The study of the nonlinear Schrödinger (NLS) equation lies at the forefront of applied mathematics and mathematical physics since it has been recognized as a generic model for describing the evolution of slowly varying wave packets in general nonlinear wave system [1,2]. It arises in a variety of physical contexts such as nonlinear optics [3][4][5], Bose-Einstein condensates [6], water waves [7] and plasma physics [8].
In the context of nonlinear optics, when the width of optical pulses is less than 1 ps, higherorder nonlinear effects have to be taken into account and the NLS equation should be modified. As a result, a generalized NLS (gNLS) equation [4] iq z + 1 2 q ττ + |q| 2 q + i (β 1 q τττ + β 2 |q| 2 q τ + β 3 q(|q| 2 ) τ ) = 0,
is derived, where β 1 , β 2 and β 3 are the parameters related to the third-order dispersion, self-steepening and stimulated Raman scattering, respectively. Due to the complexity of the gNLS equation, the study is mainly numerical. However, in some special cases, the gNLS equation becomes integrable and is available for rigorous analysis. For example, when β 1 : β 2 : β 3 = 1 : 6 : 3, it is an integrable equation called the Sasa-Satsuma (SS) equation [9,10]. Under this case, a gauge transformation
brings the SS equation into the form
It is noted that when β 1 : β 2 : β 3 = 1 : 6 : 0, equation ( 1) is also an integrable equation called Hirota equation [11].
Rogue waves, which are initially used for the description of the spontaneous and monstrous ocean surface waves, have attracted much attention in the past two decades [12][13][14]. With the experimental developments, extensive theoretical studies have also been focused on rogue waves [15][16][17][18]. The mathematical study of rogue waves originated from a rational solution of the NLS equation, which is called the Peregrine soliton [19]. Since the higher-order rogue waves were discovered by Akhmediev et al [20], significant progress on the study of higher-order rogue waves has been achieved including the multi-component generalizations of some integrable equations [21][22][23][24][25]. Based on the Riemann-Hilbert approach [26], rogue waves of infinite order have been revealed [27]. In addition, rogue waves on the periodic background [28][29][30][31] have been comprehensively investigated. Very recently, Yang and Yang [32,33] performed an in-depth study on the universal wave patterns of many integrable equations by connecting the explicit expressions of rogue waves with the Yablonskii-Vorob'ev polynomial hierarchy. There are several methods available in constructing rogue wave solutions such as the Darboux transformation [34,35] and Hirota's bilinear method [36,37]. These methods seem quite different, however, they have the same spirit to take the limit of breather solutions in which the spectral parameter tends to the branch point of the spectrum [38]. In bilinear method [37], the action of differential operators on kernel function is equivalent to the limiting process in Darboux transformation. The wave number in the kernel function is basically the spectral parameter in Lax pair.
In addition to the NLS equation, Hirota's bilinear method has been applied to find general rogue wave solutions for a variety of integrable equations such as the Ablowitz-Ladik equation [39], the Davey-Stewartson I and II equations [40,41], the Yajima-Oikawa equation [42,43], a long-wave-short-wave model of Newell type [44], the derivative NLS equation [45], the threewave equation [46], the Boussinesq equation [47] and the coupled NLS equation (Manakov system) [33]. In this method, we start with a scalar function which we call the kernel function. Then, we construct a determinant called tau function with differential operators acting on the kernel function. Under some dispersion relations, it can be shown that the tau function satisfies a set of bilinear equations. Next, through a series of reductions including dimension reduction and complex conjugate reduction, we reduce the set of bilinear equations to the bilinear form of the underlying soliton equation. Finally, we can express the general rogue wave solutions in terms of the tau functions satisfying reduction conditions. We remark here that the dimension reduction is crucial among all the reductions.
In this paper, we attempt to study the rogue wave solutions of the SS equation
by Hirota's bilinear method, where c is a real constant. Despite extensive efforts on the derivation of rogue waves, the higher-order rogue wave solutions of the SS equation have not been reported before. As far as we know, only the first-and second-order rogue wave solutions [35,[48][49][50] of the SS equation have been constructed in explicit forms. Intriguingly, as shown in [48], the first-order rogue waves of the SS equation demonstrate a distinctive feature, which is the so-called twisted-rogue wave (TRW) pair (see figure 2). In Sato theory developed by Kyoto school in 1980s, the original Kadomtsev-Petviashvili hierarchy is named Kadomtsev-Petviashvili hierarchy of A-type or AKP hierarchy according to the classification of Lie algebra, then its sub-hierarchies of B-type, C-type and D-type are called BKP, CKP and DKP hierarchies, respectively [51]. It is known that the NLS equation belongs to the AKP hierarchy while the SS equation belongs to the CKP hierarchy. In order to construct rogue wave solutions of the SS equation, we have to start with a kernel function of 2 × 2 matrix and a determinant of even order with differential operators acting on the matrix kernel function. Therefore, as explained in subsequent sections, the reduction procedure and the rogue wave solutions are much more complicated. To be specific, the tau functions associated with the first-, second-and third-order rogue wave solutions are determinants of 2 × 2, 4 × 4 and 6 × 6, respectively.
