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Rogue wave for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation.

1. Introduction

It is well known that solitary wave solutions of nonlinear evolution equations play an important role in nonlinear science fields, especially in nonlinear physical science, since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications [1]. In recent years, rogue waves, as a special type of nonlinear waves and also known as freak waves, monster waves, killer waves, extreme waves, and abnormal waves [2], have triggered much interest in various physical branches. Rogue wave is a kind of waves that seems abnormal which is first observed in the deep ocean. It always has two to three times amplitude higher than its surrounding waves and generally forms in a short time for which people think that it comes from nowhere. Rogue waves have been the subject of intensive research in oceanography [3, 4], optical fibres [5-7], superfluids [8], Bose-Einstein condensates, financial markets, and other related fields [9-13]. The first-order rational solution of the self-focusing nonlinear Schrodinger equation (NLS) was first found by Peregrine to describe the rogue waves phenomenon [14]. Recently, by using the Darboux dressing technique or Hirotas bilinear method, rogue waves solutions in complex system were obtained such as nonlinear Schrodinger equation, Hirota equation, Sasa-Satsuma equation, Davey-Stewartson equation, coupled Gross-Pitaevskii equation, coupled NLS Maxwell-Bloch equation, and coupled Schrodinger-Boussinesq equation [15-26]. In this work, we propose a homoclinic (heteroclinic) breather limit method for seeking rogue wave solution to real NEE. We consider a general nonlinear partial differential equation in the form

P (u, [u.sub.t], [u.sub.x], [u.sub.y], ... = 0, (1)

where P is a polynomial in its arguments, u : [R.sub.x] x [R.sub.y] x [R.sub.t] [right arrow] R. To determine u(t, x, y) explicitly, we take the following four steps.

Step 1. By Painleve analysis, a transformation

u = T (f) (2)

is made for some new and unknown function f.

Step 2. By using the transformation in Step 1, original equation can be converted into Hirotas bilinear form

G([D.sub.t], [D.sub.y]; f) = 0, (3)

where the D-operator [27] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Step 3. Solve the above equation to get homoclinic (heteroclinic) breather wave solution by using extended homoclinic test approach (EHTA) [28].

Step 4. Letting the period of periodic wave go to infinite in homoclinic (heteroclinic) breather wave solution, we can obtain a rational homoclinic (heteroclinic) wave and this wave is just a rogue wave.

As a example we consider (3+1)-D Yu-Toda-Sasa-Fukuyama equation which is an extension of Bogoyavlenskii-Schiff (BS) equation in higher dimension [29]. It is well known that BS equation is the reduction of the self-dual Yang-Mills equation; it is an integrable system and has an infinite number of conservation laws and N-soliton solutions [30].

The (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

It is called Yu-Toda-Sasa-Fukuyama (YTSF) equation. YTSF equation is not integrable system [29-31]; it is firstly presented by Yu et al. using the strong symmetry [30, 32]. The non-travelling wave solution was found using auto-Backlund transformation and the generalized projective Riccati equation method [32-34]. Moreover, some soliton-like solutions and periodic solutions for potential YTSF equation were obtained by Hiriota's bilinear method, the tanh-coth method, exp-function method, homoclinic test approach, and extended homoclinic test approach [33-37], respectively. Recently, some analytic solutions for the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation [38] and the (2+1)dimensional Ablowitz-Kaup-Newell-Segur equation [39] are obtained by Darvishi using the modified extended homoclinic test approach, some exact solutions of the nonlinear ZK-MEW, and the potential YTSF equations by Zayed and Arnous using the modified simple equation method [40]. Besides these, further result on soliton and its feature for (5) were not studied up to now.

This work focuses on rational breather wave and then rogue wave solutions. Applying HBLM to (3+1)-D YTSF equation we firstly get breather solitary solution and then obtain rational breather solution by letting periodic wave go to infinite in breather solitary solution. Finally, we show that this rational breather wave is just a rogue wave. This is the new physical phenomenon found out up to now.

2. Rational Homoclinic Wave (Rogue Wave)

Let [xi] - x + cz in (5); for simplicity we take constant c > 0 (c < 0 is similar), notice that [[partial derivative].sub.x] = [[partial derivative].sub.[xi]], [[partial derivative].sub.z] = c[[partial derivative].sub.[xi]] and so c = [[partial derivative].sub.z] [[partial derivative].sup.-1.sub.x], then (5) can be converted into the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Setting [eta] = [xi] - bt = x + cz - bt in (6) gives

3[u.sub.yy] + 4b[u.sub.[eta][eta]] + 3c[([u.sup.2]).sub.[eta][eta]] + c[u.sub.nn] = 0. (7)

Setting [zeta] - iy in (7) gives

3[u.sub.[zeta][zeta]] - 4b[u.sub.[eta][eta]] - 3c[([u.sup.2]).sup.[eta][eta]] - c[u.sub.[eta][eta]] = 0. (8)

It is easy to see that (7) has an equilibrium solution [u.sub.0] which is an arbitrary constant.

We suppose that

u = [u.sub.0] + 2[(ln f)[eta][eta]], (9)

where f([eta], [zeta]) is unknown real function. Substituting (9) into (8) we obtain the following bilinear form:

(3[D.sup.2.sub.[zeta]] - (4b + 6c[u.sub.0]) [D.sup.2.sub.[eta]] - c[D.sup.4.sub.[eta]] - A) f x f = 0, (10)

where A is an integration constant, [D.sup.4.sub.[eta]] f x f = 2([ff.sub.4[eta]]) - 4 [f.sub.[eta]] [f.sub.3[eta]] + 3 [f.sup.2.sub.3[eta]]), [D.sub.2.sub.[eta]] f x f = 2([f.sub.[eta][eta]] f - [f.sup.2.sub.[eta]]. With regard to (9), using the homoclinic test technique we can seek the solution in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where [p.sub.1], p, [[delta].sub.1], [[delta].sub.2] are real constants to be determined and [alpha], [beta] are constants to be determined.

