# New Rational Homoclinic and Rogue Waves for Davey-Stewartson Equation

Authors: Changfu Liu; Chuanjian Wang; Zhengde Dai; Jun Liu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A new method, homoclinic breather limit method (HBLM), for seeking rogue wave solution of nonlinear evolution equation is proposed. A new family of homoclinic breather wave solution, and rational homoclinic solution (homoclinic rogue wave) for DSI and DSII equations are obtained using the extended homoclinic test method and homoclinic breather limit method (HBLM), respectively. Moreover, rogue wave solution is exhibited as period of periodic wave in homoclinic breather wave approaches to infinite. This result shows that rogue wave can be generated by extreme behavior of homoclinic breather wave for higher dimensional nonlinear wave fields.


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## Body

## 1. Introduction

In recent years, rogue waves, as a special type of solitary waves, has been triggered much interest in various physical branches, although there is no exact definition up to now. Rogue waves is a kind of waves that seems abnormal which is first observed in the deep ocean, it has been the subject of intensive research in oceanography [1, 2], optical fibres [3–5], superfluids [6], Bose-Einstein condensates, financial markets, and related fields [7–10]. A possible mechanism for the formation of rogue waves is associated with modulation instability [11–14].The mysteriousness of rogue wave events mainly lies in the phenomenon which appears out of nowhere and disappears without trace. As is known, there are some methods to seek rogue wave such as Darboux dressing technique, Hirota bilinear method. Based on Hirota bilinear equation of nonlinear evolution equation, for Schrödinger type complex systems, there are some effective techniques such as the Peregrine breather method (PB) [11], whose representation is mathematically a ratio of two polynomials, Ma solitons [4] (MS) and Akhmediev breather methods (ABs) [3]. The main difference between these methods is the test function to Hirota bilinear equation. The test functions of PB, MS, and ABs are E1(x,t)=eiϕ(x,t)(1-((G(x,t)+iH(x,t))/D(x,t)));E2(x,t)=eiϕ(x,t)((a1cosh⁡⁡(px)+a2cos⁡⁡(kt)+ia3sin⁡(kt))/(b1cosh⁡⁡(px)+b2cos⁡⁡(kt))); and E3(x,t)=eiϕ(x,t)((a1cosh⁡⁡(αt)+a2cos⁡⁡(kx)+ia3sinh⁡⁡(αt))/(b1cosh⁡⁡(αt)+b2cos⁡⁡(kx))), respectively. Here ϕ(x,t) is real function and G(x,t),H(x,t), and D(x,t) are polynomials of (x,t), and Ei(x,t), i=2,3, may generate the rogue wave similar to E1 as k→0. The above three methods have been successfully applied to complex system such as Hirota equation, Sasa-Satsuma equation, Davey-Stewartson equation, coupled Gross-Pitaevskii equation, coupled NLS Maxwell-Bloch equation, and coupled Schrödinger-Boussinesq equation [11–18].In this work, we propose a homoclinic (heteroclinic) breather limit method for seeking rogue wave solution. We takeE(x,t)=eiϕ(x,t)((e-p(x-αt)+a1cos⁡(p1(x-βt))+a2ep(x-αt))/(e-p(x-αt)+a3cos⁡(p1(x-βt))+a4ep(x-αt))) as a test function to Hirots bilinear equation. E(x,t) can generate one or two rogue waves as p1→0. It is obvious that the E is different from E2 and E3, comparing with E2, E3, and E has more complicated structure. Now we consider the application of HBLM to Davey-Stewartson equation.Davey-Stewartson (DS) equation is written as [19]
(1)iut=-uxx-1α02uyy-2ϵα02|u|2u-2α02uv,vyy-α02vxx-2α02ϵ(|u|2)xx=0,
where u:Rx×Ry×Rt+→C, v:Rx×Ry×Rt+→R, and ϵ and α0 are constants. DS equation was introduced in a paper by Davey and Stewartson (1974) to describe the evolution of a three-dimensional disturbance in the nonlinear regime of plane Poiseuille flow. The function u(x,y,t) stands for the complex amplitude, and v(x,y,t) describes the perturbation of the real velocity. DS equation is called the DSI as ϵ=1, α0=±1 and DSII as ϵ=1, α0=±i. There are known results due to local well-posed, global existence and blow-up of some solutions, exact periodic soliton solutions, solitoff and dromion solutions [20–29]. Recently, homoclinic and heteroclinic tube solutions were obtained [29–33].We consider DSI equation:(2)iut+uxx+uyy=-2|u|2u-2uv,vxx-vyy=-2(|u|2)xx,
and DSII equation:
(3)iUt+Uxx-Uyy=2|U|2U+2UV,Vxx+Vyy=-2(|U|2)xx.