The remainder of this paper is organized as follows. In section 2, the procedure in deriving rogue wave solutions of the SS equation ( 4) by applying Hirota's bilinear method and Kadomtsev-Petviashvili hierarchy reduction technique is outlined. Then we present the explicit expression of first-order rogue waves of the SS equation and analyze their dynamics in section 3. In addition, the degeneration of the first-order rogue wave to the Peregrine soliton is discussed as well. In sections 4 and 5, we provide the explicit expressions of second-and third-order rogue waves of the SS equation respectively. In particular, the third-order solutions are expressed in terms of 6 × 6 determinants, where the associated τ -functions are polynomials of degree 24 in x and t, and to the best of our knowledge, this is the first time that the third-order rogue waves are provided in explicit form. Finally, we summarize the main results of this paper in section 6.
In this section, we use Hirota's bilinear method to derive the rogue wave solutions of the SS equation (4) based on the KP reduction technique. The main idea is given briefly below. Similar to the cases of dark soliton and breather solutions [52,53], the SS equation ( 4) is transformed into a set of three bilinear equations (see proposition 3.1 in [52])
under the non-zero boundary condition at ±∞ by the variable transformation
where κ is real, f is a real-valued function, g is a complex-valued function, q is an auxiliary tau function and D is the Hirota's bilinear operator [36] defined by
We start with a specially designed tau function of the extended KP hierarchy that has two discrete indices k and l. We introduce a kernel, which is a 2 × 2 matrix
with entries
for 1 α, β 2, where p α , q β , ξ α0 , η β0 , a, b are complex constants. Let A i and B j be differential operators of order i and j, respectively, defined by
where a k (p) (1 k i) and b l (q) (1 l j) are complex constants. Then, we define a 2N × 2N determinant
where M kl i ν j μ is a 2 × 2 matrix defined by
and (i 1 , i 2 , . . . , i N ), ( j 1 , j 2 , . . . , j N ) are arbitrary sequence of indices. A remarkable property of the function τ kl is that it satisfies eleven bilinear equations (the details are provided in appendix A).
Next, we perform a type C-reduction by requiring
and a dimension reduction by requiring
where C is a constant. By doing so, one obtains the following bilinear equations
Thirdly, we can realize the complex conjugate condition
by requiring
Notice that the C-reduction also implies that
Thus, if we define
then according to the complex conjugate condition, we have
Consequently, by taking k = l = 0 and a = iκ, the bilinear equations ( 17), ( 18) and ( 20) are reduced to exactly (5) while ( 19) is merely the complex conjugate of (18). It is commented here the key point in deriving the rogue wave solutions to the SS equation is to realize the dimension reduction (16).
In this section, we present the first-order rogue wave of the SS equation ( 4)
where
and τ kl is defined as
with
ξ and ξ * being a pair of complex conjugate roots of the quartic equation
Here, the parameters p j , q j , ξ j0 , η j0 , a (0) n , b (0) n , ( j = 1, 2, n = 0, 1) satisfy the constraints
where * denotes complex conjugation. Let us provide a brief proof. First, we introduce a notion
then since
we define
and denote a pair of complex conjugate roots for
with a = iκ, which is equivalent to (30), by ξ and ξ * . Since
where α, β = 1, 2, it then follows
which realizes the dimension reduction. Thus, the first-order rogue wave solution is approved. We note that the algebraic equation ( 30) can be rewritten as
When c c + 4κ 2 < 0, equation ( 39) has at least one pair of complex conjugate roots with nonzero real part and nonzero imaginary part. Hence, the rogue wave solution (26) exists if and only if c and κ satisfy the conditions
On the other hand, as ( 39) is a quartic equation in ξ, it has four roots (counting multiplicity). Due to the fact that all coefficients of (39) are real, these roots demonstrate a symmetric structure, that is, if ξ is a root of (39), then the other three roots are -ξ and ±ξ * . As a result, they can be explicitly expressed as
Specifically, the tau functions related to the first-order rogue wave can be simplified into the form
where (1 i, j 2)
We note that both of f and g are polynomials of degree 4 in x and t, which are more complicated than the first-order rogue wave solutions of the NLS equation where the corresponding tau functions are polynomials of degree 2 in x and t. For simplicity, we set
In particular, if we take
then we can obtain a particular solution which is plotted in figure 1 u
where
It can be seen that, compared with previous studies on first-order rogue wave solutions of the SS equation [35,48,54,55], the solution obtained here is presented in a more compact form. If we select another set of parameters
where
