Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation
In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second- and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and non-standard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order RW of the complex mKdV and the NLS equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultra-short pulse technology.
Keywords: complex MKdV equation, Darboux transformation, rogue wave.
PACS number(s): 05.45.Yv, 42.65.Tg, 03.75.Lm, 87.14.gk
The theory of nonlinear dynamics has attracted considerable interest and is fundamentally linked to several basic developments in the area of soliton theory. It is well-known that the Korteweg-de Vries (KdV) equation, modified Korteweg-de Vries (mKdV) equation, sine Gordon equation and the nonlinear Schrödinger (NLS) equation are the most typical and well-studied integrable evolution equations which describe nonlinear wave phenomena for a range of dispersive physical systems. Their stable multi-soliton solutions play an important role in the study of nonlinear waves. Further studies have also been carried out to examine the effects on these solitons due to dissipation, inhomogeneity or non-uniformity present in nonlinear media [2, 3].
The term “soliton” is a sophisticated mathematical concept that derives its name from the word “solitary wave” which is a localized wave of translation that arises from the balance between nonlinear and dispersive effects. In spite of the initial theoretical investigations, the concept of solitary wave could not gain wide recognition for a number of years in the midst of excitement created by the development of electromagnetic concepts in those times. Korteweg and de Vries (1895) developed a mathematical model for the shallow water problem and demonstrated the possibility of solitary wave generation. Next, the study of solitary waves really took off in the mid-1960s when Zabusky and Kruskal discovered the remarkably stable particle-like behaviour of solitary waves. They reported numerical experiments where solitary waves, described by the KdV equation, passed through each other unchanged in speed or shape, which led them to coin the word “soliton” to suggest such a unique property. In a follow-up study Zakharov and Shabat generalized the inverse scattering method in 1972 and also solved the nonlinear Schrödinger equation, demonstrating both its integrability and the existence of soliton solutions.
Following the above discoveries, solitary waves of all flavors advanced rapidly in many areas of science and technology. In nonlinear physics applications to many areas e.g. hydrodynamics, biophysics, atomic physics, nonlinear optics, etc., have been developed. As of now, more than a few hundreds of nonlinear evolution equations (NEEs) have been shown to admit solitons and some of these theoretical equations are also responsible for the experimental discovery of solitons[1, 7]. In general, nonlinear phenomena are often modelled by nonlinear evolution equations exhibiting a wide range of high complexities in terms of difference in linear and nonlinear effects. In the past four decades or so, the advent of high-speed computers, many advanced mathematical softwares and the development of a number of sophisticated and systematic analytical methods, which are well-supported by experiments have encouraged both theoreticians and experimentalists. Nonlinear science has experienced an explosive growth by the invention of several exciting and fascinating new concepts not just like solitons, but e.g. dispersion-managed solitons, rogue waves, similaritons, supercontinuum generation, etc.. Many of the completely integrable nonlinear partial differential equations (NPDEs) admit one of the most striking aspects of nonlinear phenomena, which describe soliton as a universal character and they are of great mathematical as well as physical interest. It is impossible to discuss all these manifestations exhaustively in this paper. We further restrict ourselves to the solitary wave manifestation in nonlinear optics. In the area of soliton research at the forefront, right now, is the study of optical solitons, where the highly sought-after goal is to use strong localized nonlinear optical pulses as the high-speed information-carrying bits in optical fibers.
Optical solitons are localized electromagnetic waves that propagate steadily in a nonlinear medium resulting from the robust balance between nonlinearity and linear broadening due to dispersion and diffraction. Existence of the optical soliton was first time found in 1973 when Hasegawa and Tappert demonstrated the propagation of a pulse through a nonlinear optical fiber described by the nonlinear Schrödinger equation. They performed a number of computer simulations demonstrating that nonlinear pulse transmission in optical fibers would be stable. Subsequently, after the fabrication of low-loss fiber, Mollenauer et al. in 1980 successfully confirmed this theoretical prediction of soliton propagation in a laboratory experiment. Since then, fiber solitons have emerged as a very promising potential candidate in long-haul fiber optic communication systems.
