda }_{1}| \phi {| }^{2}+{\bar{\lambda }}_{1}| \psi {| }^{2}}\right)}_{x}.\end{eqnarray}$

In the following, the above DT is applied to construct the breathers and rogue wave solutions of the one-component DNLSE (8).

3.2. Exact breather solutions

It can readily be shown that equation (8) has the plane wave solution
$\begin{eqnarray}{q}_{[0]}={{a}{\rm{e}}}^{-{\rm{i}}\rho x},\,\,\rho ={a}^{2},\end{eqnarray}$
where a is an arbitrary parameter.
From Step 2 in section 2, a family of solutions of Lax system (9) yield
$\begin{eqnarray*}{\rm{\Psi }}=\left(\begin{array}{c}\phi \\ \psi \end{array}\right)={\rm{\Lambda }}{ \mathcal R }{ \mathcal E }{ \mathcal Z },\end{eqnarray*}$
and
$\begin{eqnarray}\begin{array}{l}{ \mathcal R }=\exp ({\rm{i}}{\rm{\Theta }}x),\,\,{ \mathcal E }=\exp ({\rm{i}}{\rm{\Omega }}t),\\ {\rm{\Lambda }}={\rm{diag}}\left(1,{{\rm{e}}}^{{\rm{i}}\rho x}\right),\,\,{ \mathcal Z }=\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\end{array}\right),\end{array}\end{eqnarray}$
where ${ \mathcal Z }$ is an arbitrary complex vector. It then follows from (6) and (7) that
$\begin{eqnarray*}\begin{array}{l}{\rm{\Theta }}=-{\lambda }^{2}{\sigma }_{1}+{\rm{i}}a\lambda \left(\begin{array}{cc}0 & -1\\ 1 & 0\end{array}\right)-\left(\begin{array}{cc}0 & 0\\ 0 & \rho \end{array}\right),\,\,\\ {\rm{\Omega }}=-2{\lambda }^{2}{\rm{\Theta }}+\rho {\lambda }^{2}.\end{array}\end{eqnarray*}$
Using the mathematical software Maple, we can write the exponential matrix ${ \mathcal R },{ \mathcal E }$ in (12) as
$\begin{eqnarray}\begin{array}{rcl}{ \mathcal R } & = & \displaystyle \frac{1}{2\tau }\left(\begin{array}{cc}{{\rm{\Theta }}}_{1} & {{\rm{\Theta }}}_{2}\\ {\widetilde{{\rm{\Theta }}}}_{2} & {{\rm{\Theta }}}_{3}\end{array}\right){{\rm{e}}}^{-{\rm{i}}\rho x/2},\\ { \mathcal E } & = & \displaystyle \frac{1}{2\tau }\left(\begin{array}{cc}{{\rm{\Omega }}}_{1} & {{\rm{\Omega }}}_{2}\\ {\widetilde{{\rm{\Omega }}}}_{2} & {{\rm{\Omega }}}_{3}\end{array}\right),\end{array}\end{eqnarray}$
where
$\begin{eqnarray*}\left\{\begin{array}{l}{{\rm{\Theta }}}_{1}=\left(\rho -2{\lambda }^{2}\right){\rm{i}}\,\sin (\tau x)+2\tau \cos (\tau x),\\ {{\rm{\Theta }}}_{2}=-{\widetilde{{\rm{\Theta }}}}_{2}=2\lambda a\,\sin (\tau x),\,2\tau =\sqrt{{\rho }^{2}+4{\lambda }^{4}},\\ {{\rm{\Theta }}}_{3}=\left(2{\lambda }^{2}-\rho \right){\rm{i}}\,\sin (\tau x)+2\tau \cos (\tau x),\\ {{\rm{\Omega }}}_{1}=\left(\rho -2{\lambda }^{2}\right){\rm{i}}\,\sin \left(\xi t\right)+2\tau \cos \left(\xi t\right),\\ {{\rm{\Omega }}}_{2}=-{\widetilde{{\rm{\Omega }}}}_{2}=2\lambda a\,\sin \left(\xi t\right),\,\xi =2{\lambda }^{2}\tau ,\\ {{\rm{\Omega }}}_{3}=\left(2{\lambda }^{2}-\rho \right){\rm{i}}\,\sin \left(\xi t\right)+2\tau \cos \left(\xi t\right).\end{array}\right.\end{eqnarray*}$
Using the plane wave (11) as the seed solution in the DT (10), we can now obtain breather wave solutions for the one-component DNLSE (8) composed of trigonometric functions. As these solutions have tedious expressions, we omit them here. In particular, when λ = h(1 + i)/2, if $h\in (-\infty ,\sqrt{\rho })\bigcup (1,+\infty )$, the solutions contain temporally periodic nonlinear waves, named AB. Furthermore, if $h\in (-\sqrt{\rho },\sqrt{\rho })$, the solutions become spatially periodic nonlinear waves, named KM breathers. Figures 1(a) and (b) reveal the dynamics of temporally periodic and spatially periodic breathers, respectively.
Figure 1. AB and KM breathers in equation (8) (∣q∣) for the following parameters: a = 1, ρ = 1, z1 = 1, z2 = 1. (a): λ = 1.1(1 + i)/2, (b): λ = 0.9(1 + i)/2.

