Fractional Rogue Waves with Translational Coordination, Steep Crest, and Modified Asymmetry
To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger (NLS) equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It is shown that the obtained first- and second-order fractional rogue wave solutions are steeper than those of the corresponding NLS equation with integer-order derivatives. Besides, the time the second-order fractional rogue wave solution undergoes from the beginning to the end is also short. As for asymmetric fractional rogue waves with different backgrounds and amplitudes, this paper puts forward a way to construct them by modifying the obtained first- and second-order fractional rogue wave solutions.
Rogue waves are a kind of local waves, which are first found in the ocean. In the high field intensity, rogue waves may be triggered as an additional phenomenon . Though different from tsunami, the rogue waves that occur in the ocean have extraordinary destructiveness. Recently, rogue waves especially optical ones gained high attention [2–14]. It is Solli et al.  who reported the optical rogue waves generated through a generalized NLS equation. In 2010, Yan  obtained financial rogue waves by means of a nonlinear option pricing model.
The celebrated focusing and defocusing NLS equations:which are the basic models of NLS-type equations; some generalized NLS equations can be reduced to equation (1). As two examples, the first one we introduce is the generalized NLS equation with gain in the following form used in nonlinear fiber optics :where , , and are the supposed functions of propagation distance , representing the group velocity dispersion, nonlinearity, and distributed gain, respectively. And, the second one is the integrable coupled NLS equations :
Besides, the (2 + 1)-dimensional Davey–Stewartson-I (DS-I) equations :which are regarded as an extension of equation (1) to the two-dimensional space, which can be converted into an equivalent form of equation (1) by using the transformation of variables and
Fractional calculus has been applied to many fields [16–30], which shows its irreplaceable role. In 2014, Fujioka et al.  described fractional optical solitons by an extended NLS equation with fractional dispersion and fractional nonlinearity. In 2020, Liu and Chen  derived the time-space fractional cylindrical Kadomtsev–Petviashvili (cKP) and modified cKP equations which better described the propagation of ion-acoustic waves in ultrarelativistic plasmas. In order to fully understand the propagation characteristics and periodicity of dust acoustic solitary waves in dusty plasmas, Zhang et al.  derived the modifed Zakharov–Kuznetsov (mZK) equation and obtained some exact solutions of the time-fractional mZK equation. From a mathematical point of view , there are essential differences between rogue wave solutions and soliton solutions. Although rogue wave solutions are unstable, they are also different from nonautonomous solitons. The NLS-type equations not only have soliton solutions but also have rogue wave solutions. In this paper, we would like to derive a fractional-focusing NLS equation:and then, we construct its first- and second-order fractional rogue wave solutions; here, and are coprime positive integers, and is an odd number so that embedded into the fractional partial derivatives and hold (see Definition 1). Note that when , equation (6) reduces to the integer-order focusing NLS equation  which is the first mathematical model for rogue waves.
The rest of this paper is organized as follows. In Section 2, we present the conformable fractional partial derivative and its properties. In Section 3, we derive the fractional NLS equation and give its Lax representation. In Section 4, we bilinearize the fractional NLS (6) and then construct its first- and second-order fractional rogue wave solutions and show their nonlinear dynamic evolution. In Section 5, we discuss the construction of asymmetric fractional first- and second-order rogue waves by modifying the obtained fractional rogue wave solutions and conclude this paper at the same time.
2. Definition and Some Basic Properties
We first present the definition of the conformable fractional partial derivative .
Definition 1 . Let function arbitrary constant , and fractional order ; then, the conformable fractional partial derivative is defined as follows:where the positive integers and are coprime and is an odd number. As a supplement of Definition 1, we let .
When equation (7) holds, we say is differentiable with respect to If and is not related to and , then becomes the generalized version  of the conformable fractional derivative  proposed by Khalil et al.
We next list some basic properties of the conformable fractional partial derivative defined in Definition 1. The proofs of these properties are similar to the ones in [35, 36], we omit them here.
Property 1. If is differentiable with respect to , then
Property 2. If and are differentiable with respect to thenwhere and are all constants.
Property 3. If and are differentiable with respect to and respectively, then
Property 4. Fractional derivatives of some common functions have results:where and are constants.
As the end of this section, we finally show in Figure 1 four different conformable fractional partial derivatives of with respect to by setting and .
