On the Rogue Wave Solution Of the Davey-Stewartson Equation

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

On the Rogue Wave Solution of the Davey-Stewartson Equation 1 D. Prasanna1, S. Selvakumar2, Dr. P. Elangovan1 1 – PG & Research Dept. of Physics, Pachaiyappa’s College, Chennai, Tamilnadu, India 2 – PG & Research Department of Physics, Government Arts College, Ariyalur, Tamilnadu, India a – dprasanna85@gmail.com DOI 10.2412/mmse.78.59.591 provided by Seo4U.link

Keywords: Rogue wave, mathematical physics, nonlinear.

ABSTRACT. Constructing Rogue wave solution for the nonlinear evolution equations is the one of the challenging tasks for nonlinear community. Rogue wave is the deepest trough (hole) before or after the largest crest and it appears from nowhere and disappears without a trace. Benjamin-Feir Instability or Modulation Instability is responsible for the Rogue wave formation in both ocean and optics. The Mathematical model for the rogue waves is the nonlinear Schrodinger (NLS) equation. Rogue wave is one of the peculiar nonlinear waves which arises suddenly and swallow many ocean liner.

Introduction. Rogue wave is an isolated huge wave with amplitude much larger than the average wave crests around it. First observed in the ocean. Later in optics, BEC, Multi-component Plasmas and so on. Rogue wave is first recorded in 1990’s at Draupner oil platform in North Sea. Optical rogue waves are observed in nonlinear optical fibers by Solli et.al in 2007 [7].

Wave motion is a mode of transmission of energy through a medium in the form of disturbance, so wave is a disturbance which travels through space and time. Waves are important in all branches of physical and biological science. Indeed the wave concept is one of the most important unifying threads running through the entire fabric of the natural sciences. There are two broad category of waves (i). linear waves and (ii). nonlinear waves. Linear waves are governed by linear evolution equations which are linear partial differential equations (PDEs) and hence the superposition principle is valid. Wave travels with constant velocity is known as phase velocity (vp) and is independent of wave number vp ≠ vp(k) vp = ω k (1.1) and the wave group travel with the velocity (vg) which is known as group velocity which is given by, vg = dω / dk. If the group velocity is not same as the phase velocity then each component of the wave group travel with its own velocity and after certain time each component dies down in due course which is due to the dispersion phenomenon, such type waves are called as dispersive waves. This type of waves has less permanent life because of dispersion. The causes of formation of this types of waves due to earthquake, storms, and so on. Some examples for the nonlinear waves are: cyclonic waves, tsunami waves, tidal waves, electromagnetic waves in nonlinear optical fibres, solitary waves on shallow water surface and rogue waves [1].

Definition of the Rogue Wave: Rogue wave - it has deepest trough (hole) before or after the largest crest and it appears from nowhere and disappears without a trace. Soliton - localized wave, which propagate large distance without change of its size.

Causes of the Rogue wave formation: Earthquakes Ocean plates moving the quakes and the vibrations make ripples, some of which are large enough to form a rogue wave. Storms When strong winds from a storm happen to blow in the opposing direction of the ocean current the forces might be strong enough to randomly generate rogue wave.

© 2017 The Authors. Published by Magnolithe GmbH. This is an open access article under the CC BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/

MMSE Journal. Open Access www.mmse.xyz

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

Mathematical model for the Rogue waves: (1 + 1)-dimensional Nonlinear Schrodinger equation is: iqt + qxx ±2|q|2q = 0 Rogue wave solution of the NLS equation: q = [1−4( 1 + 2ix \1 + 4x2 + 4t2)]eix Mechanisms of Rogue wave formation: Breathers 1. Ma breathers, 2. Akhmediev breathers. Ma breathers are periodic in time and Akhmediev breathers are periodic along space. If we increase the modulation parameter, the temporal separation between adjacent peaks increases. When this parameter reaches to the particular critical value, the rogue wave arises.

Two soliton solution: Taylor series expansion of two-soliton solution of the DS equation.

Ma breather solution:

a)

b)

c)

d)

Fig. 1. Rogue wave solution arises from the Ma breather solution with the modulation parameter value (a) ϕ = 0:9; (b) ϕ = 0:8; (c) ϕ = 0:6; (d) ϕ = 0:1.

MMSE Journal. Open Access www.mmse.xyz

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

Akhmedive breather solution.

a)

b)

c)

d)

Fig. 2. Rogue wave solution arises from the Akhmedive breather solution with the modulation parameter (a) a= 0.1; (b) a= 0.2; (c) a=0.3; (d) a=0.4.

Rogue wave solution of the DS Equation: (2+1)-dimensional NLS equation or Davey-stewartson equation iqt + aqxx + qyy + b|q|2 q − 2qp = 0 apxx − pyy − ab(|q|2 )xx = 0

(1) (2)

where a= ± 1; b – constant; q – complex field; p – real field Two-soliton solution of the DS Equation: The bilinear transformation is: q = g/ f, p = −2a (log f)xx

(3)

Boundary condition is: |q|2 → qo2 substituting the equation (3) into the equation (1) and (2) We get the bilinear form of the DS equation (iDt + aDx2 + Dy2 − bqo2) g .f = 0 MMSE Journal. Open Access www.mmse.xyz

(4)

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

(aDx2− Dy2 − bqo2) f .f = −bgg ∗

(5)

We introduce Hirota expansion for the function g and f as, f = (1 + ϵ1 f1 + ϵ2 f2 + ϵ3 f3 + ...)