3.3. Rogue wave solution

If τ = 0 (i.e. λ4 + ρ2 = 0), the function $\exp ({\rm{i}}{\rm{\Theta }}x+{\rm{i}}{\rm{\Omega }}t)$ yields a combination of exponential and polynomial functions of x and t. In particular, by substituting $\lambda =\tfrac{\sqrt{\rho }}{2}(1+{\rm{i}})$ into (13) and using the following Taylor expansion formulas:
$\begin{eqnarray}\sin (x)=x-\displaystyle \frac{{x}^{3}}{3!}+\displaystyle \frac{{x}^{5}}{5!}-\ldots ,\,\,\cos (x)=1-\displaystyle \frac{{x}^{2}}{2!}+\displaystyle \frac{{x}^{4}}{4!}-\ldots ,\end{eqnarray}$
we arrive at the family of solutions
$\begin{eqnarray*}q=a{{\rm{e}}}^{-{\rm{i}}\rho x}+\displaystyle \frac{4{\rm{i}}}{\sqrt{\rho }}{\left(\displaystyle \frac{{s}_{1}{\bar{s}}_{2}}{(1+{\rm{i}})| {s}_{1}{| }^{2}+(1-{\rm{i}})| {s}_{2}{| }^{2}}\right)}_{x},\end{eqnarray*}$
where
$\begin{eqnarray*}\left\{\begin{array}{l}{{ \mathcal R }}_{0}=\left(\begin{array}{cc}1+(1+{\rm{i}})\rho x/2 & (1+{\rm{i}})\rho x/2\\ -(1+{\rm{i}})\rho x/2 & 1-(1+{\rm{i}})\rho x/2\end{array}\right),\\ {{ \mathcal E }}_{0}=\left(\begin{array}{cc}1+({\rm{i}}-1){\rho }^{2}t/2 & ({\rm{i}}-1){\rho }^{2}t/2\\ (1-{\rm{i}}){\rho }^{2}t/2 & 1-({\rm{i}}-1){\rho }^{2}t/2\end{array}\right),\\ \left(\begin{array}{c}{s}_{1}\\ {s}_{2}\end{array}\right)={{\rm{e}}}^{-{\rm{i}}\rho x}{\rm{\Lambda }}{{ \mathcal R }}_{0}{{ \mathcal E }}_{0}{{ \mathcal Z }}_{0},\,\,{{ \mathcal Z }}_{0}=\left(\begin{array}{c}{\widetilde{z}}_{1}\\ {\widetilde{z}}_{2}\end{array}\right).\end{array}\right.\end{eqnarray*}$
This is a first-order rogue wave solution. Figure 2 is plotted for the rogue wave ∣q∣ for equation (8) using suitable parameters, it is localized both in time (t) and space (x), thus revealing the usual rogue wave features. In particular, in figures 1 and 2, we observe that a rogue wave can arise from the extreme behaviour of a breather wave.
Figure 2. First-order rogue wave obtained from equation (8) (∣q∣) for the parameters: $\rho =1,{\widetilde{z}}_{1}=1,{\widetilde{z}}_{2}=1$.

4. Example 2

If we choose the following matrix function:
$\begin{eqnarray*}{ \mathcal U }=({q}_{1},{q}_{2}),\end{eqnarray*}$
then system (1) can be reduced to the coupled derivative NLSE (CDNLSE)
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{i}}{q}_{1,t}+{q}_{2,{xx}}+{\rm{i}}{\left[\left(| {q}_{1}{| }^{2}+| {q}_{2}{| }^{2}\right){q}_{1}\right]}_{x}=0,\\ {\rm{i}}{q}_{2,t}+{q}_{2,{xx}}+{\rm{i}}{\left[\left(| {q}_{2}{| }^{2}+| {q}_{2}{| }^{2}\right){q}_{2}\right]}_{x}=0.\end{array}\right.\end{eqnarray}$
Equation (15) is important in plasma physics and the ultra-short pulse field.

4.1. Darboux transformation

Equation (15) is completely integrable; its Lax pairs are:
$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Psi }}}_{x}=U{\rm{\Psi }}=-{\rm{i}}{\lambda }^{2}{\sigma }_{2}+\lambda Q,\\ {{\rm{\Psi }}}_{t}=V{\rm{\Psi }}=-2{\rm{i}}{\lambda }^{4}{\sigma }_{2}+2{\lambda }^{3}Q-{\rm{i}}{\lambda }^{2}{Q}^{2}{\sigma }_{2}+\lambda {Q}^{3}-{\rm{i}}\lambda {Q}_{x}{\sigma }_{2},\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray*}{\sigma }_{2}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1\end{array}\right),\,\,Q=\left(\begin{array}{ccc}0 & {q}_{1} & {q}_{2}\\ -{\bar{q}}_{1} & 0 & 0\\ -{\bar{q}}_{2} & 0 & 0\end{array}\right).\end{eqnarray*}$
Here, equation (15) can be deduced directly from the equation UtVx + [U, V] = 0.
As in [55], we construct the DT of equation (15) as follows:
$\begin{eqnarray*}{\rm{\Psi }}[1]=T{\rm{\Psi }},\,\,T={M}_{0}{\lambda }^{2}+{M}_{1}\lambda -I.\end{eqnarray*}$
In addition, the transformation between potential functions is given by
$\begin{eqnarray}\left\{\begin{array}{l}{q}_{1,[1]}={q}_{1,[0]}-\displaystyle \frac{{\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}}{| {\lambda }_{1}{| }^{2}}{\left(\displaystyle \frac{\phi \bar{\psi }}{{\lambda }_{1}| \phi {| }^{2}+{\bar{\lambda }}_{1}\left(| \psi {| }^{2}+| \varphi {| }^{2}\right)}\right)}_{x},\\ {q}_{2,[1]}={q}_{2,[0]}-\displaystyle \frac{{\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}}{| {\lambda }_{1}{| }^{2}}{\left(\displaystyle \frac{\phi \bar{\varphi }}{{\lambda }_{1}| \phi {| }^{2}+{\bar{\lambda }}_{1}\left(| \psi {| }^{2}+| \varphi {| }^{2}\right)}\right)}_{x},\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray*}\begin{array}{l}I={\rm{diag}}(1,1,1),\\ {M}_{0}=\displaystyle \frac{1}{{\bar{\lambda }}_{1}}I+\displaystyle \frac{{\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}}{{\bar{\lambda }}_{1}{\lambda }_{1}}\left(\begin{array}{cc}\displaystyle \frac{\phi \bar{\phi }}{{\rm{\Delta }}} & 0\\ 0 & \displaystyle \frac{{{PP}}^{\dagger }}{\bar{{\rm{\Delta }}}}\end{array}\right),\\ {M}_{1}=\displaystyle \frac{{\bar{\lambda }}_{1}^{2}-{\lambda }_{1}^{2}}{{\bar{\lambda }}_{1}{\lambda }_{1}}\left(\begin{array}{cc}0 & \displaystyle \frac{\phi {{\rm{\nabla }}}^{\dagger }}{{\rm{\Delta }}}\\ \displaystyle \frac{{\rm{\nabla }}\bar{\phi }}{\bar{{\rm{\Delta }}}} & 0\end{array}\right),\\ {\rm{\Delta }}={\lambda }_{1}| \phi {| }^{2}+{\bar{\lambda }}_{1}\left(| \psi {| }^{2}+| \varphi {| }^{2}\right),\,\,{\rm{\nabla }}=\left(\begin{array}{c}\psi \\ \varphi \end{array}\right).\end{array}\end{eqnarray*}$
Here, $\Psi$ = (φ, ψ, φ) is the solution of the Lax system (16) under the condition that λ = λ1.