## 2. Homoclinic Breather and Rogue Wave Solution of DSI

Making transformationu=(a/2)exp⁡(ia2t)Q, v=-φ/2 and substituting it into (2), we can get
(4)iQt+Qxx+Qyy=-a2(|Q|2-1)Q+Qφ,φxx-φyy=2a2(|Q|2)xx,
where Q=Q(x,y,t) is a complex function and φ is a real. By the dependent variable transformation
(5)Q=GF,φ=-4(ln⁡⁡F)xx,
with G being a complex and F being a real, then (4) can be converted into the form
(6)iGtF-iFtG+GxxF-2GxFx+GFxx+GyyF-2GyFy+GFyy-(a2+B)GF=0,2(FyyF-Fy2-FxxF+Fx2)-BF2-a2GG*=0,
where B is an integration constant and an asterisk denotes the complex conjugation.By means of the extended homoclinic test approach [33], we take the test function as follows:
(7)G=e-p(x+y/2+αt)+a1cos⁡(p1(x+2y-αt))+a2ep(x+y/2+αt),F=e-p(x+y/2+αt)+a3cos⁡(p1(x+2y-αt))+a4ep(x+y/2+αt),
where all of a3, a4, p, p1, β, β1, and α are real and a1,a2 are complex. Substituting (7) into (6) and equating the coefficients of all powers of ejp(x+y/2+αt)cos⁡(p1(x+2y-αt)),ejp(x+y/2+αt)sin(p1(x+2y-αt)) and e±2(p(x+y/2+αt))(j=0,±1) to zero, we can obtain a set of algebraic equations for p, p1, β, β1, α, and aj, j=1, 2, 3, 4, with
(8)B=-a2,(4pp1-ip1α)a1+(4pp1+ip1α)a3=0,(-ip1α-4pp1)a4a1+(ip1α-4pp1)a3a2=0,(ipα+5p24-5p12)a1+(5p24-5p12-ipα)a3=0,(5p24-ipα-5p12)a4a1+(ipα+5p24-5p12)a3a2=0,-10a1a3p12+(5p2+2ipα)a2+(5p2-2ipα)a4=0,a2(a32-a1a1*)=0,a2(a42-a2a2*)=0,(2a2-3p22-6p12)a3-a2(a1+a1*)=0,(2a2-3p22-6p12)a3a4-a2(a1a2*+a1*a2)=0,(2a2-6p2)a4-6a32p12-a2(a2+a2*)=0.
Solving these equations, we obtain the relations between the parameters as
(9)B=-a2,p12=21p220,p2=320a2-39α2624,a1=(iα+4p)a3iα-4p,a2=(iα+4p)2a4(iα-4p)2,a32=4(21α2-80p2)a421(α2+16p2).Fromp2≥0 and a32≥0 in (9), we have (800a2/507)≤α2≤(320a2/39). Substituting (9) into (7) and then (5) and taking a4>0, we obtain the solution for DSI equation as
(10)u=a2ei(θ+a2t)2cosh⁡[ξ+γ+iθ]+(a3/a4)cos⁡(η)2cosh⁡(ξ+γ)+(a3/a4)cos⁡(η),v=2(2a3a4(p2-p12)cos⁡⁡(η)cosh⁡⁡(ξ+γ)+4a3pp1a4sin⁡(η)sinh⁡⁡(ξ+γ)+4a4p2-a32p12a4)×([a3cos⁡⁡(η)+2a4cosh⁡⁡(ξ+γ)]2)-1,
where ξ=p(x+y/2+αt), η=p1(x+2y-αt), γ=ln⁡a4, eiθ=(iα+4p)/(iα-4p), and p, p1, α, a3, and a4, are given by (9). Note that if (u(x,y,t),v(x,y,t)) is the solution of DSI equation, then (u(x,-y,t),v(x,-y,t)) is the solution as well. So, we also obtain solution of DSI equation:
(11)u1=a2ei(θ+a2t)2cosh⁡[ξ1+γ+iθ]+(a3/a4)cos⁡(η1)2cosh⁡(ξ1+γ)+(a3/a4)cos⁡(η1),v1=2(2a3a4(p2-p12)cos⁡⁡(η1)cosh⁡⁡(ξ1+γ)+4a3pp1a4sin⁡(η1)sinh⁡⁡(ξ1+γ)+4a4p2-a32p12)×([a3cos⁡⁡(η1)+2a4cosh⁡⁡(ξ1+γ)]2)-1,
