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

Abstract

A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (3+1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation is used as an example to illustrate the effectiveness of the suggested method. A new family of two-wave solution, rational breather wave solution, is obtained by extended homoclinic test method, and it is just a rogue wave solution. This result shows rogue wave can come from extreme behavior of breather solitary wave for (3+1)-dimensional nonlinear wave fields.

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 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 [15–26]. 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 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,…=0, (1) where 𝑃 is a polynomial in its arguments, 𝑢∶𝑅_𝑥×𝑅_𝑦×𝑅_𝑡→𝑅. To determine 𝑢(𝑡,𝑥,𝑦) explicitly, we take the following four steps.

Step 1. By Painlev’e analysis, a transformation 𝑢=𝑇𝑓 (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 Hirota’s bilinear form 𝐺𝐷_𝑡,𝐷_𝑦;𝑓=0, (3) where the 𝐷-operator [27] is defined by 𝐷^𝑚_𝑡𝐷^𝑛_𝑦𝑓=𝜕𝑡,𝑦⋅𝑔𝑡,𝑦−𝜕𝜕𝑡𝜕𝑡^′𝑚𝜕−𝜕𝜕𝑦𝜕𝑦^′𝑛𝑓𝑡,𝑦𝑔𝑡^′,𝑦^′_{𝑡′=𝑡,𝑦′=𝑦}. (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 −4𝑢_𝑡+Φ(𝑢)𝑢_𝑧𝑥+3𝑢_𝑦𝑦Φ(𝑢)=0,=𝜕²_𝑥+4𝑢+2𝑢_𝑥𝜕_𝑥⁻¹. (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 nontravelling 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 𝜉=𝑥+𝑐𝑧 in (5); for simplicity we take constant 𝑐>0 (𝑐<0 is similar), notice that 𝜕_𝑥=𝜕_𝜉, 𝜕_𝑧=𝑐𝜕_𝜉 and so 𝑐=𝜕_𝑧𝜕_𝑥⁻¹, then (5) can be converted into the following form: −4𝑢_𝜉𝑡+𝑐𝑢⁴_𝜉+3𝑐𝑢²_𝜉𝜉+3𝑢_𝑦𝑦=0. (6) Setting 𝜂=𝜉−𝑏𝑡=𝑥+𝑐𝑧−𝑏𝑡 in (6) gives 3𝑢_𝑦𝑦+4𝑏𝑢_𝜂𝜂+3𝑐𝑢²_𝜂𝜂+𝑐𝑢_𝜂𝜂=0. (7) Setting 𝜁=𝑖𝑦 in (7) gives 3𝑢_𝜁𝜁−4𝑏𝑢_𝜂𝜂−3𝑐𝑢²_𝜂𝜂−𝑐𝑢_𝜂𝜂=0. (8) It is easy to see that (7) has an equilibrium solution 𝑢₀ which is an arbitrary constant.

We suppose that 𝑢=𝑢₀+2ln𝑓_𝜂𝜂, (9) where 𝑓(𝜂,𝜁) is unknown real function. Substituting (9) into (8) we obtain the following bilinear form: 3𝐷²_𝜁−4𝑏+6𝑐𝑢₀𝐷²_𝜂−𝑐𝐷⁴_𝜂−𝐴𝑓⋅𝑓=0, (10) where 𝐴 is an integration constant, 𝐷⁴_𝜂𝑓⋅𝑓=2(𝑓𝑓_4𝜂−4𝑓_𝜂𝑓_3𝜂+3𝑓²_2𝜂), 𝐷²_𝜂𝑓⋅𝑓=2(𝑓_𝜂𝜂𝑓−𝑓²_𝜂). With regard to (9), using the homoclinic test technique we can seek the solution in the form 𝑓=𝑒^{−𝑝1(𝜂−𝛼𝜁)}+𝛿₁cos𝑝𝜂+𝛽𝜁+𝛿₂𝑒^{𝑝1(𝜂−𝛼𝜁)}, (11) where 𝑝₁,𝑝,𝛿₁,𝛿₂ are real constants to be determined and 𝛼,𝛽 are constants to be determined.