4.2. Exact breather solutions

It is easy to find that equation (15) admits the plane wave solution
$\begin{eqnarray}{q}_{1,[0]}={a}_{1}{{\rm{e}}}^{-{\rm{i}}{\sigma }^{2}x},\,\,{q}_{2,[0]}={a}_{2}{{\rm{e}}}^{-{\rm{i}}{\sigma }^{2}x},\,\,\sigma =\sqrt{{a}_{1}^{2}+{a}_{2}^{2}},\end{eqnarray}$
where a1, a2 are arbitrary parameters.
Similarly, the corresponding solutions of the Lax system (16) read
$\begin{eqnarray*}{\rm{\Psi }}=\left(\begin{array}{c}\phi \\ \psi \\ \varphi \end{array}\right)=\widetilde{{\rm{\Lambda }}}\widetilde{{R}}\widetilde{{E}}{ \mathcal Z },\,\,\,\,{ \mathcal Z }=\left(\begin{array}{c}{\mu }_{1}\\ {\mu }_{2}\\ {\mu }_{3}\end{array}\right),\end{eqnarray*}$
and
$\begin{eqnarray}\begin{array}{l}\widetilde{{R}}=\exp ({\rm{i}}\widetilde{{\rm{\Theta }}}x),\,\,\widetilde{{E}}=\exp ({\rm{i}}\widetilde{{\rm{\Omega }}}t),\\ \widetilde{{\rm{\Lambda }}}={\rm{diag}}\left(1,{{\rm{e}}}^{{\rm{i}}{\sigma }^{2}x},{{\rm{e}}}^{{\rm{i}}{\sigma }^{2}x}\right),\end{array}\end{eqnarray}$
where $\widetilde{{Z}}$ is an arbitrary complex vector. We then have
$\begin{eqnarray*}\begin{array}{l}\widetilde{{\rm{\Theta }}}=-{\lambda }^{2}{\sigma }_{2}-\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & {\sigma }^{2} & 0\\ 0 & 0 & {\sigma }^{2}\end{array}\right)\\ -{\rm{i}}\lambda \left(\begin{array}{ccc}0 & {a}_{1} & {a}_{2}\\ -{a}_{1} & 0 & 0\\ -{a}_{2} & 0 & 0\end{array}\right),\,\,\widetilde{{\rm{\Omega }}}=-2{\lambda }^{2}\widetilde{{\rm{\Theta }}}-{\sigma }^{2}{\lambda }^{2}.\end{array}\end{eqnarray*}$
The $\widetilde{{R}}$ in (19) can now be written as
$\begin{eqnarray*}\widetilde{{R}}=\displaystyle \frac{1}{2\widetilde{\tau }}\left(\begin{array}{ccc}{\psi }_{1} & {\psi }_{2} & {\psi }_{3}\\ {\widetilde{\psi }}_{2} & {\psi }_{4} & {\psi }_{5}\\ {\widetilde{\psi }}_{3} & {\psi }_{5} & {\psi }_{6}\end{array}\right){{\rm{e}}}^{-{\rm{i}}{\sigma }^{2}x/2},\end{eqnarray*}$
where
$\begin{eqnarray*}\left\{\begin{array}{l}{\psi }_{1}=\left({\sigma }^{2}-2{\lambda }^{2}\right)\mathrm{isin}\left(\widetilde{\tau }x\right)+2\widetilde{\tau }\cos \left(\widetilde{\tau }x\right),\\ {\psi }_{2}=-{\widetilde{\psi }}_{2}=2{a}_{1}\lambda \sin \left(\widetilde{\tau }x\right),\,\,2\widetilde{\tau }=\sqrt{{\sigma }^{4}+4{\lambda }^{4}},\\ {\psi }_{3}=-{\widetilde{\psi }}_{3}=2{a}_{2}\lambda \sin (\widetilde{\tau }x),\\ {\psi }_{4}={\sigma }^{-2}\left({a}_{1}^{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)\mathrm{isin}\left(\widetilde{\tau }x\right)+2\widetilde{\tau }{a}_{1}^{2}\cos \left(\widetilde{\tau }x\right)+2\widetilde{\tau }{a}_{2}^{2}{{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)x}\right),\\ {\psi }_{5}={a}_{1}{a}_{2}{\sigma }^{-2}\left(\left(2{\lambda }^{2}-{\sigma }^{2}\right)\mathrm{isin}\left(\widetilde{\tau }x\right)+2\widetilde{\tau }\cos \left(\widetilde{\tau }x\right)-2\widetilde{\tau }{{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)x}\right),\\ {\psi }_{6}={\sigma }^{-2}\left({a}_{2}^{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)\mathrm{isin}\left(\widetilde{\tau }x\right)+2\widetilde{\tau }{a}_{2}^{2}\cos \left(\widetilde{\tau }x\right)+2\widetilde{\tau }{a}_{1}^{2}{{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)x}\right).\end{array}\right.\end{eqnarray*}$
In addition, the $\widetilde{{E}}$ in (19) can be written as
$\begin{eqnarray*}\widetilde{{E}}=\displaystyle \frac{1}{2\widetilde{\tau }}\left(\begin{array}{ccc}{\phi }_{1} & {\phi }_{2} & {\phi }_{3}\\ {\widetilde{\phi }}_{2} & {\phi }_{4} & {\phi }_{5}\\ {\widetilde{\phi }}_{3} & {\phi }_{5} & {\phi }_{6}\end{array}\right),\end{eqnarray*}$
where
$\begin{eqnarray*}\left\{\begin{array}{l}{\phi }_{1}=\left({\sigma }^{2}-2{\lambda }^{2}\right)\mathrm{isin}\left(\widetilde{\xi }t\right)+2\widetilde{\tau }\cos \left(\widetilde{\xi }t\right),\\ {\phi }_{2}=-{\widetilde{\phi }}_{2}=2{a}_{1}\lambda \sin \left(\widetilde{\xi }t\right),\,\,\xi =2{\lambda }^{2}\widetilde{\tau },\\ {\phi }_{3}=-{\widetilde{\phi }}_{3}=2{a}_{2}\lambda \sin (\widetilde{\xi }t),\\ {\phi }_{4}={\sigma }^{-2}\left({a}_{1}^{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)\mathrm{isin}\left(\widetilde{\xi }t\right)+2\widetilde{\tau }{a}_{1}^{2}\cos \left(\widetilde{\xi }t\right)+2\widetilde{\tau }{a}_{2}^{2}{{\rm{e}}}^{2{\rm{i}}{\lambda }^{4}t}\right),\\ {\phi }_{5}={a}_{1}{a}_{2}{\sigma }^{-2}\left(\left(2{\lambda }^{2}-{\sigma }^{2}\right)\mathrm{isin}\left(\widetilde{\xi }t\right)+2\widetilde{\tau }\cos \left(\widetilde{\xi }t\right)-2\widetilde{\tau }{{\rm{e}}}^{2{\rm{i}}{\lambda }^{4}t}\right),\\ {\phi }_{6}={\sigma }^{-2}\left({a}_{2}^{2}\left(2{\lambda }^{2}-{\sigma }^{2}\right)\mathrm{isin}\left(\widetilde{\xi }t\right)+2\widetilde{\tau }{a}_{2}^{2}\cos \left(\widetilde{\xi }t\right)+2\widetilde{\tau }{a}_{1}^{2}{{\rm{e}}}^{2{\rm{i}}{\lambda }^{4}t}\right).\end{array}\right.\end{eqnarray*}$
Similarly, using the plane wave (18) as the seed solution in the DT (17), we can find a new rational solution for equation (15) that is made up of trigonometric functions and exponential functions. In fact, when $\widetilde{h}=\sigma (1+{\rm{i}})/2$, if $\widetilde{h}\in (-\infty ,-\sigma )\bigcup (\sigma ,+\infty )$, the solutions become temporally periodic. Besides, when $\widetilde{h}\in (-\sigma ,\sigma )$, the solutions become spatially periodic. As depicted in figures 35, we easily find that q1 and q2 have different structures. In the ∣q1∣ component, the temporally periodic breathers are displayed on a bright soliton background in figures 3(a)–3(c). On the other hand, in the q2 component, the solutions become AB waves (see figures 3(d)–3(f)). In figure 4, we can easily see that q1 and q2 have similar structures. In addition, the interaction phenomenon can change the propagation direction of the AB wave. As shown in figure 5, the propagation direction of the breather wave is perpendicular to that of the soliton. In this case, because the bright soliton rapidly decreases to zero, the spatially periodic breather is hardly identifiable in the component q1. In addition, in the component ∣q2∣, the spatially periodic breather is only observed in the region x > 0. Interestingly, in the q2 component, the bright soliton wave turns into a dark soliton wave in the course of its evolution.
Figure 3. Hybrid solution to equation (15) for the parameters: a1 = 0, a2 = 1, λ = 0.6(1 + i), μ1 = 1, μ2 = 1. (a) μ3 = 5, (b) μ3 = 25, (c) μ3 = 80.
Figure 4. Hybrid solution to equation (15) for the parameters: a1 = 1, a2 = −1, λ = 0.6(1 + i), μ1 = μ2 = μ3 = 1.
Figure 5. Hybrid solution to equation (15) for the parameters: a1 = 0, a2 = 1, 2λ = 0.9(1 + i), μ1 = μ2 = μ3 = 1.

