## Abstract

To show the existence and properties of matter rogue waves in an *F*=1 spinor Bose–Einstein condensate (BEC), we work on the three-component Gross–Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright–dark–bright and bright–bright–bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an *F*=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.

## 1. Introduction

Bose–Einstein condensates (BECs) have been demonstrated for weakly interacting, dilute gases of alkali metal and hydrogen atoms [1–5]. Magnetic traps have been used to confine the condensates, but they have the drawback that spin flips in the atoms lead to untrapped states. For that reason, the spin orientation of the trapped alkali atoms cannot be regarded as a degree of freedom [1–3].

As a sodium atom BEC has been confined in an optical dipole trap, the restrictions of the magnetic traps can be eliminated [4,5]. In the condensate held by an optical potential, the spin of atoms has been proved to be free [4–7]. Laboratory experiments and theoretical approaches have been used to study the properties of spinor BECs [4–9].

The assembly of atoms in the *F*=1 state is characterized by a vectorial-order parameter: *Φ*(*x*,*t*)=[*Φ*_{+1}(*x*,*t*),*Φ*_{0}(*x*,*t*),*Φ*_{−1}(*x*,*t*)]^{T} with the components corresponding to the three values of the vertical spin projection, *m*_{F}=1,0,−1 [6,9–13]. The dynamics of the spinor condensates can be described by the multi-component Gross–Pitaevskii (GP) equations within the mean field approximation [6,9–13]
*a*and
*b*where *Φ*_{j}(*j*=+1,0,−1) are the functions of the space *x* and time *t*, *m* is the atomic mass, asterisk denotes the complex conjugate, i is the imaginary unit, the subscripts, respectively, denote the partial derivatives with respect to *x* and *t*, *c*_{0} and *c*_{2} stand for the mean field and spin-exchange collisions, respectively, given by *c*_{0}=(*g*_{0}+2*g*_{2})/3 and *c*_{2}=(*g*_{2}−*g*_{0})/3, with *g*_{0} and *g*_{2} being coupling constants [14,15],
*a*_{f} is the *s*-wave scattering lengths in the total hyperfine spin *f* channel, *a*_{⊥} is the size of the transverse ground state, *ζ* is the Riemann zeta-function [6,9,10,14,15].

In this work, we will consider the system with the coupling constants *c*_{0}=*c*_{2}=−*c*<0 [9–13]. This situation corresponds to attractive mean field interaction and ferromagnetic spin-exchange interaction [9–13]. Introducing the dimensionless form, *a*
*b*
*c*where *φ*_{j}(*j*=+1,0,−1) are the functions of space *x* and time *t*. We note that equation (1.3) represents a (1+1)-dimensional system. Kawaguchi & Ueda [9] discussed the physics of spinor BECs. The soliton solutions and modulational instability (MI) of equation (1.3) have been obtained with the Darboux transformation (DT) [10]. With the inverse scattering method, the interactions between two-soliton solutions of equation (1.3) have been discussed [11]. Bright–dark soliton complexes in spinor BECs have been studied in [12]. Some types of rational solutions have been obtained with the DT [13].

Oceanic rogue waves are extreme events that appear with low probability but have a dramatic impact [16–25]. The notion of a rogue wave has moved to the physics of BECs [26–29]. Laboratory experiments and theoretical approaches have been performed to study rogue matter waves [26–29]. In BECs, the management of the Feshbach resonance for nonlinearity and a tunable atomic trapping potential also provide us with a tool for manipulating rogue waves [26–29]. One type of mathematical description of a rogue wave is provided by the Peregrine soliton [30–38]. Peregrine solitons have been experimentally observed in an optical system [35]. The Darboux-dressing transformation (DDT) has been used to search for all bounded rational solutions [39–43]. The DDT was introduced in [39–43].

The mechanism of a rogue wave is still an open question [44–47]. MI can be viewed as a mechanism that can lead to rogue wave formation. MI, also known as the Benjamin–Feir instability, has been studied in the water wave context [48,49]. It describes the exponential growth of an initially sinusoidal long-wave perturbation of a plane wave solution [44,48,49]. Recent studies have shown that not every kind of MI leads to rogue wave formation [50,51]. Baseband MI has been defined as the condition where the MI gain band contains the zero-frequency perturbation as a limiting case, while the passband MI has been defined as the condition where the MI gain band does not contain the zero-frequency perturbation as a limiting case [50,51]. Baronio *et al.* [50,51] have shown that the existence condition of rogue wave solutions in different nonlinear wave models, such as the defocusing vector nonlinear Schrödinger equation, Fokas–Lenells equation and long-wave–short-wave resonance equation, coincides with the condition of baseband MI.

In this paper, we will concentrate on the generalized vector rational solutions of equation (1.3) and search the conditions for the existence of rogue wave solutions of equation (1.3) based on the DDT. The link between the existence of rogue wave solutions and the presence of baseband MI will also be discussed. We note that such generalized solutions can generate more rogue wave solutions. In §2, using the DDT, we will derive the vector rational solutions of equation (1.3) for the purpose of modelling extreme events. In §3, for certain special parameter values, we will give examples of such solutions. The link between the existence of rogue wave solutions and the presence of baseband MI will be discussed, and in §4 we will present our conclusions.