Substituting (10) into (9) we can get an algebraic equation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then equating the coefficients of all powers of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (j = -1, 0, 1) to zero, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Take [p.sub.1] = p; then (12) can be reduced into the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Solving (13) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where b, c, [[delta].sub.2] are some free real constants and [alpha], [beta] are some free constants. Setting c = -1 in (14) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Choosing [u.sub.0] [not equal to] 2b/3 and [[delta].sub.2] > 0, we get from (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Substituting (15)-(16) into (11), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [alpha], [beta] are some free constants. Substituting (17) into (9) yields the solutions of (8) as follows, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where [m.sub.0] = 9 ([[beta].sup.2] - [[alpha].sup.2])/6([[beta].sup.2] - 2 (2b - 3[u.sub.0]) - 3 [[alpha].sup.2]), [m.sub.1] = [square root of 6[[alpha].sup.2] - 2 (2b - 3[u.sub.0]) - 3 [[beta].sup.2]]/[square root of 6[[beta].sup.2] - 2 (2b - 3[u.sub.0]) - 3 [[alpha].sup.2]] < 1, and p = [+ or -] ([square root of 3[[alpha].sup.2] - [[beta].sup.2])/2).

Taking [zeta] - iy, [alpha] - ai, and [beta] = [omega]i into (18) yields the solutions of (7) as follows, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

where a, [omega] are some free real constants, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The solution [u.sub.1] ([eta], y) (resp., [u.sub.2]([eta], y)) shows a new family of two-wave, breather solitary wave, which is a solitary wave and meanwhile is a periodic wave whose amplitude periodically oscillates with the evolution of time. It shows elastic interaction between a left-propagation (backwarddirection) periodic wave with speed b and homoclinic wave of different direction with speed 2(2b - 3[u.sub.0])/3[omega].

Taking [eta] = [xi] - bt = x - z - bt into (19) gives and yields the breather-type soliton solutions of the (3+1)-D YTSF equation as follows, respectively (see Figures 1 and 2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where a, [omega] are some free real constants, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Use (19) and take [[delta].sub.2] = 1; then (1/2) ln([[delta].sub.2]) = 0 in u2. So, solution [u.sub.2] can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where [m.sub.0] = 12 [p.sup.2]/(4[p.sup.2] + 2(2b - 3 [u.sub.0]) + 3 [[omega].sup.2]) and [m.sub.1] = [square root of 3[a.sup.2] + 2(2b - 3[u.sub.0]) - 4 [p.sup.2]]/[square root of 4[p.sup.2] + 2(2b - 3[u.sub.0]) + 3 [[omega].sup.2]].

Now we consider a limit behavior of [u.sub.(1).sub.2] as the period 2[pi]/p of periodic wave cos(p([eta] - [omega]y)) goes to infinite; that is, p [right arrow] 0. By computing, we obtain the following result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where A = 1/(3[[omega].sup.2] +2(2b - 3[u.sub.0])); here we have used [m.sub.1] [right arrow] 1 and [omega] = a as p [right arrow] 0.

U contains two waves with different velocities and directions. It is easy to verify that [U.sub.rogue wave] is a rational solution of (7). Moreover, we can show that [U.sub.rogue wave] also is breather-type solution. In fact, U [right arrow] 0 for fixed [eta] as y [right arrow] [+ or -] [varies]. So, U is not only a rational breather solution but also a rogue wave solution which has two to three times amplitude higher than its surrounding waves and generally forms in a short time. It is an example that the rogue wave can come from breather solitary wave solution for real equation. One can think whether the energy collection and superposition of breather solitary wave in many many periods leads to a rogue wave or not.

Taking [eta] = [xi] - bt = x - z - bt into (21), we obtain the rogue wave solutions of the (3+1)-D YTSF equation as follows (see Figure 3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

where A = 1/(3[[omega].sup.2] +2(2b - 3[u.sub.0])); here we have used [m.sub.1] [right arrow] 1 and [omega] = a as p [right arrow] 0.

3. Conclusion

In this paper, we propose a new method for seeking rogue wave, homoclinic (heteroclinic) breather limit method (HBLM). Applying this method to the real (3+1)-D YTSF equation, we obtain a family of homoclinic breather solution and rational homoclinic solution. Furthermore, rational homoclinic solution obtained here is just a rogue wave solution, and then we obtain the rogue wave solutions of the (3+1)-D YTSF equation. In future, we intend to study the interaction between breather wave and solitary wave. What is more, can we obtain similar results to another integrable or nonintegrable system with homoclinic or heteroclinic breather wave? How can one use the homoclinic breather wave to obtain rogue wave under contained conditions?

http://dx.doi.org/10.1155/2014/378167

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Chinese Natural Science Foundation Grants nos. 11372294 and 11361048 and the Sichuan Educational science Foundation Grant no. 09zc008.

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Hanlin Chen, (1) Zhenhui Xu, (2) and Zhengde Dai (3)

(1) Joint Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology, Mianyang 621010, China

(2) Applied Technology College, Southwest University of Science and Technology, Mianyang 621010, China J School of Mathematics and Physics, Yunnan University, Kunming 650091, China

Correspondence should be addressed to Zhenhui Xu; xuzhenhuil9@163.com

Received 27 March 2014; Accepted 3 July 2014; Published 17 July 2014

Academic Editor: Qi-Ru Wang
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Title Annotation:Research Article
Author:Chen, Hanlin; Xu, Zhenhui; Dai, Zhengde
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2014
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