It can be seen that the smaller the fractional order is, the larger the value of its derivative function is. This naturally raises the question of whether we can construct such rogue wave solutions whose widths and crests are dependent on the fractional order so that they are steeper and more destructive than those of the corresponding integer-order case. The following work will clarify the existence of such a fractional rogue wave solution for equation (6). In addition, we will construct, in Section 4, another fractional rogue wave solution of equation (6), which takes less time than the integer order one from the occurrence to the end. Steep crest or short duration is closer to one of the characteristics of rogue waves, that is, it appears and disappears very suddenly. Besides, it is worth noting that the rogue wave solutions to construct in this paper possess coordinate parameters. Such rogue wave solutions are no longer limited to the coordinate origin as usual, and their positions are selective. Besides, after a modification, such fractional rogue wave solutions can also show asymmetry.
3. Lax Representation of the Fractional NLS Equation
Theorem 1. Fractional NLS equation (6) is Lax integrable, which has the Lax representation:withwhere denotes the complex conjugate.
Proof. A direct computation tellsAt the same time, it is easy to verify the relation:Substituting equations (21) and (22) into equation (19) yieldswhich is exactly equivalent to equation (6). The proof is ended.
4. Fractional Rogue Wave Solutions of the Fractional NLS Equation
This section consists of three parts for the rogue wave solutions of equation (6), one part is the bilinear forms of equation (6) and the other three parts are the first- and second-order fractional rogue wave solutions and their nonlinear dynamic analysis.
Theorem 2. Suppose thatwhere and are real- and complex-valued functions of and respectively, then fractional NLS equation (6) has the following bilinear forms:where and are the fractional version  of Hirota’s bilinear operator :
Proof. Firstly, by the transformationwe write equation (6) asSecondly, we suppose thatthen equation (29) can be reduced toFurther, lettingWith the help of the fractional bilinear operator (27), we finally can rewrite equations (31) and (32) as equations (25) and (26), respectively. Thus, considering equations (28) and (30), we have completed the proof.
Based on the fractional bilinear forms (25) and (26), the fractional N-soliton solution of equation (6) can be obtained. However, the starting point of this paper is to construct rogue wave solutions, and we omit them. In what follows, we shall employ equation (31) to construct fractional rouge wave solutions of equation (6).
4.2. First-Order Fractional Rogue Wave Solution
For the first-order fractional rogue wave solution, we have Theorem 3 below.
Theorem 3. Fractional NLS equation (6) has the following first-order fractional rogue wave solution:and the maximum crest value of which is
Proof. We first suppose thatwhere , and are all undetermined real constants. Substituting equations (34) and (35) into equation (31), canceling the common denominator, and then collecting the coefficients of the same powers of the real part and imaginary part, we drive a set of algebraic equations about these undermined real constants. Further, solving the derived set of algebraic equations yieldswhere is a nonzero real constant, and all the other undetermined real constants not mentioned here are all zeros. Finally, from equations (28), (30), and (34)–(36), we arrive at first-order rogue wave solution (33).
Sinceandwe then have , and hence, complete the proof.
4.3. Second-Order Fractional Rogue Wave Solution
For the second-order fractional rogue wave solution, we have Theorem 4 below.
Theorem 4. Fractional NLS equation (6) has the following second-order fractional rogue wave solution:and the crest value at the point is Here,
Proof. Inspired by the work  and equation (39), we suppose thatwhere are all real constants to be further undetermined. By a series of operations, such as substituting equations (40) and (41) into equation (31), canceling the common denominator, separating the real and imaginary parts, and collecting the coefficients of the same powers , then a set of algebraic equations about these undermined constants is derived. Solving this set of algebraic equations, we can determine equations (40) and (41) byBy using equations (28), (30), and (42)–(46), we can easily obtain equation (39). When and equations (40) and (41) give and , respectively. We then obtain The proof is ended.
4.4. Nonlinear Dynamic Evolution and Comparison
In this section, we show first- and second-order fractional rogue wave solutions (33) and (39) for the nonlinear dynamic evolution. At the same time, for the sake of comparison, we also show the rogue wave solutions of corresponding NLS equation (6) with the integer order.
In Figures 2–5 , first-order rogue wave solution (33) with translational coordination and its comparison reference without translational coordination are shown. It can be seen that the width of fractional rogue wave (33) along the x-axis is much narrower than that of the corresponding integer-order rogue wave.