4.3. Rogue wave solution

Similarly, by choosing the spectral parameter $\lambda =\tfrac{\sigma }{2}(1+{\rm{i}})$ and utilizing the Taylor expansion formulas (14), we obtain the first-order rogue wave solution of equation (15)
$\begin{eqnarray*}\begin{array}{l}\left(\begin{array}{c}{q}_{1}\\ {q}_{2}\end{array}\right)\,=\,\left(\begin{array}{c}{a}_{1}\\ {a}_{2}\end{array}\right){{\rm{e}}}^{-{\rm{i}}{\sigma }^{2}x}\\ +\displaystyle \frac{4{\rm{i}}}{\sigma }{\left(\displaystyle \frac{{s}_{1}}{(1+{\rm{i}})| {s}_{1}{| }^{2}+(1-{\rm{i}})\left(| {s}_{2}{| }^{2}+| {s}_{3}{| }^{2}\right)}\left(\begin{array}{c}{\bar{s}}_{2}\\ {\bar{s}}_{3}\end{array}\right)\right)}_{x},\end{array}\end{eqnarray*}$
where
$\begin{eqnarray*}\left(\begin{array}{c}{s}_{1}\\ {s}_{2}\\ {s}_{3}\end{array}\right)\,=\,{{\rm{e}}}^{-{\rm{i}}{\sigma }^{2}x/2}\widetilde{{\rm{\Lambda }}}{\widetilde{{R}}}_{0}{\widetilde{{E}}}_{0}{\widetilde{{Z}}}_{0},{\widetilde{{Z}}}_{0}=\left(\begin{array}{c}{\widetilde{\mu }}_{1}\\ {\widetilde{\mu }}_{2}\\ {\widetilde{\mu }}_{3}\end{array}\right),\end{eqnarray*}$
and
$\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Lambda }}}_{0} & = & \left(\begin{array}{ccc}{\sigma }^{2}(1+{\rm{i}})x/2+1 & {a}_{1}(1+{\rm{i}})\sigma x/2 & {a}_{2}(1+{\rm{i}})\sigma x/2\\ -{a}_{1}(1+{\rm{i}})\sigma x/2 & {\sigma }^{-2}\left({a}_{1}^{2}{{\rm{\Gamma }}}_{1}+{a}_{2}^{2}{{\rm{e}}}^{-{\sigma }^{2}(1+{\rm{i}})x/2}\right) & {a}_{1}{a}_{2}{\sigma }^{-2}\left({{\rm{\Gamma }}}_{1}-{{\rm{e}}}^{-{\sigma }^{2}(1+{\rm{i}})x/2}\right)\\ -{a}_{2}(1+{\rm{i}})\sigma x/2 & {a}_{1}{a}_{2}{\sigma }^{-2}\left({{\rm{\Gamma }}}_{1}-{{\rm{e}}}^{-{\sigma }^{2}(1+{\rm{i}})x/2}\right) & {\sigma }^{-2}\left({a}_{2}^{2}{{\rm{\Gamma }}}_{1}+{a}_{1}^{2}{{\rm{e}}}^{-{\sigma }^{2}(1+{\rm{i}})x/2}\right)\end{array}\right),\\ {{ \mathcal E }}_{0} & = & \left(\begin{array}{ccc}{\sigma }^{4}({\rm{i}}-1)t/2+1 & {a}_{1}{\sigma }^{3}({\rm{i}}-1)t/2 & {a}_{2}{\sigma }^{3}({\rm{i}}-1)t/2\\ {a}_{1}{\sigma }^{3}(1-{\rm{i}})t/2 & {\sigma }^{-2}\left({a}_{1}^{2}{{\rm{\Gamma }}}_{2}+{a}_{2}^{2}{{\rm{e}}}^{-{\rm{i}}{\sigma }^{4}t/2}\right) & {a}_{1}{a}_{2}{\sigma }^{-2}\left({{\rm{\Gamma }}}_{2}-{{\rm{e}}}^{-{\rm{i}}{\sigma }^{4}t/2}\right)\\ {a}_{2}{\sigma }^{3}(1-{\rm{i}})t/2 & {a}_{1}{a}_{2}{\sigma }^{-2}\left({{\rm{\Gamma }}}_{2}-{{\rm{e}}}^{-{\rm{i}}{\sigma }^{4}t/2}\right) & {\sigma }^{-2}\left({a}_{2}^{2}{{\rm{\Gamma }}}_{2}+{a}_{1}^{2}{{\rm{e}}}^{-{\rm{i}}{\sigma }^{4}t/2}\right)\end{array}\right),\\ {{\rm{\Gamma }}}_{1} & = & -(1+{\rm{i}}){\sigma }^{2}x/2+1,\,\,{{\rm{\Gamma }}}_{2}=(1-{\rm{i}}){\sigma }^{4}t/2+1.\end{array}\end{eqnarray*}$
If a1 = a2 = 1, μ1 = 0, μ2 = μ3 = 1, the first-order hybrid solution degenerates to an eye-shaped rogue wave in two components. Here, we omit these figures.
If the parameters μj ≠ 0, and the backgrounds are all non-vanishing, we see that a first-order rogue wave interacts with a breather in two components (see figure 8). By choosing different parameters for aj and μj, we can obtain various arrangements of the two components, such as a first-order rogue wave and an amplitude-varying soliton, a first-order rogue wave, and a breather, etc. Here, we discuss the dynamics of the first-order hybrid solution.
Figures 6(a)–(c) demonstrate that a hybrid solution between a rogue wave and a soliton exists in the q1 component. In particular, if the value of μ3 in the q1 component is increased, the first-order rogue wave cannot be easily observed, since it appears from the zero plane background. When the value μ1 is changed, it reveals that a hybrid solution between a first-order rogue wave and an amplitude-varying soliton exists in the q2 component, and a hybrid solution between a first-order rogue wave and a bright soliton exists in the q1 component shown in figure 7. Figures 7(d)–(f) show that the soliton in the q2 component is an anti-dark soliton if t < 0 and becomes a dark soliton if t > 0. In particular, this kind of amplitude-varying