then we may express the corresponding solution in the same form as (46). As depicted in figure 2, this solution demonstrates a structure called TRW pair which was first reported in [48] and is a distinctive characteristic of the SS equation in contrast with many other integrable equations. It bears the name TRW pair due to the feature that it comprises two extended rogue-wave components bending toward each other and displaying an identical but antisymmetric structure. Further analysis indicates that this TRW pair possesses four zero-amplitude points P 1 , P 2 , Q 1 , Q 2 , which are located at (-0.540 049, 0.026 462), (-5.033 412, 2.141 300), (-0.230 825, 0.363 748) and (-5.342 638, 1.804 014) respectively, and has the maximum amplitude 2.758 536, which is attained at the points M 1 and M 2 that are located at (0.471 911, 0.009 464) and (-6.044 867, 2.158 196) respectively. Interestingly, by tuning the values of the free parameter a = iκ, the first-order rogue wave solutions display four types of intrinsic structures. To illustrate this, we set c = -1/2. The wave profiles are shown in figure 3 at three values of a = 0.5i, 1.04i and 1.84i respectively. Specifically, starting from the TRW pair depicted in figure 2 (a = 0.45i), the two zero-amplitude points Q 1 , Q 2 first move toward each other and then merge into the same point Q as κ increases, whereas the other two zero-amplitude points P 1 , P 2 remain, thereby giving rise to a rogue wave solution with three zero-amplitude points (see figures 3(a) and (d)). Furthermore, increasing κ will cause the zero-amplitude point Q disappears but remains to be a local minimum before becoming a saddle point (see figures 3(b) and (e)). Afterward, the rogue wave solution exhibits a relatively simpler structure, which consists of two different types. Both types contain two zero-amplitude points while as the increase of κ, the two maximum points move toward each other and eventually merge into a single point, generating a rogue wave resembling the Peregrine soliton (see figures 3(c) and (f )). In addition, figure 3 also indicates that the value of κ influences the duration of rogue wave, which decreases as κ (> 0) increases. Finally, we discuss the reduction of first-order rogue wave solutions of the SS equation. Let c = -1 and apply the transformation
where x = τz/(12 ), t =z, then the SS equation (equation (1) with
by taking the limit → 0. Furthermore, we take κ = -1/(6 ). As → 0, the solution (26) reduces to the following solution of the NLS equation
which is the so-called Peregrine soliton [19] with maximum amplitude 3 that is attained at (τ , z) = (1/2, 0) and two zero-amplitude points located at (τ , z) = ((1 ± √ 3)/2, 0). Clearly this indicates that the Peregrine soliton is the limiting case of rogue wave solutions of the SS equation.
The tau function for second-order rogue wave is a 4 × 4 determinant with
where and p i , q j , ξ i0 , η j0 , a (0) n , b (0) n are constants that satisfy the same constraints as the first-order rogue waves. It is easily shown that the following relations hold D r,s,x A (1) 1 (p α )B (1) 1 (q β )m kl
3 (p α )B (0) 3 (q β )m kl
which lead to
Thus, the second-order rogue waves are approved. Let us show their dynamics. For simplicity, we may set Recently Yang and Yang [32,33] have systematically studied the rogue wave patterns of several integrable equations such as the NLS equation, the Bossinesq equation and the Manakov system. To be more precise, they have shown that when one of the internal parameters in the rogue wave solutions is large enough, by studying the asymptotic behaviors of solutions, the rogue wave patterns can be described by the root structure of the Yablonskii-Vorob'ev polynomial hierarchy via certain linear transformation. In particular, they have shown that the second-order rogue wave may consist of three separating fundamental rogue waves which are far away from the origin. It turns out that this also occurs in the second-order rogue wave solutions of the SS equation. As shown in figure 4, the second-order rogue waves contain three fundamental rogue waves which separate from each other when a (0) 3 is large enough. As there are four types of fundamental rogue waves, we have totally four types of second-order rogue waves that contain three separating fundamental rogue waves. On the other hand, when a (0) 3 is small, the three fundamental rogue waves may merge together (see figure 5). In addition, the argument of a (0) 3 strongly affects the orientation of these rogue waves while the values of a and c determine the type and duration of rogue waves.
Similar to the first-order rogue wave solutions, with κ = -1/( 6), the second-order rogue wave solutions of the SS equation degenerate to the second-order rogue wave solutions of the NLS equation as → 0. It is also noted that the corresponding tau functions of these rogue wave solutions are polynomials of degree 12 in x and t whereas the second-order rogue wave solutions of the NLS equation have a much simpler structure as their τ -functions are of degree 6 in x and t [37].