A similar difference also occurs in second-order rogue wave solution (39) with translational coordination and its comparison reference without translational coordination, but the widths of fractional rogue wave (39) along x-axis and t-axis are all much narrower than those of the corresponding integer-order rogue wave, see Figures 6–9 . This shows that, in addition to the steep crest, the time from the beginning to the end of the fractional rouge wave is also short. With the decrease of fractional order , the high and subhigh crests of second-rogue wave solution (39) are getting closer and closer until almost together, always keeping at the point (see Figure 10).
5. Discussion and Conclusion
In summary, we have derived fractional NLS equation (6) with Lax integrability by introducing conformable fractional partial derivative (7). Based on transformed equation (31), first- and second-order fractional rogue wave solutions (33) and (39) with translational coordination of equation (6) are obtained. The translation coordinate provides a certain degree of freedom for adjusting the position of fractional rogue wave solutions (33) and (39) on the coordinate plane.
First-order fractional rogue wave solution (33) can also obtained by using the limits as did in  and equation (28); here,which is a fractional homoclinic solution of equation (29). Using the limits of similar fractional homoclinic solutions like equation (47) to construct second-order fractional rogue wave solution (39) is worthy of study.
In addition to steepness and instantaneity, asymmetry is also a remarkable characteristic of the rogue wave. It is also worth exploring to construct asymmetric rogue waves by means of fractional models. In Figures 11–18 , two asymmetry (see Figure 13(a) for an example) first- and second-order fractional rogue waves with different backgrounds (see Figures 13(b), 14(c), and 18(b) for , while ) and large amplitudes (see Figures 14(b) and 18(a) for and ) are shown based on solutions (33) and (39), and the assumption that the value of in Definition 1 always returns a complex number for all x<a when q is an even number. Computer simulation results through Mathematica 8.0 show that, with the decrease of fractional order , one of the four subhigh waves in the second-order fractional rogue waves gradually exceeds the highest crest. This is different from the usual second-order rogue waves. However, under this case, such first- and second-order rogue wave solutions do not satisfy equation (6). This is because that when with is a complex number, is no longer a real valued function as previously assumed, and and are not the solutions of equation (31). In fact, when it cannot transform equation (6) into equation (29), but is valid. In spite of this, it may be important that this paper provides a way to construct asymmetric fractional rogue waves by modifying the obtained first- and second-order fractional rogue wave solutions.
The data used in this paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Liaoning BaiQianWan Talents Program of China (2019), National Science Foundation of China (11971475), Natural Science Foundation of Education Department of Liaoning Province of China (LJ2020002), and Natural Science Foundation of Xinjiang Autonomous Region of China (2020D01B01).
B. L. Guo and L. M. Ling, “Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations,” Chinese Physics Letters, vol. 28, no. 11, Article ID 110202, 2011.View at: Google Scholar
Y. Ohta and J. K. Yang, “Rouge waves in the davey-stewartson I equations,” Physical Review E, vol. 86, Article ID 036604, 2012.View at: Google Scholar
Y. F. Yue, L. L. Huang, and Y. Chen, “Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrodinger equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 89, Article ID 105284, 2020.View at: Google Scholar
H. Y. Zhang and Y. F. Zhang, “Analysis on the M-rogue wave solutions of a generalized (3 + 1)-dimensional KP equation,” Applied Mathematics Letters, vol. 102, Article ID 106145, 2019.View at: Google Scholar
V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Physical Review Letters, vol. 90, no. 11, Article ID 113902, 2003.View at: Google Scholar
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999.
Y. Wang and Q. G. Deng, “Fractal derivative model for tsunami travelling,” Fractals, vol. 27, Article ID 1950017, 2019.View at: Google Scholar
W. H. Zhang, H. W. Yang, and D. Z. Gao, “Time fractional order mZK model and its exact solution in dust plasma,” Mathematical Modeling and Its Applications, vol. 9, no. 38, pp. 66–73, 2020, (in Chinese).View at: Google Scholar
B. L. Guo, L. X. Tian, Z. Y. Yan, and L. M. Ling, Rouge Wave and its Mathematical Theory, Zhejiang science and Technology Press, Hangzhou, China, 2015, in Chinese.
B. XU, Y. F. Zhang, and S. Zhang, “Line soliton interactions for shallow ocean-waves and novel solutions with peakon, ring, conical, columnar and lump structures based on fractional KP equation,” Advances in Mathematical Physics, vol. 2021, Article ID 6664039, 15 pages, 2021.View at: Publisher Site | Google Scholar