soliton is annihilated when t = 0. If μ1 becomes larger, the distance between a first-order rogue wave and a soliton increases. When the value of a1 selected in figure 8 is one, instead of zero in figures 6 and 7, the first-order rogue wave can merge with a breather. In figure 8, we observe that a first-order rogue wave appears with a breather in two components. In addition, we can see that μ1 can control the distance between a first-order rogue wave and a breather. In particular, as a1 → 0, the breather wave yields a bright soliton in the q1 component (see figures 6(a) and 8(a)) and the breather wave disappears in the q2 component (see figures 6(d) and 8(d)).
Figure 6. Hybrid solution of equation (15) for the parameters: a1 = 0, a2 = 1, μ1 = 1, μ2 = 1. (a), (d) μ3 = 2, (b), (e) μ3 = 10, (c), (f) μ3 = 50.
Figure 7. Hybrid solution of equation (15) for the parameters: a1 = 0, a2 = 1, μ2 = 1, μ3 = 1. (a), (d) μ1 = 2, (b), (e) μ1 = 10, (c), (f) μ1 = 40.
Figure 8. Hybrid solution of equation (15) for the parameters: a1 = 1, a2 = 1, μ2 = 1, μ3 = 1. (a), (d) μ1 = 1, (b), (e) μ1 = 50, (c), (f) μ1 = 100.
If we do not consider the different arrangements of the two components, the interactions of the localized waves in the coupled system (15) can be completely classified into three types. Our results provide evidence of some obvious interactions between rogue waves and solitons or breathers. To the best of our knowledge, these types of dynamic patterns in the coupled system (15) have never emerged in the corresponding uncoupled systems. In this section, we generalized Baronio’s results [60] to obtain these kinds of mixed interactions of localized waves. In addition, we constructed mixed interactions of localized waves in the two-component system (15) through a variable separation technique. However, these mixed interactions cannot be obtained in single-component systems using the variable separation technique. Therefore, a conclusion can be drawn that these kinds of mixed interactions of localized waves can only be obtained by the variable separation technique in nonlinear systems with more than two components with the corresponding Lax pair including matrices larger than 2 × 2.

5. Example 3

If the matrix function ${ \mathcal U }$ is chosen to be a symmetric matrix
$\begin{eqnarray}{ \mathcal U }(x,t)=\left(\begin{array}{cc}{q}_{1} & {q}_{0}\\ {q}_{0} & {q}_{-1}\end{array}\right),\end{eqnarray}$
the above system (1) becomes the Hermitian symmetric space derivative NLSE (HSS-DNLSE) [5659]
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{i}}{q}_{1,t}+{q}_{1,{xx}}+{\rm{i}}{\left[\left(| {q}_{1}{| }^{2}+2| {q}_{0}{| }^{2}\right){q}_{1}\right]}_{x}+{\rm{i}}{\left({q}_{0}^{2}{\bar{q}}_{-1}\right)}_{x}=0,\\ {\rm{i}}{q}_{-1,t}+{q}_{-1,{xx}}+{\rm{i}}{\left[\left(2| {q}_{0}{| }^{2}+| {q}_{-1}{| }^{2}\right){q}_{-1}\right]}_{x}+{\rm{i}}{\left({q}_{0}^{2}{\bar{q}}_{1}\right)}_{x}=0,\\ {\rm{i}}{q}_{0,t}+{q}_{0,{xx}}+{\rm{i}}{\left[\left(| {q}_{1}{| }^{2}+| {q}_{0}{| }^{2}+| {q}_{-1}{| }^{2}\right){q}_{0}\right]}_{x}+{\rm{i}}{\left({q}_{1}{\bar{q}}_{0}{q}_{-1}\right)}_{x}=0,\end{array}\right.\end{eqnarray}$
where the potentials q±, q0 are three complex-valued functions. The Lax pair of equations (21) can be expressed as
$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Psi }}}_{x}=U{\rm{\Psi }}=\left(-{\rm{i}}{\lambda }^{2}{\sigma }_{3}+\lambda {U}_{1}\right){\rm{\Psi }},\\ {{\rm{\Psi }}}_{t}=V{\rm{\Psi }}=\left(-2{\rm{i}}{\lambda }^{4}{\sigma }_{3}+2{\lambda }^{3}{U}_{1}-{\rm{i}}{\lambda }^{2}{U}_{1}^{2}{\sigma }_{3}+\lambda {U}_{1}^{3}-{\rm{i}}\lambda {U}_{1,x}{\sigma }_{3}\right){\rm{\Psi }},\end{array}\right.\end{eqnarray}$
where
$\begin{eqnarray*}{\sigma }_{3}=\left(\begin{array}{cc}I & 0\\ 0 & -I\end{array}\right),\,\,{U}_{1}=\left(\begin{array}{cc}0 & { \mathcal U }\\ -{{ \mathcal U }}^{\dagger } & 0\end{array}\right),\end{eqnarray*}$
λ is a constant spectral parameter, and I is a 2 × 2 identity matrix.