Further, in addition to several important developments in soliton theory, the concept of modulational instability (MI) has also been widely used in many nonlinear systems to explain why experiments involving white coherent light supercontinuum generation (SCG), admit a triangular spectrum which can be described by the analytical expressions for the spectra of Akhmediev breather solutions at the point of extreme compression. In the case of the NLS equation, Peregrine already in  had identified the role of MI in the formation of patterns resembling high-amplitude freak waves or rogue wave (RW). RWs have recently been also reported in different areas of science. In particular, in photonic crystal fibre RWs are well-established in connection with SCG . This actually has stimulated research for RWs in other physical systems and has paved the way for a number important applications, including the control of RWs by means of SCG[11, 12], as well as studies in e.g. superfluid Helium , Bose Einstein condensates , plasmas [15, 16], microwave , capillary phenomena , telecommunication data streams , inhomogeneous media , water experiments , and so on. Recently, Kibler et al.  using a suitable experiment with optical fibres were able to generate femtosecond pulses with strong temporal and spatial localization and near-ideal temporal Peregrine soliton characteristics.
For the past couple of years, several nonlinear evolution equations were shown to exhibit the RW-type rational solutions [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. From the above listed works, it is clear that one of the possible generating mechanisms  for the higher-order RW is the interaction of multiple breathers possessing identical and very particular frequency of the underlying equation. Though the theory of solitons and many mathematical methods have been well-used in connection with soliton theory for the past four decades or so, to the best of our knowledge, the dynamics of multi-rogue wave evolutions has not yet been systematically investigated in integrable nonlinear systems .
Very recently, considering the propagation of few-cycle optical pulses in cubic nonlinear media and by developing multiple scaling approach to the Maxwell-Bloch-Heisenberg equation up to the third-order in terms of expansion parameter, the complex mKdV equation was derived[42, 43]. Circularly polarized few-cycle optical solitons were found which are valid for long pulses. Thus, it is more than worthy to systematically investigate the existence of the few-cycle optical rogue waves for this model, and this is the main purpose of the present paper.
The organization of this paper is as follows. In Section 2, based on the parameterized Darboux transformation (DT) of the mKdV equation, the general formation of the solution is given. In Section 3, we construct the higher-order rogue waves from a periodic seed with constant amplitude and analyze their structures in detail by choosing suitable system parameters. We provide detailed discussion about the obtained results in Sections 4 and 5.
2. The Darboux Transformation
For our analysis, we begin with coupled complex mKdV equations of the form of
Under a reduction condition , the above coupled equations reduce to the complex mKdV
The complex mKdV equation is one of the well-known and completely integrable equations in soliton theory, which possesses all the basic characters of integrable models. From a physical point of view, the above equation has been derived for, e.g. the dynamical evolution of nonlinear lattices, plasma physics, fluid dynamics, ultra-short pulses in nonlinear optics, nonlinear transmission lines and so on. The Lax pair corresponding to the coupled mKdV equations is given by, i.e.
Here, is an arbitrary complex spectral parameter or also called eigenvalue, and is the eigenfunction corresponding to of the complex mKdV equation. From the compatibility condition , one can easily obtain the coupled equations (1) and (2). Furthermore, set be a gauge transformation by
This implies that, in order to prove that the mKdV equation is invariant under the gauge transformation (6), it is important to look for determine the such that , have the same forms as , . Meanwhile, the seed solutions (, ) in spectral matrixes , are mapped into the new solutions (, ) in terms of transformed spectral matrixes , .