(6)

g = g0 (1 + ϵ1g1 +ϵ2g2+ ϵ3g3 + ...)

(7)

Substituting g and f in the bilinear equation (4) and (5) we have, ϵ0: (i∂t + a∂2x + ∂2y − bq02) (g0.1) = 0 (a∂x2 − ∂y2 – bq02)(1.1) = −bg0g 0*

(8a) (8b)

ϵ1: [i(Dt + 2kDx + 2lDy) + aDx2 + Dy2] (1.f1 + g1.1) = 0

(9a)

[(aDx2 – Dy2 − bq02)] (1.f1 + f1.1) = −bq02 (1.g1* + g1.1)

(9b)

ϵ2: [i(Dt + 2kDx + 2lDy ) + aDx2 + Dy2] + (1.f2 + g2.1 + g1.f1) = 0

(10a)

[aDx2 – Dy2 − bq02)] (1.f2 + f2.1 + f1.f1) = −bq02 (1.g2* + g1.g1* + g2.1)

(10b)

From the equation (8b) we get g0 = q0e iξ ξ = kx + ly − ωt + ξ (0) Substituting g0 in the equation (8a) we get the dispersion relation is ω = ak2 + l2 − bq02

(11)

for the one soliton solution from the equation (9a) and (9b) we choose f1 = e η1 g1 = eη1+iφ1 For the two soliton solution by assuming: f1 = eη1 + eη2 g1 = eη1+iφ1 + eη2+iφ2 Substituting the above equation into the equations (10a) and (10b) we get, f2 = Deη1+η2 g2 = Deη1+η2+i(φ1+φ2) Therefore f = 1 + eη1 + eη2 + Deη1+η2 g = q0eiξ (1 + eη1+iφ1 + eη2+iφ2 + Deη1+η2+i(φ1+φ2) MMSE Journal. Open Access www.mmse.xyz

(12) (13)

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

where, ηj = Kjx + Ljy − Ωjt + ηj0 Ωj = 2akKj + 2lLj − (aK j2 + Lj2 ) cot φj /2

(14)

sin2 (φj/ 2) = aKj2 – Lj2/ 2bq02; j = 1, 2.

(15)

D = [ (−2bq02 sin(φ1 /2 ) cos(φ1−φ2/ 2) + aK1K2 + L1L2) / (−2bq02 sin(φ1 /2 ) cos(φ1+φ2/ 2) + aK1K2 + L1L2)] (16) Rogue wave solution of the DS equation: Putting K1 = K2* = iϵc , L1 = L2* = iϵd and η10 = η20* = ϵ(iθ ˜ − σ˜ ) + iπ. φ1 = φ2 = ±2ϵ d2 − ac2 /2bq02

(17)

Ω = ϵ [γ˜ + (ac2 + d2) 2bq02 d2 − ac2]

(18)

γ = ϵ (2akc + 2ld) = ϵγ˜

(19)

D = 1 + ϵ2 (d2 − ac2 2bq02) + o(ϵ3 ) = 1 + ϵ2α2 + o(ϵ3)

(20)

g = q0e i(kx+ly−ωt)[ ϵ2( ξ2 + η2 + α2 − 4α(α ± iη) + o(ϵ3)

(21)

f = ϵ2 (ξ2 + η2+ α2) + o (ϵ3)

(22)

where ξ = cx + dy − γ˜t + θ ˜ η = Ω˜t + ˜σ α2 = d2 − ac2 /2bq02 Rogue wave solutions: Substituting the equation (21) and (22) into the equation (3). q = q0e i(kx+ly−ωt) [1 – (4α(α ± iη)) /(ξ2 + α2 + η2)]

(23)

−4ac [(η + α − ξ ) / (ξ + η + α ) ]

(24)

2

2

2

2

2

2

2 2

MMSE Journal. Open Access www.mmse.xyz

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

a)

(b)

Fig. 3. The Rogue waves solution of the Davey-Stewartson equation with the parameter values a = 1, r = 1; k = 1, l = 1, c = 1 2, and d = √ 1/ 2. (a) t= - 0.05; (b) t=-0.2.

Summary. We have obtained two-soliton solution of the DS equation through bilinear method. By choosing wave number as pure imaginary and by making Taylor expansion of two-soliton solution we obtained first order Rogue wave solution of the DS equation.

References [1] C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue wave in the Ocean, Springer-Verky Berlin Heidelberg, 2009.

[2] M. Tajiri, T. Arai, Growing-and-decaying mode solution to the Davey-Stewartson equation, Phys.

Rev E 60 (1999).

[3] J. Satsuma, M.J. Ablowitz, Two‐dimensional lumps in nonlinear dispersive systems, J.Math. Phys 20 (1979), DOI 10.1063/1.524208 [4] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004 [5] N. Akhmedive, A.Ankiewicz, N. Akhmedive, A.Ankiewicz, Phys. Rev E 80 (2009), Phys. Rev E 80 (2009), DOI 10.1103/PhysRevE.80.026601 [6] M. Lakshmanan, S. Rajasekar, Nonlinear Dynamics, Integrability, Chaos and Pattertns, SpringerVerlag, Berlin 2003 [7] D. R.Solli, C. Ropers, P. Koonath & B. Jalali, Optical Rogue Waves, Nature 450, 1054-1057 (2007), DOI 10.1038/nature06402

MMSE Journal. Open Access www.mmse.xyz