5.1. Breather wave solution

It can easily be found that equation (20) has the plane wave solution
$\begin{eqnarray}\begin{array}{l}{ \mathcal U }={\mathbb{A}}{{\rm{e}}}^{-{\rm{i}}{\epsilon }^{2}x},\,\,\epsilon =\sqrt{{c}_{1}^{2}+{c}_{2}^{2}},\\ {\mathbb{A}}=\left(\begin{array}{cc}{c}_{1} & {c}_{2}\\ {c}_{2} & -{c}_{1}\end{array}\right),\end{array}\end{eqnarray}$
where c1, c2 are real numbers.
In a similar way, the corresponding solution of the Lax system (22) reaches
$\begin{eqnarray*}\begin{array}{l}{\rm{\Psi }}=\left(\begin{array}{c}\phi \\ \psi \\ \varphi \\ \chi \end{array}\right)\,=\,\check{{\rm{\Lambda }}}\check{{R}}\check{{E}}{\mathscr{Z}},\\ {\mathscr{Z}}=\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right),\end{array}\end{eqnarray*}$
and
$\begin{eqnarray}\begin{array}{l}\check{{R}}=\exp ({\rm{i}}\check{{\rm{\Theta }}}t),\,\,\check{{E}}=\exp ({\rm{i}}\check{{\rm{\Omega }}}t),\\ \check{{\rm{\Lambda }}}={\rm{diag}}\left(1,1,{{\rm{e}}}^{{\rm{i}}{\epsilon }^{2}x},{{\rm{e}}}^{{\rm{i}}{\epsilon }^{2}x}\right),\end{array}\end{eqnarray}$
where ${\mathscr{Z}}$ is an arbitrary complex vector. In addition, the matrix functions $\check{{\rm{\Theta }}}$, $\check{{\rm{\Omega }}}$ can be obtained as follows:
$\begin{eqnarray*}\begin{array}{l}\check{{\rm{\Theta }}}=\left(\begin{array}{cc}{\lambda }^{2}I & {\rm{i}}\lambda {\mathbb{A}}\\ -{\rm{i}}\lambda {\mathbb{A}} & \left({\epsilon }^{2}-{\lambda }^{2}\right)I\end{array}\right),\\ \check{{\rm{\Omega }}}={\lambda }^{2}\left(2\hat{{\rm{\Theta }}}-{\hat{a}}^{2}\right).\end{array}\end{eqnarray*}$
The exponential matrix $\check{{R}}$ in (24) can now be written as
$\begin{eqnarray*}\check{{R}}\,=\,\displaystyle \frac{1}{2\kappa }\left(\begin{array}{cc}{\kappa }_{1}I & \lambda {\mathbb{A}}{\kappa }_{2}\\ -\lambda {\mathbb{A}}{\kappa }_{2} & {\kappa }_{3}I\end{array}\right){{\rm{e}}}^{-{\rm{i}}{\epsilon }^{2}x/2},\end{eqnarray*}$
where
$\begin{eqnarray*}\left\{\begin{array}{l}{\kappa }_{1}=\left({\epsilon }^{2}-2{\lambda }^{2}\right){\rm{i}}\,\sin (\kappa x)+2\kappa \cos (\kappa x),\\ {\kappa }_{2}=2\sin (\kappa x),\,\,2\kappa =\sqrt{{\epsilon }^{4}+4{\lambda }^{4}},\\ {\kappa }_{3}=\left(2{\lambda }^{2}-{\epsilon }^{2}\right){\rm{i}}\,\sin (\kappa x)+2\kappa \cos (\kappa x).\end{array}\right.\end{eqnarray*}$
We can also write the exponential matrix $\check{{E}}$ in (24) as
$\begin{eqnarray*}\check{{E}}\,=\,\displaystyle \frac{1}{2\kappa }\left(\begin{array}{cc}{\gamma }_{1}I & \lambda {\mathbb{A}}{\gamma }_{2}\\ -\lambda {\mathbb{A}}{\gamma }_{2} & {\gamma }_{3}I\end{array}\right),\end{eqnarray*}$
where
$\begin{eqnarray*}\left\{\begin{array}{l}{\gamma }_{1}=\left({\epsilon }^{2}-2{\lambda }^{2}\right){\rm{i}}\,\sin (\gamma t)+2\kappa \cos (\gamma t),\\ {\gamma }_{2}=2\sin (\gamma t),\,\,\gamma =2{\lambda }^{2}\kappa ,\\ {\gamma }_{3}=\left(2{\lambda }^{2}-{\epsilon }^{2}\right){\rm{i}}\,\sin (\gamma t)+2\kappa \cos (\gamma t).\end{array}\right.\end{eqnarray*}$
In the following, utilizing the plane wave (23) as the seed solution in the following DT [59]:
$\begin{eqnarray*}\begin{array}{l}\left(\begin{array}{c}{q}_{1}\\ {q}_{0}\\ {q}_{-1}\end{array}\right)\,=\,\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\\ -{c}_{1}\end{array}\right){{\rm{e}}}^{-{\rm{i}}{\epsilon }^{2}x}-\displaystyle \frac{{\bar{\lambda }}^{2}-{\lambda }^{2}}{| \lambda {| }^{2}}\\ \times \,{\left(\displaystyle \frac{\phi }{\lambda \left(| \phi {| }^{2}+| \psi {| }^{2}\right)+\bar{\lambda }\left(| \varphi {| }^{2}+| \chi {| }^{2}\right)}\left(\begin{array}{c}\bar{\psi }\\ \bar{\varphi }\\ \bar{\chi }\end{array}\right)\right)}_{x},\end{array}\end{eqnarray*}$
we can obtain a new breather solution for equation (20). Figures 9 and 10 reveal the dynamics of temporally periodic (also called AB) and spatially periodic breathers (also called KM breathers), respectively.
Figure 9. AB wave in equation (20) for the parameters: c1 = 0, c2 = 1, z1 = z2 = z3 = z4 = 1, 2λ = 1.1(1 + i).
Figure 10. KM breathers in equation (20) for the parameters: c1 = 0, c2 = 1, z1 = z2 = z3 = z4 = 1, 2λ = 0.9(1 + i).