## 2. Darboux-dressing transformation-based algorithm and vector rational solutions of equation (1.3)

The Lax pair of equation (1.3) can be expressed as [10,11]
*Ψ* is a 4×4 square matrix that depends on the variables *x* and *t*, the subscripts, respectively, denote the partial derivatives with respect to *x* and *t*, *λ* is the complex spectral parameter of Lax pair (2.1), **I**_{2×2} is the 2×2 unit matrix, **O** is the 2×2 zero matrix and † denotes the conjugate transpose. We note that *Ψ* could be a 4×4 square matrix or a 4×2 matrix. Here, to construct the DDT, we choose that the *Ψ* is a 4×4 square matrix. The *Ψ* has been chosen as a 4×2 matrix for the DT.

Hereby, we introduce the steps towards the vector rogue wave solutions from the plane wave solutions. The DDT for equation (1.3) can be expressed as [10]
*Ψ* is mapped into another solution of Lax pair (2.1) *Ψ*[1], *Q*_{1} is mapped into *Q*_{1}[1], which is another solution of equation (1.3), *Ψ*_{111},*Ψ*_{113},*Ψ*_{211},*Ψ*_{213})^{T}=*ΨZ*_{0}, *Z*_{0} is a 4×1 matrix and can be expressed as (*Z*_{11}, *Z*_{21}, *Z*_{31}, *Z*_{41})^{T}, the superscript T denotes the matrix transpose and *Z*_{s1}(*s*=1,2,3,4) are arbitrary complex numbers.

To obtain the rogue wave solutions of equation (1.3), we start with the seed solutions of equation (1.3) as
*k*, *a*_{1} and *a*_{2} are real numbers. Then, we can get the corresponding *U*_{1} in Lax pair (2.1) as
*F* can be expressed as

Motivated by expression (2.4), we set
*Φ* is a 4×4 square matrix that depends on the variables *x* and *t*,
*Ψ* and (*Ψ*_{1,1},*Ψ*_{2,1})^{T} can be written as

Expressions (2.8) and (2.9) show that, if the two matrices *A*(*λ*) and *B*(*λ*) are not diagonalizable but are similar to a Jordan form, the matrix (*Ψ*_{1,1},*Ψ*_{2,1})^{T} is a linear combination of the polynomial functions [39–42]. Our aim is to find those particular values of *λ*. Solving the characteristic polynomial of *A*(*λ*), i.e. *P*(*Λ*)=det[*Λ*−*A*(*λ*)], we obtain the four roots of the characteristic polynomial *P*(*Λ*)

(1) *The case* *Λ*_{1}=*Λ*_{2}=*Λ*_{3}=*Λ*_{4}: If the four roots of the characteristic polynomial are the same, we obtain
*A*(*λ*)] means the trace of *A*(*λ*).

In the case of *Λ*−*A*(*λ*)] yields the elementary divisors

### Proposition 2.1

*If the form of elementary divisors satisfies expression (2.11), then A(λ) is similar to a Jordan form A*_{J}*,
**where P=[P*_{1}*,P*_{2}*,P*_{3}*,P*_{4}*] denotes the similarity transformation matrix with each P*_{m} *(m*=1,2,3,4) *being a* 1×4 *vector, and the Jordan form A*_{λ} *can be expressed as
*

(2) *The case* *Λ*_{1}=*Λ*_{2}≠*Λ*_{3}=*Λ*_{4}: If *Λ*_{1}=*Λ*_{2}≠*Λ*_{3}=*Λ*_{4}, using the element transform of determinant det[*Λ*−*A*(*λ*)] yields the elementary divisors
*A*(*λ*) is similar to a diagonal form of the similarity transformation matrix. So, in case 2, the matrix (*Ψ*_{1,1},*Ψ*_{2,1})^{T} is a linear combination of the exponential functions, which implies that solutions cannot be rational.

Based on the above analysis, we know that the rogue wave solutions of equation (1.3) can be obtained if *k*=0, *a*_{1}=1 and *a*_{2}=1. Then, we will obtain the rogue wave solutions of equation (1.3).

Based on expression (2.9), we can get *Ψ*_{1,1} and *Ψ*_{2,1}. As mentioned above, *Ψ*_{1,1} and *Ψ*_{2,1} are the 2×2 matrices and can be expressed as
*G*_{111}, *F*_{111}, *G*_{113}, *F*_{113}, *G*_{211}, *F*_{211}, *G*_{213} and *F*_{213} are given in appendix A.