Substituting (10) into (9) we can get an algebraic equation of 𝑒^{𝑝1(𝜂−𝛼𝜁)}. Then equating the coefficients of all powers of 𝑒^{𝑗𝑝(𝜉−𝛼𝜁)}(𝑗=−1,0,1) to zero, we get 2𝛿₁𝑐𝑝𝑝³₁+4𝑏+6𝑐𝑢₀𝛿₁𝑝𝑝₁+3𝛿₁𝛼𝛽𝑝𝑝₁−2𝛿₁𝑐𝑝³𝑝₁=0,4𝑏+6𝑐𝑢₀𝛿₁𝛿₂𝑝𝑝₁+2𝛿₁𝛿₂𝑝𝑝³₁+3𝛿₁𝛿₂𝛼𝛽𝑝𝑝₁−2𝛿₁𝛿₂𝑐𝑝³𝑝₁=0,12𝛿₂𝛼²𝑝²₁−4𝛿²₁𝑐𝑝⁴+4𝑏+6𝑢₀𝛿²₁𝑝²−3𝛿²₁𝛽²𝑝²−44𝑏+6𝑐𝑢₀𝛿₂𝑝²₁−16𝛿₂𝑐𝑝⁴₁=0,6𝛿₁𝑐𝑝²𝑝²₁−4𝑏+6𝑢₀𝛿₁𝑝²₁+4𝑏+6𝑢₀𝛿₁𝑝²−𝛿₁𝑐𝑝₄−𝛿₁𝑐𝑝⁴₁+3𝛿₁𝛼²𝑝²₁−3𝛿₁𝛽²𝑝²=0,−𝛿₁𝛿₂𝑐𝑝⁴₁+6𝛿₁𝛿₂𝑐𝑝²𝑝²₁−4𝑏+6𝑐𝑢₀𝛿₁𝛿₂𝑝²₁−3𝛿₁𝛿₂𝛽²𝑝²−𝛿₁𝛿₂𝑐𝑝⁴+4𝑏+6𝑐𝑢₀𝛿₁𝛿₂𝑝²+3𝛿₁𝛿₂𝛼²𝑝²₁=0. (12) Take 𝑝₁=𝑝; then (12) can be reduced into the following: 4𝑏+6𝑐𝑢₀𝛿₁𝑝²+3𝛿₁𝛼𝛽𝑝²=0,4𝑏+6𝑐𝑢₀𝛿₁𝛿₂𝑝²+3𝛿₁𝛿₂𝛼𝛽𝑝²=0,4𝛿₁𝑐𝑝⁴+3𝛿₁𝛼²𝑝²−3𝛿₁𝛽²𝑝²=0,4𝛿₁𝛿₂𝑐𝑝⁴−3𝛿₁𝛿₂𝛽²𝑝²𝛿₁+3𝛿₁𝛿₂𝛼²𝑝²=0,12𝛿₂𝛼²𝑝²−4𝛿²₁𝑐𝑝⁴+4𝑏+6𝑐𝑢₀𝛿²₁𝑝²−3𝛿₁𝛽²𝑝²−44𝑏+6𝑐𝑢₀𝛿₂𝑝²−16𝛿₂𝑐𝑝⁴=0. (13) Solving (13) yields 𝛿₁=±26𝛼²−22𝑏−3𝑢₀−3𝛽²𝛿₂6𝛽²−22𝑏−3𝑢₀−3𝛼²,1𝛼𝛽=−34𝑏+6𝑐𝑢₀,𝑝²=3𝛽4𝑐²−𝛼², (14) where 𝑏,𝑐,𝛿₂ are some free real constants and 𝛼,𝛽 are some free constants. Setting 𝑐=−1 in (14) gives 𝛿₁=±26𝛼²−22𝑏−3𝑢₀−3𝛽²𝛿₂6𝛽²−22𝑏−3𝑢₀−3𝛼²,2𝛼𝛽=−32𝑏−3𝑢₀,𝑝²=34𝛼²−𝛽². (15) Choosing 𝑢₀≠2𝑏/3 and 𝛿₂>0, we get from (15) 𝛽>|𝛼|2,𝛼=−2𝑏−3𝑢₀,𝛼3𝛽₂>22𝑏−3𝑢₀,𝑢3𝛽⁰<2𝑏3or>2𝑏3or𝛼²<2𝑏3⋅𝑢₀>2𝑏3. (16) Substituting (15)-(16) into (11), we have 𝑓₁𝜂,𝜁=2𝛿₂𝑝2cosh𝜂+2𝑏−3𝑢₀𝜁+13𝛽2ln𝛿₂+ℎ₁𝑝,𝑓cos𝜂+𝛽𝜁₂𝜂,𝜁=2𝛿₂𝑝2cosh𝜂+2𝑏−3𝑢₀𝜁+13𝛽2ln𝛿₂−ℎ₁cos𝑝𝜂+𝛽𝜁, (17) where ℎ₁=2(6𝛼²−2(2𝑏−3𝑢₀)−3𝛽²)𝛿₂/6𝛽²−2(2𝑏−3𝑢₀)−3𝛼², 𝑝=±(3(𝛼²−𝛽²)/2), and 𝛼, 𝛽 are some free constants. Substituting (17) into (9) yields the solutions of (8) as follows, respectivly: 𝑢₁𝜂,𝜁=𝑢₀+2𝑝²𝑚₀+2𝑚₁𝑝2sinh𝜂+2𝑏−3𝑢₀𝜁+13𝛽2ln𝛿₂𝑝×𝑝2×sin𝜂+𝛽𝜁cosh𝜂+2𝑏−3𝑢₀𝜁+13𝛽2ln𝛿₂+𝑚₁cos𝑝𝜂+𝛽𝜁²⁻¹,𝑢₂𝜂,𝜁=𝑢₀+2𝑝²𝑚₀−2𝑚₁𝑝2sinh𝜂+2𝑏−3𝑢₀𝜁+13𝛽2ln𝛿₂𝑝×𝑝2×sin𝜂+𝛽𝜁cosh𝜂+2𝑏−3𝑢₀𝜁+13𝛽2ln𝛿₂−𝑚₁cos𝑝𝜂+𝛽𝜁²⁻¹, (18) where 𝑚₀=9(𝛽²−𝛼²)/(6𝛽²−2(2𝑏−3𝑢₀)−3𝛼²), 𝑚₁=6𝛼²−2(2𝑏−3𝑢₀)−3𝛽²/6𝛽²−2(2𝑏−3𝑢₀)−3𝛼²<1, and 𝑝=±(3(𝛼²−𝛽²)/2).