where ξ1=p(x-y/2+αt), η1=p1(x-2y-αt), and γ=ln⁡a4. Solution (11) is the homoclinic solution of DSI equation. Indeed, we have
(12)(u1,v1)⟶(a2exp⁡(i(a2t+2θ)),0),kkkkkkkkkkkkkkkkast⟶+∞;(u1,v1)⟶(a2exp⁡(ia2t),0),ast⟶-∞,
where 2θ is a phase shift and (aexp⁡(ia2t),0) is a fixed circle of DSI [30]. Note that solution (11) contains not only a periodic wave cos⁡(p1(x-2y-αt)), so its amplitude periodically oscillates with the evolution of time (the breather effect), but also a solitary wave 1/cosh⁡(p(x-y/2+αt)+γ), which shows that interaction between a solitary wave and a periodic wave with the same velocity α and opposite propagation direction can form a new family of homoclinic solution. This is a new phenomenon of evolution of a three-dimensional disturbance in the nonlinear regime of plane Poiseuille flow (Figure 1).(a) Homoclinic breather wave of|u1| in DSI. (b) Homoclinic breather wave of v1 in DSI.
(a)
(b)In the above two cases, seta3=-(2/21)(441r2-1680p2)/(16p2+r2) and a4=1, when α=(8/39)195a. Let p→0, and we can obtain two rogue wave solutions for DSI as follows (Figures 2 and 3):
(13)u1=a2exp⁡(ia2t)×(12(861α4t2+(-1344yt-42xt)α3kk+(2184xy+861x2+1869y2)α2kk-18560α4t2)×(861α4t2+(-1344yt-42xt)α3k+(2184xy+861x2+1869y2)α2k+8320α4t2)-1+i(-6720α2t+(-3360y-6720x)α)×(861α4t2+(-1344yt-42xt)α3k+(2184xy+861x2+1869y2)α2k+8320861α4t2+(-1344yt-42xt)α3)-112),v1=-84α2(-35259α4t2kkkk+(50736yt-1722xt)α3kkkk+(35301x2+89544xy+36939y2)α2kkkk-341120α4t2)×(12(861α4t2+(-1344yt-42xt)α3kk+(2184xy+861x2+1869y2)α2kk+8320α4t2)212)-1,(14)u2=a2exp⁡(ia2t)×(12(861α4t2+(1344yt-42xt)α3k+(-2184xy+861x2+1869y2)α2k-18560α4t2)×(861α4t2+(1344yt-42xt)α3k+(-2184xy+861x2+1869y2)α2k+8320α4t2)-1+i(-6720α2t+(3360y-6720x)α)×(861α4t2+(1344yt-42xt)α3k+(-2184xy+861x2+1869y2)α2k+8320α4t2)-112),v2=-84α2(-35259α4t2kkkk+(-50736yt-1722xt)α3kkkk+(35301x2-89544xy+36939y2)α2kkkk-341120α4t2)×(12(861α4t2+(-1344yt-42xt)α3kk+(-2184xy+861x2+1869y2)α2kk+8320α4t2)212)-1,
where α=(8/39)195a.(a) Homoclinic rogue wave|u1| in solution (13). (b) Homoclinic rogue wave v1 in solution (13).
(a)
(b)(a) Behaviour of|u2| in solution (14). (b) Behaviour of v2 in solution (14).
(a)
(b)Both(u1,v1) and (u2,v2) are rational homoclinic (rogue) wave solutions of DSI equation. In fact, we have (uj,vj)→((a/2)exp⁡(ia2t),0) as t→∞, j=1,2.