With DDT (2.2) and expression (2.9), the rogue wave solutions of equation (1.3) can be expressed as
*a*
*b*
*c*where [*S*_{12}]_{ij}’s (*i*,*j*=1,2) represent the *S*_{12} entry in row *i* and column *j*, *Ψ*_{1,1} and *Ψ*_{2,1} have been given in expression (2.15). Detailed expressions of solution (2.16) are provided in appendix B (see also the electronic supplementary material). All analytical results can be verified by the Mathematica software.

## 3. Discussion

In §2, we obtained the vector rational solutions and conditions for the existence of rogue wave solutions of equation (1.3). In this section, we will show the localized nonlinear structure with multiple peaks localized in *x* and *t*.

Figures 1 and 2 show that, in the *φ*_{+1} and *φ*_{−1} components, the bright rogue waves with two humps appear, while in the *φ*_{0} component the dark rogue wave with four valleys appears. By decreasing the value of *Z*_{31}, we can see that the two peaks of the rogue waves merge in the *φ*_{+1} and *φ*_{−1} components, while the valleys of the dark rogue wave merge in the *φ*_{0} component, as shown in figures 3 and 4.

In figures 5 and 6, we show the bright vector rogue waves with *Z*_{11}=1, *Z*_{21}=1, *Z*_{31}=4 and *Z*_{41}=1. With the decrease in the value of *Z*_{31} in each component, these rogue waves have different structures: in the *φ*_{+1} and *φ*_{−1} components, the bright rogue waves are still bright rogue waves, while in the *φ*_{0} component a bright rogue wave with one hump in the course of evolution turns into a dark rogue wave with three valleys (figures 7*b* and 8*b*). In figure 9, we find that, in the *φ*_{+1} and *φ*_{−1} components, the bright rogue waves appear, while in the *φ*_{0} component the dark rogue wave with two valleys appears. Interestingly, the bright rogue wave in the *φ*_{−1} component has no valley, as shown in figures 9*c* and 10*c*. As shown in figure 7, we find that the rogue waves in the *φ*_{+1} and *φ*_{−1} components are bright, while in the *φ*_{0} component the dark rogue wave with three valleys appears. We note that effective energy exchanges take place between waves *φ*_{+1}, *φ*_{0} and *φ*_{−1}. Effective energies can be given by [43]

We know that the rogue wave solutions of equation (1.3) can be obtained if *a*_{1}, *a*_{2} and *k*, which implies that the rogue waves can exist in the whole parametric space. According to the baseband MI theory [50,51], if one can show that for equation (1.3) the MI, if present, is of baseband type only; if this is the case, the link between the baseband MI and rogue wave will be confirmed. With the baseband MI theory [50,51], one can set the solutions to the perturbation term in the amplitudes of the plane waves as [10,50,51]
*u*_{1} and *u*_{2} are *x* periodic with frequency *Ω*, i.e. *k*=0. With *k*≠0, the four roots of the characteristic polynomial can be expressed as

When an eigenvalue *Λ* has a negative imaginary part, MI occurs. As the gain band (where *G*(*Ω*)≠0) can be expressed as 0 ≤ *Ω*_{1} < *Ω* < *Ω*_{2}, the baseband MI is derived when *Ω*_{1}=0. If *M* has four real roots and no MI occurs. On the other hand, if *M* has four complex conjugate roots and equation (1.3) shows the baseband MI. We note that, for equation (1.3), if there is MI, it is of baseband type only, confirming our analytic predictions mentioned above.

## 4. Conclusion

In this paper, we have studied the existence and properties of matter rogue waves in an *F*=1 spinor BEC based on the three-component GP equations, i.e. equation (1.3).

We have shown that the rogue wave solutions of equation (1.3) can be obtained if *A*(*λ*) and *B*(*λ*)] and their Jordan form. We have shown that the rogue wave solutions exist if the baseband MI is present. Recently, Frisquet *et al.* [52] provided the first experimental observation of the dark rogue wave. We expect that the broad family of rogue waves presented here could be observed in the spinor BEC.

## Data accessibility

All the mathematical results are in analytic form and are reproducible.

## Authors' contributions

W.R.-S. obtained the vector rogue wave solutions and wrote the paper. L.W. discussed the dynamics of rogue waves and MI. Both authors gave their final approval for publication.

## Competing interests

We have no competing interests.

## Funding

This work has been supported by the National Natural Science Foundation of China under grant nos. 61705006 and 11305060, and by the Fundamental Research Funds of the Central Universities.

## Acknowledgements

We express our sincere thanks to the editor and reviewers for their valuable comments.

## Appendix A

## Appendix B

We give some examples of solution (2.16):

*Case* 1: *Z*_{11}=*i*, *Z*_{21}=3, *Z*_{31}=2, *Z*_{41}=4 (figure 1)
*Case* 2: *Z*_{11}=1, *Z*_{21}=1, *Z*_{31}=4 and *Z*_{41}=1 (figure 5)

*Case* 3: *Z*_{11}=1, *Z*_{21}=1, *Z*_{31}=2 and *Z*_{41}=1 (figure 7)

## Footnotes

Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.3957454.

- Received April 13, 2017.
- Accepted November 29, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.