5.2. Rogue wave solution

In what follows, by choosing the spectral parameter $\lambda =\tfrac{\epsilon }{2}(1+{\rm{i}})$ and using the formulas (14), we can obtain the first-order rogue wave solution of equation (20)
$\begin{eqnarray*}\begin{array}{l}\left(\begin{array}{c}{q}_{1}\\ {q}_{0}\\ {q}_{-1}\end{array}\right)=\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\\ -{c}_{1}\end{array}\right){{\rm{e}}}^{-{\rm{i}}{\epsilon }^{2}x}+\displaystyle \frac{4{\rm{i}}}{\epsilon }\\ \times \,{\left(\displaystyle \frac{r}{(1+{\rm{i}})\left(| r{| }^{2}+| {s}_{1}{| }^{2}\right)+(1-{\rm{i}})\left(| {s}_{2}{| }^{2}+| {s}_{3}{| }^{2}\right)}\left(\begin{array}{c}{\bar{s}}_{1}\\ {\bar{s}}_{2}\\ {\bar{s}}_{3}\end{array}\right)\right)}_{x},\end{array}\end{eqnarray*}$
where
$\begin{eqnarray*}\begin{array}{l}\left(\begin{array}{c}r\\ {s}_{1}\\ {s}_{2}\\ {s}_{3}\end{array}\right)={{\rm{e}}}^{-{\rm{i}}{\epsilon }^{2}x/2}\check{{\rm{\Lambda }}}{\check{{R}}}_{0}{\check{{E}}}_{0}{{\mathscr{Z}}}_{0},\,\,{{\mathscr{Z}}}_{0}=\left(\begin{array}{c}{\nu }_{1}\\ {\nu }_{2}\\ {\nu }_{3}\\ {\nu }_{4}\end{array}\right),\\ {\check{{R}}}_{0}=\left(\begin{array}{cc}\left((1+{\rm{i}}){\epsilon }^{2}x/2+1\right)I & (1+{\rm{i}})\epsilon x/2{\mathbb{A}}\\ -(1+{\rm{i}})\epsilon x/2{\mathbb{A}} & \left(-(1+{\rm{i}}){\epsilon }^{2}x/2+1\right)I\end{array}\right),\\ {\check{{E}}}_{0}=\left(\begin{array}{cc}\left(({\rm{i}}-1){\epsilon }^{4}t/2+1\right)I & ({\rm{i}}-1){\epsilon }^{3}/2{\mathbb{A}}\\ -({\rm{i}}-1){\epsilon }^{3}/2{\mathbb{A}} & \left((1-{\rm{i}}){\epsilon }^{4}t/2+1\right)I\end{array}\right).\end{array}\end{eqnarray*}$
In what follows, we will display the propagation characteristics of the novel rogue wave using two images. Figures 1113 present the dynamics of the novel rogue wave; this is made possible by the selection of suitable parameters which are helpful for enriching the dynamical behaviors of the nonlinear wave solutions. Interestingly, a wave with three peaks emerges without valleys in the q1 component; in the q0 component, a wave with a peak emerges without valleys. In particular, a wave with two peaks and two valleys emerges (also called a four-petaled rogue wave) in the q−1 component. To the best of the authors’ knowledge, the same dynamic patterns have not emerged from the scalar NLS equation and the Hirota equation to date.
Figure 11. Rogue waves obtained from equation (20) for the parameters: c1 = 0, c2 = 1, ν1 = ν2 = ν3 = ν4 = 1.
Figure 12. Density plot of figure 12.
Figure 13. The same parameters as those used in figure 12, except for c1 = 1.

6. Conclusions

In this work, we have shown how to construct breathers, rogue waves, and mixed interactions in the derivative NLSEs using the DT combined with an asymptotic expansion. These obtained solutions can be explicitly expressed in a ‘separation of variables’ form. These solutions exhibit a range of interesting and complicated dynamics, discussed by varying the available parameters. These include the AB, KM breathers, the Peregrine soliton, breathers and rogue waves that interact with dark and bright solitons, a first-order rogue wave that interacts with a breather, a three-peaked rogue wave without valleys, a one-peaked rogue wave without valleys, a four-petaled rogue wave, etc. These new spatiotemporal patterns reveal the potential rich dynamics in rogue wave solutions. Our results show that multicomponent coupled systems admit more abundant dynamical behaviors than the scalar case, which further helps us to explore different dynamics in related fields, such as Bose–Einstein condensates, optical fibers, superfluids, etc. Reference [60] presents the experimental conditions used to observe the mixed interactions of localized waves in vector NLSEs. Therefore, we expect that the new spatiotemporal patterns obtained in this work will be verified and observed in physical experiments in the near future.
Some scholars have studied the breathers and rogue wave solutions of the derivative NLSEs using DT formation [55, 59]. In contrast to their work, we obtained these solutions for the derivative NLSEs by adopting a variable separation and Taylor expansion technique. Through comparison, we find that the differences between these works are mainly reflected by two aspects: (i) the other papers that used DT to study the derivative NLSEs did not consider a variable separation technique, as we did; (ii) our work contains many new phenomena that are different from those of the other papers that using DT; these phenomena greatly enrich the properties of the derivative NLSEs.

We would like to thank the editor and referees for the valuable suggestions and comments that improved the manuscript.