Recently, using the generalized Darboux transformation, th-order
rogue wave solutions for the complex mKdV equation have been
proposed in e.g.. However, in our work, we shall
systematically analyze the evolution of the different patterns of
higher-order rogue waves by suitably choosing the parameters in the
rational solutions. In addition, it is worth to note that the
obtained results are in agreement with our recently published
developments about the method of generating higher-order rogue waves
2.1 One-fold Darboux Transformation
From the knowledge of the known form of the DT for the nonlinear Schrödinger equation [44, 45, 46, 47, 48, 49], we assume that a trial Darboux matrix in eq. (6) has the following form
where , , , , , , , are functions of and . From
by comparing the coefficients of , , it yields
From the coefficients of , we conclude that and are functions of only. Similarly, from
and by comparing the coefficients of , , we obtain the following set of equations
In order to obtain the non-trivial solutions of the complex mKdV equation, we provide the Darboux transformation under the condition . Without loss of generality, and based on eqs. (12) and (14), we observe that the Darboux matrix T admits the following form
Here, , , , are functions of and , which could be expressed by two eigenfunctions corresponding to and . To begin with, we introduce eigenfunctions and associated distinct eigenvalues as follows
We assume from now on that even number eigenfunctions and eigenvalues are given by odd ones as the following rule ():
For convenience and simplicity of our mathematical manipulations, we propose the following theorems:
Theorem 1. The elements of a one-fold Darboux matrix are presented with the eigenfunction corresponding to the eigenvalue as follows
with , and then the new solutions and are given by
and the new eigenfunction corresponding to is
Proof. Note that , and is derived from the functional form of , then and is derived from the functional form of . So, and are arbitrary constants, and hence, we let for simplicity for later calculations. By transformation defined by eq. (12) and eq. (14), new solutions are given by
By making use of the general property of the DT, i.e.,
, after some manipulations, eq. (18) is obtained.
Next, substituting given in
eq. (18) into eq. (22), the new solutions are
given as in eq. (20). Furthermore, by using the explicit
matrix representation eq. (19) of , then
is given by
It is trivial to confirm by using the special choice on and in eq. (17). This means generates a new solution of the complex mKdV from a seed solution . Note that for .
2.2 n-fold Darboux transformation
By -times iteration of the one-fold DT , we obtain -fold DT of the complex mKdV equation with the special choice on and in eq. (17). To save space, we omit the tedious calculation of and its determinant representation. Under the above conditions, the reduction condition is preserved by , so we just give in the following theorem.
Theorem 2. Under the choice of eq. (17), the n-fold DT generates a new solution of the complex mKdV equation from a seed solution as
By making use of Theorem 2 with a suitable seed solution, we can generate the multi-solitons, multi-breathers, and multi-rogue waves of the complex mKdV equation. As the multi-soliton and multi-breather solutions are well-known and completely explored for the complex mKdV equation, next, we shall concentrate mainly on the systematic construction of the higher-order rogue waves from the double degeneration  of the DT. Though the construction of higher-order rogue wave solutions is quite cumbersome, one can still validate the correctness of these solutions with the help of modern computer tools such as a simple symbolic calculation or equivalent, and also by a direct numerical computation.
3. Higher-order rogue waves
In this section, starting with a non-zero seed , , we shall present higher-order rogue waves of the complex mKdV equation. If , a constant, which is just a seed solution to generate soliton. So, in this paper, we choose . By using the principle of superposition of the linear differential equations, then, the new eigenfunctions corresponding to can be provided by
Here, are the arbitrary constants, is an infinitesimal parameter.
We are now in a position to consider the double degeneration of
to obtain higher-order rogue wave as in our earlier
investigations . It is trivial to check that
in eq. (24), which means that
these eigenfunctions are degenerate at .
Setting and substituting
defined by eq. (24)
back into eq. (23), the double degeneration, i.e. eigenvalue
and eigenfunction degeneration, occurs in . Next,
now becomes an indeterminate form . Set
and set be given
by eq. (24), we obtain -th order rogue wave
solutions by higher-order Taylor expansion of with respect
Theorem 3. An n-fold degenerate DT with a given eigenvalue is realized in the degenerate limit of . This degenerate n-fold DT yields a new solution of the mKdV equation starting with the seed solution ,where
Here, , denotes the floor function of .
In the following, to avoid the tedious mathematical steps we encountered, we only present the expressions of the 1st, 2nd and 3rd order rogue waves by using Theorem 3. In each case, the solution describes the envelope of the rogue wave, and its square modulus contains information such as e.g. wave evolution above water surface, or the intensity of few-cycle optical wave, etc.