The tau function associated to the third-order rogue wave solution of the SS equation ( 4) is a 6 × 6 determinant
with
The corresponding differential operators are defined as follows
A
It can be shown that (i, j = 2, 3, μ, ν = 0, 1)
Based on above relations, we can verify
which completes the proof. With the explicit third-order rogue wave solutions, we can explore their dynamics in detail. For the NLS equation, it has been shown graphically by Ohta and Yang [37] that its third-order rogue waves may be comprised of six individual first-order rogue waves. In this case, there are two possible configurations: one is a symmetric triangle while the other is like a pentagon. The analytic proof of these patterns is given by Yang and Yang [32] by connecting rogue wave solutions of the NLS equation with the root structure of the Yablonskii-Vorob'ev polynomial hierarchy and assuming certain parameter is large enough. Surprisingly, these patterns also appear in the third-order rogue wave solutions of many other integrable systems [33]. Since the NLS equation is a limiting case of the SS equation, we may expect that the SS equation shares analogous features. As depicted in figures 6-9, this is indeed the case. Our computation indicates that the corresponding tau functions of third-order rogue wave solutions are polynomials of degree 24 in x and t, which are much more complicated than the tau functions (degree 12 in x and t) of third-order rogue waves of the NLS equation. Therefore, in addition to the two configurations, four types of intrinsic structures occur in the third-order rogue waves of the SS equation as shown in figures 6-9. When a (0) 0 = 120, a (0) 3 is large enough and other parameters are zero, the third-order rogue wave consists of six first-order rogue waves that feature a triangle. In contrast, when a (0) 0 = 120, a (0) 5 is large enough and other parameters are zero, the third-order rogue waves are comprised of six first-order rogue waves which constitute a distorted pentagon. It can also be seen from figures 6-9 that the degree of distortion depends on the values of the parameter a which also affects the type and duration of rogue waves.
In the present paper, the rogue wave solutions are shown to exist for the SS equation ( 4) when c < 0 and are constructed by means of Hirota's bilinear method. We have presented the first-, second-and third-order rogue wave solutions of the SS equation in explicit forms. In particular, we have confirmed that the TRW and four types of intrinsic structures exist in all orders of rogue wave solutions of the SS equation. While the explicit first-and second-order rogue wave solutions have been reported previously, to the best of our knowledge, the explicit expressions of the third-order rogue wave solutions are relatively new. We have shown that the degree of the tau functions corresponding to the third-order rogue wave solutions is 24, which is much higher than that of the NLS equation (degree 12). We have also shown that there are totally four types of third-order rogue waves consisting of six individual first-order rogue waves, and each type can be classified into two patterns. One pattern is a triangle whereas the other is a distorted pentagon. The types of rogue waves and the degree of distortion of the pentagons are determined by the choices of free parameters in the solutions. In addition, the degeneration of first-order rogue wave solutions of the SS equation to its limiting case, the NLS equation, was discussed in detail. This indicates that the rogue wave solutions of the SS equation are generalizations of rogue wave solutions of the NLS equation. We expect that the results obtained in this work could deepen our understanding on rogue waves in the deep ocean or optical fibers. As a further topic, we will explore the Nth-order rogue wave solutions of the SS equation and their universal patterns. Lemma 1. Suppose m kl i j , ϕ kl i , ψ kl j , which are functions of x, y, t, r, s, satisfy the differential and difference relations
which imply the relations
then the determinant
satisfies the following bilinear equations in the KP hierarchy
Proof. The proof of this lemma is based on properties of the Gram-type determinant [56].
For convenience, we only prove the τ kl defined in (13) satisfies the bilinear equation (87) as other cases can be treated by using similar techniques. We first recall two properties of determinants, i.e.,
and
where Δ i j is the (i, j)-cofactor of the matrix (a i j ). By using these properties, we may rewrite the derivatives and shifts of the τ function as below
Then it follows from the identities above and equation ( 66) that
where α, β = 1, 2, and p α , q β , ξ α0 , η β0 are constants, then direct computations indicate that m kl αβ , ϕ kl α , ψ kl β satisfy the differential and difference relations ( 64 ψ kl 2 j-1 , ψ kl 2 j = B j (q 1 )ψ kl 1 , B j (q 2 )ψ kl 2 ,
where i, j = 1, 2, . . . , N and
a i-n (p) n! (p∂ p ) n , B j (q) = j n=0 b j-n (q) n! (q∂ q ) n , then the functions m kl i j , ϕ kl i , ψ kl j would satisfy the differential and difference relations (64)-(70) as the operators A i and B j commute with the partial differentials with respect to x, y, t, r, s. Hence, using lemma 1, we conclude that the determinant
satisfies the bilinear equations ( 77)-(87) for any sequence of indices (i 1 , i 2 , . . . , i N ; j 1 , j 2 , . . . , j N ).