Taking 𝜁=𝑖𝑦, 𝛼=𝑎𝑖, and 𝛽=𝜔𝑖 into (18) yields the solutions of (7) as follows, respectivly: 𝑢₁𝜂,𝑦=𝑢₀+2𝑝²𝑚₀+2𝑚₁𝑝2sinh𝜂+2𝑏−3𝑢₀𝑦+13𝜔2ln𝛿₂𝑝×𝑝2×sin𝜂−𝜔𝑦cosh𝜂+2𝑏−3𝑢₀𝑦+13𝜔2ln𝛿₂+𝑚₁cos𝑝𝜂−𝜔𝑦²⁻¹,𝑢₂𝜂,𝑦=𝑢₀+2𝑝²𝑚₀−2𝑚₁𝑝2sinh𝜂+2𝑏−3𝑢₀𝑦+13𝜔2ln𝛿₂𝑝×𝑝2×sin𝜂−𝜔𝑦cosh𝜂+2𝑏−3𝑢₀𝑦+13𝜔2ln𝛿₂−𝑚₁cos𝑝𝜂−𝜔𝑦²⁻¹, (19) where 𝑎,𝜔 are some free real constants, 𝑚₀=9(𝜔²−𝑎²)/(6𝜔²+2(2𝑏−3𝑢₀)−3𝑎²), 𝑚₁=6𝑎²+2(2𝑏−3𝑢₀)−3𝜔²/6𝜔²+2(2𝑏−3𝑢₀)−3𝑎²<1, and 𝑝=±(3(𝜔²−𝑎²)/2).

The solution 𝑢₁(𝜂,𝑦) (resp., 𝑢₂(𝜂,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 (backward-direction) periodic wave with speed 𝑏 and homoclinic wave of different direction with speed 2(2𝑏−3𝑢₀)/3𝜔.

