J. Math. Anal. Appl. 380 (2011) 689–696

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Journal of Mathematical Analysis andApplications

www.elsevier.com/locate/jmaa

Rogon-like solutions excited in the two-dimensional nonlocal nonlinearSchrödinger equation

Zhenya Yan

Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 July 2010Available online 15 February 2011Submitted by P. Broadbridge

Keywords:Nonlocal NLS equationSimilarity transformationRational-like solutionsRogue wavesRogons

We report the analytical one- and two-rogon-like solutions for the two-dimensionalnonlocal nonlinear Schrödinger equation by means of the similarity transformation. Theseobtained solutions can be used to describe the possible physical mechanisms for rogue-like wave phenomenon. Moreover, the free function of space y involved in the obtainedsolutions excites the abundant structures of rogue-like wave propagations. The Hermite–Gaussian function of space y (normalized function) is, in particular, chosen to depict thedynamical behaviors for rogue-like wave phenomenon.

© 2011 Elsevier Inc. All rights reserved.

1. Introduction

Recently, the rogue wave (RW, also known as ‘rogon’ for the special case [1]) phenomena have attracted more andmore attention from the different point of views such as marine observations, laboratory experiments, and theoreticalanalysis [2–14], since they exist in many fields of nonlinear science, such as the ocean (see a collection of rogue waveobservations [3,9]), the optical fibers [14–23], and the Bose–Einstein condensates [17,24,25], the atmosphere [26], and eventhe finance [27,28]. Moreover, numerous news media successively reported the RW phenomenon such as Nature News, BBCNews, PhysOrg, Reuters, ScienceDaily, Physicsworld, Physics, the Financial Express, ScientificAmerican, etc. It is so importantto further understand and to even manage the physical mechanisms for RW phenomenon in order to suppress them in theharmful aspects (e.g. oceanic RWs [3,9]) and to excite them in the useful fields (e.g. optical RWs to stimulate supercontinuumgeneration [14,16] or more useful examples in future).

RWs are also named as freak waves, monster waves, killer waves, giant waves, huge waves, super waves, gigantic waves,or extreme waves, etc. [4]. There is not a uniform definition for the mentioned above rogue waves. It is often defined as:the height of the rogue wave should exceed the significant wave height in 2–2.2 times [9]. Based on the nonlinear theorymechanisms of RW occurrence, which can more effectively explain the RWs than the linear mechanisms, the basic nonlinearmodel describing the RW phenomenon is, in the dimensionless form, the focusing nonlinear Schrödinger (NLS) equation [4]

i∂ψ

∂t+ ∂2ψ

∂x2+ 2|ψ |2ψ = 0, (1)

whose first exact rational (rogon) solution (also called Peregrine breather or algebraic breather) related to the rogue wavephenomenon was found by Peregrine [29] in the envelope rational function

ψ(x, t) = e2it[

1 − 4(1 + 4it)

1 + 4x2 + 16t2

], (2)

E-mail address: zyyan@mmrc.iss.ac.cn.

0022-247X/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2011.01.071

690 Z. Yan / J. Math. Anal. Appl. 380 (2011) 689–696

by considering a double Taylor series expansion about the amplitude peak of the Ma solitons [30]. Moreover, the Akhme-diev breathers [31,32] were also proposed for Eq. (1) by using the Darboux transformation (see [33] for details). The 2D,3D NLS equation and the coupled NLS equations were also used to illustrate the possible formation mechanisms for RWphenomenon [4,15,34–39].

In 2007, Solli et al. [16] showed that the RW phenomenon could be observed in the optical fiber and presented theconcept of the optical rogue waves by studying the generalized NLS equation [40]

∂ψ

∂z− i

∑m=2

imβm

m!∂mψ

∂tm= iγ

[|ψ |2ψ + i

ω0

∂

∂t

(|ψ |2ψ) − T Rψ∂|ψ |2

∂t

](3)

which is the extension of the NLS equation (1) with the self-steeping term and self-frequency shift, where βm ’s denotethe fiber dispersion coefficients, γ is the nonlinear coefficient, ω0 is the central carrier frequency of the field, and T Rdescribes the delayed nonlinear response of silica fiber. More recently, Akhmediev et al. [19] developed a direct approachfor generating the whole hierarchy of rational (rogon) solutions of NLS equation to describe the possible mechanisms for RWphenomenon [20–22]. Ankiewicz et al. [23] also extended the idea to illustrate the rogue wave solutions of the higher-orderperturbing NLS equation [40,41]

i∂ψ

∂x+ 1

2

∂2ψ

∂t2+ |ψ |2ψ + iε

[a∂3ψ

∂t3+ b

∂

∂t

(|ψ |2ψ) + cψ∂|ψ |2

∂t

]= 0, (4)

which is the generalization of the NLS equation (1) with the three-order dispersion term a(∂3ψ)/(∂t3), self-steeping termb(∂(|ψ |2ψ))/(∂t) and self-frequency shift cψ(∂|ψ |2)/(∂t). Recently, another experiment also showed that optical roguewaves existed in nonlinear optical fiber [18].