1
Akhmediev N Ankiewicz A Taki M 2009 Phys. Lett. A 373 675 678

DOI

2
Draper L 1964 Oceanus 10 13 15

3
Müller P Garrett C Osborne A 2005 Oceanography 18 66 75

DOI

4
Solli D R Ropers C Koonath P Jalali B 2007 Nature 450 1054 1057

DOI

5
Chen S Baronio F Soto-Crespo J M Grelu P Mihalache D 2017 J. Phys. A: Math. Theor. 50 463001

DOI

6
Bludov Y V Konotop V V Akhmediev N 2009 Phys. Rev. A 80 033610

DOI

7
Moslem W M Shukla P K Eliasson B 2011 Europhys. Lett. 96 25002

DOI

8
Onorato M Residori S Bortolozzo U Montina A Arecchi F T 2013 Phys. Rep. 528 47 89

DOI

9
Mihalache D 2021 Rom. Rep. Phys. 73 403

10
Yan Z Y 2010 Commun. Theor. Phys. 54 947 949

DOI

11
Zakharov V E Shabat A B 1972 Sov. Phys. JETP 34 62 69

12
Peregrine D H 1983 J. Austral. Math. Soc. B 25 16 43

DOI

13
Akhmediev N Eleonskii V M Kulagin N E 1987 Theor. Math. Phys. 72 809 818

DOI

14
Kuznetsov E 1977 Sov. Phys. Dokl. 22 507 508

15
Ma Y C 1979 Stud. Appl. Math. 60 43 58

DOI

16
Vanderhaegen G Naveau C Szriftgiser P Kudlinski A Conforti M Mussot A Onorato M Trillo S Chabchoub A Akhmediev N 2021 PNAS 118 e2019348118

DOI

17
Akhmediev N Ankiewicz A 1993 Phys. Rev. A 47 3213 3221

DOI

18
Mihalache D Lederer F Baboiu D M 1993 Phys. Rev. A 47 3285 3290

DOI

19
Mihalache D Panoiu N C 1993 J. Phys. A: Mat. Gen. 26 2679 2697

DOI

20
Gagnon L 1993 J. Opt. Soc. Am. B 10 469 474

DOI

21
Grimshaw R Tovbis A 2013 Proc. Royal Soc. 469 20130094

22
Chabchoub A Hoffmann N P Akhmediev N 2011 Phys. Rev. Lett. 106 204502

DOI

23
Kibler B Fatome J Finot C Millot G Dias F Genty G Akhmediev N Dudley J M 2010 Nat. Phys. 6 790 795

DOI

24
Kibler B Fatome J Finot C Millot G Genty G Wetzel B Akhmediev N Dias F Dudley J M 2012 Sci. Rep. 2 463

DOI

25
Guo B Ling L Liu Q P 2012 Phys. Rev. E 85 026607

DOI

26
Yan Z 2011 Phys. Lett. A 375 4274 4279

DOI

27
Ma W X Zhou Y 2018 J. Differ. Equ. 264 2633 2659

DOI

28
Mu G Qin Z Grimshaw R 2015 SIAM J. Appl. Math. 75 1 20

DOI

29
Dai C Q Zhou G Q Zhang J F 2012 Phys. Rev. E 85 016603

DOI

30
Wang X B Tian S F Zhang T T 2018 Proc. Amer. Math. Soc. 146 3353 3365

DOI

31
Wang X B Han B 2020 J. Phys. Soc. Jpn. 89 014001

DOI

32
Akhmediev N Ankiewicz A Soto-Crespo J M 2009 Phys. Rev. E 80 026601

DOI

33
He J S Tao Y S Porsezian K Fokas A S 2013 J. Nonlinear Math. Phys. 20 407 419

DOI

34
Dong M J Tian L X 2021 Commun. Theor. Phys. 73 025001

DOI

35
Wang X B Han B 2019 Commun. Theor. Phys. 71 152 160

DOI

36
Xu T Chen Y 2018 Nonlinear Dyn. 92 2133 2142

DOI

37
Sun W R Wang L 2018 Proc. R. Soc. A 474 20170276

DOI

38
Ma W X 2021 Commun. Theor. Phys. 73 065001

DOI

39
Liu X Zhou Q Biswas A Alzahrani A K Liu W 2020 J. Adv. Research 24 167 173

DOI

40
Dai C Q Wang Y Y Zhang J F 2020 Nonlinear Dyn. 102 379 391

DOI

41
Han H B Li H J Dai C Q 2021 Appl. Math. Lett. 120 107302

DOI

42
Dai C Wu G Li H J Wang Y Y 2021 Fractals 29 2150192

43
Du Z Tian B Qu Q X Chai H P Zhao X H 2020 Chaos Solitons & Fractals 130 109403

DOI

44
Fei J Cao W 2020 Waves Random Complex Medium 30 54 64

DOI

45
Zhao L C Li S C Ling L M 2014 Phys. Rev. E 89 023210

DOI

46
Nakata I 1991 J. Phys. Soc. Jpn. 60 3976 3977

DOI

47
Daniel M Veerakumar V 2002 Phys. Lett. A 302 77 86

DOI

48
Nakata I Ono H Yosida M 1993 Prog. Theor. Phys. 90 739 742

DOI

49
MjØlhus E 1976 Plasma Phys 16 321 334

DOI

50
Mio K Ogino T Minam K Taketa S 1976 J. Phys. Soc. Jpn 41 265 271

DOI

51
Fedun V Ruderman M S Erdélyi R 2008 Phys. Lett. A 372 6107 6110

DOI

52
Ruderman M S 2002 J. Plasma Phys. 67 271 276

DOI

53
Tzoar N Jain M 1981 Phys. Rev. A 23 1266 1270

DOI

54
Anderson D Lisak M 1983 Phys. Rev. A 27 1393 1398

DOI

55
Ling L M Liu Q P 2010 J. Phys. A Math. Theor. 43 434023

DOI

56
Fordy A P 1984 J. Phys. A 17 1235 1245

DOI

57
Saleem U Hassan M U 2017 J. Phys. Soc. Jpn. 86 064002

DOI

58
Xu Y Zhou R G 2013 Appl. Math. Comput. 219 4551 4559

59
Shen J Geng X Xue B 2019 Commun. Nonlinear Sci. Numer. Simulat 78 104877

DOI

60
Baronio F Degasperis A Conforti M Wabnitz S 2012 Phys. Rev. Lett. 109 044102

DOI

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