Taking 𝜂=𝜉−𝑏𝑡=𝑥−𝑧−𝑏𝑡 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): 𝑢₁𝑡,𝑥,𝑦,𝑧=𝑢₀+2𝑝²𝑚₀+2𝑚₁𝑝2sinh𝑥+2𝑏−3𝑢₀+13𝜔𝑦−𝑧−𝑏𝑡2ln𝛿₂𝑝×𝑝2×sin𝑥−𝜔𝑦−𝑧−𝑏𝑡cosh𝑥+2𝑏−3𝑢₀+13𝜔𝑦−𝑧−𝑏𝑡2ln𝛿₂+𝑚₁cos𝑝𝑥−𝜔𝑦−𝑧−𝑏𝑡²⁻¹,𝑢₂𝑡,𝑥,𝑦,𝑧=𝑢₀+2𝑝²𝑚₀−2𝑚₁𝑝2sinh𝑥+2𝑏−3𝑢₀+13𝜔𝑦−𝑧−𝑏𝑡2ln𝛿₂𝑝×𝑝2×sin𝜂−𝜔𝑦−𝑧−𝑏𝑡cosh𝑥+2𝑏−3𝑢₀+13𝜔𝑦−𝑧−𝑏𝑡2ln𝛿₂−𝑚₁cos𝑝𝑥−𝜔𝑦−𝑧−𝑏𝑡²⁻¹, (20) where 𝑎,𝜔 are some free real constants, 𝑚₀=9(𝜔²−𝑎²)/(6𝜔²+2(2𝑏−3𝑢₀)−3𝑎²), 𝑚₁=6𝑎²+2(2𝑏−3𝑢₀)−3𝜔²/6𝜔²+2(2𝑏−3𝑢₀)−3𝑎²<1, and 𝑝=±(3(𝜔²−𝑎²)/2).

Figure 1 

The figure of 𝑢1(𝑡,𝑥,𝑦,𝑧) as 𝑢0=−2 and 𝑏=3.

Figure 2 

The figure of 𝑢2(𝑡,𝑥,𝑦,𝑧) as 𝑢0=−2, 𝑏=3, 𝜔=3.5, 𝑝=0.6092400758, 𝑥=−1, and 𝑧=−2.

Use (19) and take 𝛿₂=1; then (1/2)ln(𝛿₂)=0 in 𝑢₂. So, solution 𝑢₂ can be rewritten as follows: 𝑢₂⁽¹⁾𝜂,𝑦=𝑢₀+2𝑝²𝑚₀−2𝑚₁𝑝2sinh𝜂+2𝑏−3𝑢₀𝑦𝑝×𝑝23𝜔×sin𝜂−𝜔𝑦cosh𝜂+2𝑏−3𝑢₀𝑦3𝜔−𝑚₁cos𝑝𝜂−𝜔𝑦²⁻¹, (21) where 𝑚₀=12𝑝²/(4𝑝²+2(2𝑏−3𝑢₀)+3𝜔²) and 𝑚₁=3𝑎²+2(2𝑏−3𝑢₀)−4𝑝²/4𝑝²+2(2𝑏−3𝑢₀)+3𝜔².

Now we consider a limit behavior of 𝑢₂⁽¹⁾ as the period 2𝜋/𝑝 of periodic wave cos(𝑝(𝜂−𝜔𝑦)) goes to infinite; that is, 𝑝→0. By computing, we obtain the following result: 𝑈_roguewave=𝑢₀+166𝐴−𝜂+22𝑏−3𝑢₀/3𝜔𝑦𝜂−𝜔𝑦𝜂+22𝑏−3𝑢₀/3𝜔𝑦²+𝜂−𝜔𝑦²+8𝐴², (22) where 𝐴=1/(3𝜔²+2(2𝑏−3𝑢₀)); here we have used 𝑚₁→1 and 𝜔=𝑎 as 𝑝→0.

𝑈 contains two waves with different velocities and directions. It is easy to verify that 𝑈_roguewave is a rational solution of (7). Moreover, we can show that 𝑈_roguewave also is breather-type solution. In fact, 𝑈→0 for fixed 𝜂 as 𝑦→±∝. 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 𝜂=𝜉−𝑏𝑡=𝑥−𝑧−𝑏𝑡 into (21), we obtain the rogue wave solutions of the (3+1)-D YTSF equation as follows (see Figure 3): 𝑈_(ytsf)roguewave=𝑢₀+2166𝐴−𝑥+2𝑏−3𝑢₀××23𝜔𝑦−𝑧−𝑏𝑡𝑥−𝜔𝑦−𝑧−𝑏𝑡𝑥+2𝑏−3𝑢₀3𝜔𝑦−𝑧−𝑏𝑡²+𝑥−𝜔𝑦−𝑧−𝑏𝑡²+8𝐴²⁻¹, (23) where 𝐴=1/(3𝜔²+2(2𝑏−3𝑢₀)); here we have used 𝑚₁→1 and 𝜔=𝑎 as 𝑝→0.

(a)

(b)

Figure 3 

The figure of 𝑈(ytsf)roguewave as 𝑎=𝜔, 𝑢0=1, 𝑏=2, 𝜔=√6/3, 𝑥=−1, and 𝑧=−2.

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?

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.