## 3. Homoclinic Breather and Rogue Wave Solution of DSII

As we know, the(ae-2a2it,0) is hyperbolic fixed cycle of DSII equation when the period of y is larger than the period of x [30]. Similar to the argument in [30], we can analyze the linear stability of fixed cycle (ae-2|a|2it,0). Similar to the dealing with process of (2), by means of transformation of functions
(15)U=GF,V=-2(ln⁡F)xx,
Equation (3) can be converted into the bilinear form
(16)(iDt+Dx2-Dy2)G·F=λG·F,(Dx2+Dy2+λ)F·F=2GG*,
where G is a complex function and F is a real. Now, we take the following ansatz:
(17)G=ae-2a2it[e-p2(mx+ny+αt)+b1cos⁡p1(kx+ly-αt)kkkkkkk+b2ep2(mx+ny+αt)],F=e-p2(mx+ny+αt)+b3cos⁡p1(kx+ly-αt)+b4ep2(mx+ny+αt),
where p1, p2, m, n, k, l, α, b3, and b4 are real and b1, b2 are complex. Substituting (17) into (16), then we obtain
(18)λ=2a2,p22=(218(k2-l2)(km-nl)2a2k+(l3n-(km+m2)l2kkk+k2nl+n2k2-k3m)α212)×(124(k3m-k2nl-n2k2kkkkk+l2km+m2l2-l3n)kk×(km-nl)212)-1,p12=(m2-n2-2nl+2km)p22k2-l2,b1=b3(iα+2p2mk-2p2nl)iα-2p2mk+2p2nl,b2=b4(iα+2p2mk-2p2nl)2(iα-2p2mk+2p2nl)2,b32=-4b4×(12(m2-n2)(mk-nl)2p22kkl-α2(2mk-n2-2nl+m2)12)×(12(2mk-n2-2nl+m2)kk×((4m2k2-8nlmk+4n2l2)p22+α2)12)-1,kkkkkkkkkkkkkkkkkkkkkkkkknl+km=0.
So we can obtain breather solution of DSII equation as follows:
(19)U=ae-i(2a2t-θ)×(122b4cosh⁡(p2(mx+ny+αt)+iθ+ϕ)kk+b3cos⁡(p1(kx+ly-αt))12)×(122b4cosh⁡(p2(mx+ny+αt)+ϕ)kkk+b3cos⁡(p1(kx+ly-αt))12)-1,V=-2(2b4kkklk×cosh⁡(p2(mx+ny+αt)+ϕ)p22m2kkklk-b3cos⁡(p1(kx+ly-αt))p12k2b4)×(122b4cosh⁡(p2(mx+ny+αt)+ϕ)kk+b3cos⁡(p1(kx+ly-αt))12)-1+2((2b4sinh⁡(p2(mx+ny+αt)+ϕ)p2mkkkk-b3sin(p1(kx+ly-αt))p1kb4)2)×(12(122b4cosh⁡(p2(mx+ny+αt)+ϕ)kkkl+b3cos⁡(p1(kx+ly-αt))122b4cosh⁡(p2(mx+ny+αt)+ϕ))212)-1,
where θ=arctanb1andϕ=ln⁡(b4).The rogue wave of the DSII system is derived when the period of periodic wave goes to infinite. Indeed, by lettingp1→0 in solution (19), solution (19) becomes rogue waves:
(20)U=ae2ia2t×(12-(mx+ny+αt)2+2(kx+ly-αt)2kll+16i(mx+ny+αt)δ1δ2+δ412)×(122(mx+ny+αt)2δ1kk+2(kx+ly-αt)2+δ312)-1,V=-2(4m2δ1+4k2)×(122(mx+ny+αt)2δ1kk+2(kx+ly-αt)2+δ312)-1+2(124(mx+ny+αt)δ1mkkll+4(kx+ly-αt)k12)2×(+2(kx+ly-αt)2+δ312)212(122(mx+ny+αt)2δ1kkl+2(kx+ly-αt)2+δ312)212)-1,
where
(21)δ1=k2-l2m2-n2-2ln+2mk,δ2=mk-lnα,δ3=16(k2-l2)(mk+m2-n2-ln)(mk-ln)2(m2-n2-2ln+2mk)2α2,δ4=32(k2-l2)(5mk+3m2-5ln-3n2)(mk-ln)2(m2-n2-2ln+2mk)2α2,m=1312α29(41α2+512a2),n=-1640α29(41α2+512a2).Figure4(a): The dynamical evolution of bright breather |U(x,t)| in solution (19) is plotted with parameters a=α=k=2, b4=1, and l=1.6.(a) Bright breather structure|U| in DSII. (b) Bright rogue wave |U| in DSII.
(a)
(b)Figure4(b): The dynamical evolution of bright rogue wave |U(x,t)| in solution (20) is plotted with parameters a=α=k=2, b4=1, and l=1.6.Figure5(a): The dynamical evolution of dark breather V(x,t) in solution (19) is plotted with parameters a=α=k=2, b4=1, and l=1.6.(a) Dark breather structureV in DSII. (b) Dark rogue wave V in DSII.
(a)
(b)Figure5(b): The dynamical evolution of dark rogue wave V(x,t) in solution (20) is plotted with parameters a=α=k=2, b4=1, and l=1.6.It is easy to verify that both solution (19) and (20) are solutions of DSII. Similar to DSI, we can show that solution (20) is a rational homoclinic rogue waves.


## 4. Conclusion

In summary, based on Hirota bilinear form, applying homoclinic breather limit method to DSI and DSII equations, we obtain a new kind of homoclinic solutions with locally oscillatory structure and rational homoclinic rogue wave solutions. We also investigate and exhibit the different homoclinic rogue wave structures of solutions. These results show the complexity and variety of dynamical behavior of the DS system. Following these ideas in this work, the problem needed to further study is other types of nonlinear evolution equations whether have this kind of rational homoclinic solutions.



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*Source: 572863-2014-04-16.xml*