Dynamics of rogue waves on a multi-soliton background in a vector nonlinear Schrödinger equation
Abstract. General higher order rogue waves of a vector nonlinear Schrödinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The th order semi-rational solutions containing free parameters are expressed in separation of variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. The study of our results would be of much importance in understanding and predicting rogue wave phenomena arising in nonlinear and complex systems, including optics, fluid dynamics, Bose-Einstein condensates and finance and so on.
Keywords. Vector nonlinear Schrödinger equations, rogue waves, Darboux-dressing transformation
AMS subject classification. 22E46, 53C35, 57S20
It is well known that many nonlinear wave equations of physical interest support solitons, which are localised waves arising from a balance between dispersion and nonlinearity, and which can propagate steadily for a long time. Recently, over the last two decades, it has been recognised that another class of solutions, namely breathers, are also of fundamental importance. Breathers propagate steadily, and are localised in either time or space, while being periodic in either space or time. Further, due to their localisation properties, breathers have been invoked as models of rogue waves, also called freak waves, which are large amplitude waves which apparently appear without warning, and then disappear without trace. While they have been most often found in the context of water waves -, they have also been found in other physical contexts such as optical fibres -.
The breather solutions of the focusing nonlinear Schrödinger equation (NLSE) have been widely invoked as models of rogue waves, see the references above and Akhmediev  for instance. The NLSE equation is integrable  and many kinds of exact solutions have been found. In particular, the Peregrine breather, the Akhmediev breather (AB)  and the Kuznetsov-Ma breathers (KM) - have been associated with rogue waves, as the potential outcome of the modulational instability of a plane wave. AB is periodic in space and localized in time, while KM is periodic in time and localised in space. The Peregrine breather is especially considered as a rogue wave prototype because it is localised in both time and space, and so captures the fundamental features of rogue waves. Importantly it has a peak amplitude which is exactly three times the background. Also, it is the generic outcome of a wide class of modulated plane waves . When the spatial or temporal period is taken to be infinite, AB or KM become Peregrine breather in the limit. Peregrine breather of NLSE has been observed experimentally in water wave tanks  and also in nonlinear fibre optics -.
The Peregrine breather is the lowest order item in a hierarchy of rational solutions for NLSE. Since these higher-order forms may also be invoked as models for rogue waves, with higher amplitudes, it is of interest to find explicit expressions for these higher-order rogue wave solutions. For example, Akhmediev  presented some low-order rational solutions using a Darboux transformation method. Ohta and Yang  obtained general high-order rogue waves in terms of determinants using the Hirota bilinear method. Guo - constructed the -order rogue wave solutions using a generalized Darboux transformation. Dubard et al.  discussed the quasi-rational solutions via algebraic-geometry method. As well as for the NLSE, rational solutions have also been explored for some other nonlinear wave equations in , , -.
Here we shall extend a recent study of the integrable vector nonlinear Schrödinger equations (VNLSE), or Manakov system, by Baronio et al.  This can be expressed in dimensionless form as,
Here the subscripts stand for partial differentiation with resect to , an evolution variable, and , a spatial variable. The dependent variables represent wave envelopes, whose physical meaning depends on the particular context. In particular the system (1) has applications in nonlinear optics - and in Bose-Einstein condensates -. Also note that Eqs. (1) correspond to the self-focusing (or anomalous dispersion) regime. The fundamental vector rogue wave solutions have been recently reported by several authors , , . Then the first-, second- and third-order rogue wave solutions of VNLSE were explicitly presented . Here our aim is to find the general -th order rogue wave solutions. We will construct hierarchies of semi-rational solutions of the VNLSE (1), using an asymptotic expansion method. Our obtained solutions are an extension of the results of Baronio et al.  to higher-order rogue wave solutions, and the pure rational solutions of our results can be identified with the ones obtained by Zhai et al.  As the representations obtained are a combination of rational and exponential functions, some interesting structures can be observed, such as (1) the coexistence of higher-order rogue waves and bright-dark multi-soliton solutions, and (2) the coexistence of multi-rogue waves and bright-dark multi-soliton solutions. In general our obtained solutions indicate the complexity that can arise when rogue waves interact with solitons.
The paper is organised as follows. In section 2, The Lax pair and Darboux-dressing transformation of VNLSE (1) are briefly reviewed. In section 3 the dynamic behaviour of a family of fundamental rogue wave solutions is examined. Then in section 4, we present some novel periodic breathers. In section 5, we use an asymptotic expansion method to obtain -th order rogue wave solutions of VNLSE (1). Then, in section 6, we give some examples to illustrate the a range of dynamic behaviour of our obtained rogue wave solutions. We conclude with a summary in section 7.
2 Asymptotic expansion of Darboux-dressing transformation
The VNLSE (1) is integrable, and is a condition for the compatibility of the Lax pair,
Here is a matrix variable, is the complex spectral parameter, is a constant diagonal matrix and is the matrix variable
It is straightforward to check the condition of compatibility between the two equation in (2)
leads directly to the VNLSE(1).
Here, and is the fundamental solution for the Lax equations (2) corresponding to and for the seed solutions of the VNLSE (1). The constant parameter is complex while is an arbitrary nonzero complex 3-dimensional constant vector. Next, it is useful to note that the Darboux-dressing transformation (4) can be replaced with the alternative form
Since is also a Darboux transformation of NLSE, it follows that
In general, for a Darboux transformation, the zero seed solution allows for the construction of a hierarchy of multisoliton solutions , while a plane wave seed solution results in a hierarchy of breather-type solutions related to modulation instability. However, here we note that
This means that the Darboux-dressing transformation (5) cannot be iterated continuously for the same spectral parameter.
In order to eliminate this limitation, we introduce the following expansion theorem
which can then be used to produce new solutions for the same spectral parameter.
Theorem Let be a solution of the Lax pair system (2) corresponding to the spectral parameter and a seed solution . If has an expansion at
are solutions of the Lax pair system (2) corresponding to the same spectral parameter and solution
Remark: In the above Theorem, the denotation means that are replaced correspondingly by but are left unchanged in . In addition, we introduce the denotation
The meaning of is same as . Concretely, let us illustrate these denotations by two simplest examples: for , . For , .
Generally, employing the iteration relation (11) and with the help of the following expansion
Then could be rewritten in a more explicit form
It is natural to obtain
Meanwhile, the corresponding solution of the VNLSE is given by
Thus we have shown that the Theorem holds for .
Next we use mathematical induction for integers . Assume the Theorem holds for , that is,
Employing the following properties
the left side of (20) is given by
Comparing this with the right side of (20), we get that
Factoring out the factor in (2), we arrive at
The assumption (20) allows us to construct the next step of the Darboux dressing transformation which satisfies
Next, we need to show that
Similarly, for the temporal flow, we can show that
This completes the proof by induction.
In addition, by a slight adjustment of the above proof of our Theorem, it will provide a justifiable way to prove the Theorem 2 in Guo et al. .
3 Exact breather solutions of the Lax pair system
Before using the expansion Theorem to construct higher-order rogue wave solutions, we use the Darboux-dressing transformation (2) to construct a new class of breather solutions. It is readily shown that the VNLSE (1) admit the plane wave background solution,
where and are arbitrary parameters which, without loss of generality, are taken as real, and is a wavenumber.
The corresponding solution of the Lax system (2) is sought in the form
and is an arbitrary complex vector, and . Here, it is required that the constant matrices and satisfy
Then the exponential matrices in (36) can be written as
Similarly the exponential matrices in (36) can be written as