Rogue Wave for the ( 3 + 1 )-Dimensional Yu-Toda-Sasa-Fukuyama Equation

and Applied Analysis 3 Take p 1 = p; then (12) can be reduced into the following: (4b + 6cu 0 ) δ 1 p 2 + 3δ 1 αβp 2 = 0, (4b + 6cu 0 ) δ 1 δ 2 p 2 + 3δ 1 δ 2 αβp 2 = 0, 4δ 1 cp 4 + 3δ 1 α 2 p 2 − 3δ 1 β 2 p 2 = 0, 4δ 1 δ 2 cp 4 − 3δ 1 δ 2 β 2 p 2 δ 1 + 3δ 1 δ 2 α 2 p 2 = 0, 12δ 2 α 2 p 2 − 4δ 2 1 cp 4 + (4b + 6cu 0 ) δ 2 1 p 2 − 3δ 1 β 2 p 2 − 4 (4b + 6cu 0 ) δ 2 p 2 − 16δ 2 cp 4 = 0. (13)


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][6][7], superfluids [8], Bose-Einstein condensates, financial markets, and other related fields [9][10][11][12][13].The first-order rational solution of the self-focusing nonlinear Schrödinger 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 Schrödinger equation, Hirota equation, Sasa-Satsuma equation, Davey-Stewartson equation, coupled Gross-Pitaevskii equation, coupled NLS Maxwell-Bloch equation, and coupled Schrödinger-Boussinesq equation [15][16][17][18][19][20][21][22][23][24][25][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 where  is a polynomial in its arguments,  :   ×   ×   → .To determine (, , ) explicitly, we take the following four steps.
Step 1.By Painlev' e analysis, a transformation is made for some new and unknown function .
Step 2. By using the transformation in Step 1, original equation can be converted into Hirota's bilinear form where the -operator [27] is defined by =,  = . (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 -soliton solutions [30].
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.
The solution  1 (, ) (resp.,  2 (, )) 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  and homoclinic wave of different direction with speed 2(2 − 3 0 )/3.
Taking  =  −  =  −  −  into (19) gives and yields the breather-type soliton solutions of the (3+1)-D YTSF equation as follows, respectivly (see Figures 1 and 2): ) ,  2 (, , , ) where ,  are some free real constants, and  = ±(√3( 2 −  2 )/2).Use (19) and take  2 = 1; then (1/2) ln( 2 ) = 0 in  2 .So, solution  2 can be rewritten as follows: (1)  2 (, ) where Now we consider a limit behavior of  (1)  2 as the period 2/ of periodic wave cos(( − )) goes to infinite; that is,  → 0. By computing, we obtain the following result: where  = 1/(3 2 + 2(2 − 3 0 )); here we have used  1 → 1 and  =  as  → 0.  contains two waves with different velocities and directions.It is easy to verify that  rogue wave is a rational solution of (7).Moreover, we can show that  rogue wave also is breathertype solution.In fact,  → 0 for fixed  as  → ± ∝.So,  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.

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