Firstly, we set , and take the form of determinants. By using the 1st-order Taylor expansion with respect to in terms of elements of and through , we determined and by equating the coefficient of , and then obtained the explicit expression for the 1st-order rogue wave as
Its evolution is presented in Figure 1 (left) with the condition and the Taylor expansion at , in order to compare this with higher-order rogue waves. It is trivial to find that when and . This means that the asymptotic plane of has the height . Particularly, let and , is a soliton propagating along a line with a non-vanishing boundary. Set and , then gives of ref..
When , we construct the 2nd-order rogue waves under the assumption from Theorem 3. An explicit form of is constructed as
Here, and are two degree 6 polynomials in and , which are given in appendix A. From Figure 1 (right), one finds that under the assumption , or equivalently , the second-order rational solution admits a single high maximum at the origin. By suitably adjusting the parameter one could control the decaying rate of the profile in the -plane. This is a fundamental pattern. Furthermore, as is shown in Figure 6, when taking and , the large peak of the 2nd rogue wave is completely separated and forms a set of three first-order rational solution for sufficiently large meanwhile , and actually forms an equilateral triangle.
When , and set and , then Theorem 3 yields an explicit formula of the 3rd rogue wave with parameters . Set , we have
Here, and are two degree 12 polynomials in and , which are given in appendix B. This is the fundamental pattern of the 3rd-order rogue wave, which is plotted in Figure 2(left) with a different value of .
In general, Theorem 3 provides an efficient tool to produce analytical forms of higher-order rogue waves of the complex mKdV equation. Actually, we have also constructed the analytical formulas for 4th, 5th and 6th -order rogue waves. However, because of their long expressions describing these solutions, we do not present them here but would provide upon request. The validity of all these higher-order rogue waves has been verified by symbolic computation. According to the explicit formulae of the th-order rogue waves under fundamental patterns, we find that their maximum amplitude is by setting and in , and the height of the asymptotic plane is , which is the same as that of the rogue wave of the NLS equation. This fact can be easily verified through Figs.(1-3). All figures in this paper are plotted based on these explicit analytical formulas of the solutions. Once the explicit analytical higher-order rogue waves are known, our next aim is to generate and understand underlying the dynamics of the obtained different patterns by suitably selecting the value of .
4. Results and discussion
The above discussion is a clear manifestation of the evolution of the
higher-order rogue waves from the Taylor expansion of the degenerate
breather solutions. A brief discussion about the generating mechanism
of higher-order rogue waves from the nonlinear evolution equation has
already been reported by . For our
purpose now, we customize our discussion only up to 6th-order rogue
waves, since higher-order rogue waves are difficult to construct owing
to the extreme complexity and tedious mathematical calculations.
It is quite obvious from our numerical analysis that the choice of
parameters such as and actually do generate three
different basic patterns of rogue wave solutions.
Let us discuss these patterns now.
Fundamental patterns: When, e.g. , or equivalently in , the rational solutions of any order have a similar structure. In addition, there are local maxima on each side of the line at . Starting from , before the central optimum high amplitude, there is a sequence of peaks with gradual increase in height. Here, one can observe that the number of first peaks is , then there is a row of symmetric peaks with respect to time as shown in Figs. (1-3) for 6 rogue waves.
There are only two parameters and in the explicit forms of the rogue waves under fundamental patterns. It is a challenge problem to illustrate analytically the role of and in the control of the profile for the higher-order RWs due to the extreme complexity of the explicit forms of the nth-order RWs (). So, we only study this problem for the first-order RW . To this end, we introduce a method, i.e., the contour line method, to analyze the contour profile of the red bright spots in the density plot of Fig. 4, which intuitively shows the localization characters such as length and width of the RW. On the background plane with height , a contour line of with is a hyperbola
which has two asymptotes
and two non-orthogonal axes:
There are two fixed vertices: on () plane of all value of . Here, is also a median of one triangle composed of above two asymptotes and a parallel line of -axis except . We combine the density plots and the above three lines in Fig.4 with different values of . At height , a contour line of with is given by a quartic polynomial
which has two end points
and along -direction.
Moreover, there are two fixed points expressed by ,
on () plane of all value of . At height , a contour line of with is also given by a quartic polynomial
which is defined on interval [-] of . For this contour line, there are four fixed points: on () plane of all value of . Two centers of valleys of given by