Subsequently, the matter and discrete RWs were also considered numerically in [24,25]. Furthermore, we introduced theconcept of the rogons (or freakons) for the rogue waves (or freak waves) with the property of the elastic interaction and alsoobtained the nonautonomous rogon solutions of the inhomogeneous NLS equation with variable coefficients to describe thepossible formation mechanisms for rogue-like wave phenomenon [1]. Moreover, three-dimensional rogue waves have alsobeen obtained for the three-dimensional Gross–Pitaevskii (or NLS) equation with variable coefficients [17].

2. Nonlocal nonlinear model

The model under the consideration is to focus on the two-dimensional nonlocal NLS equation [42,43]

i∂ A

∂t= −∂2 A

∂x2− σ A

∞∫−∞

|A|2 dy, (5)

where A ≡ A(x, y, t) is a two-dimensional field envelope and σ > 0 is the nonlinearity coefficient. Recently, based on thebilinear transformation [44], the multi-soliton solutions of Eq. (5) with σ = 2 and t → −t were given in terms of Gram-typedeterminant [43]. Eq. (5) is, in fact, a nonlinear differential-integral equation. To the best of our knowledge, the analyticalrogon-like solutions of Eq. (5) were not reported before. A natural problem is whether Eq. (5) possesses rogon-like solutionssimilar to the NLS equation. In this paper, we would like to answer the problem so that the rogon-like solutions of Eq. (5)are obtained by using the generalized form of the similarity reduction [45–49] and the direct ansatz [1,17,19,22,27].

3. Similarity reduction and rogon-like solutions

To construct the analytical rogon-like solutions of Eq. (5), we employ the two-dimensional similarity transformation[45,49]

A(x, y, t) = g(y)eiϕ(x,y,t)φ(η(x, t), τ (t)

), (6)

to Eq. (5) and obtain

iτt∂φ

∂τ+ η2

x∂2φ

∂η2+ σ |φ|2φ

∞∫−∞

∣∣g(y)∣∣2

dy + ηxx∂φ

∂η+ i(ηt + 2ϕxηx)

∂φ

∂η− (

ϕt + ϕ2x

)φ + iϕxxφ = 0, (7)

where g(y), η(x, t), τ (t), ϕ(x, y, t) and φ(η, τ ) are functions of the indicated variables to be determined.If we require that the unknown functions η(x, t), τ (t), ϕ(x, y, t) and g(y) satisfy the following constraints

ηxx = 0, ϕxx = 0, ηt + 2ϕxηx = 0, ϕt + ϕ2x = 0, τt = η2

x , (8a)

σ

∞∫ ∣∣g(y)∣∣2

dy = Gη2x , (8b)

−∞

Z. Yan / J. Math. Anal. Appl. 380 (2011) 689–696 691

Fig. 1. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A1|2 given in Eq. (15) forα = ω = 1.0, n = 0 with k = 0. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

where G is a constant and sign(G) = sign(σ ), then Eq. (7) reduces to the one-dimensional standard NLS equation withconstant coefficients

i∂φ(η, τ )

∂τ+ ∂2φ(η, τ )

∂η2+ G

∣∣φ(η, τ )∣∣2

φ(η, τ ) = 0, (9)

whose multi-rogon solutions have been presented for G > 0 such as one-rogon (Peregrine breather, algebraic breather,or first-order rational) solution [1,17,20,29], two-rogon (second-order rational) solutions [1,17,19,20,22], and multi-rogon(higher-order rational) solutions [19].

It follows from system (8) that after some algebra, these two new independent variables (η(x, t), τ (t)) and phaseϕ(x, y, t) are given by

η(x, t) = α(x − 2kt), τ (t) = α2t, ϕ(x, y, t) = kx − k2t + c(y), (10)

and the wave function g(y) satisfies the following integral equation

∞∫−∞

∣∣g(y)∣∣2

dy = Gα2/σ , (11)

i.e. the integration of |g(y)|2 is a bounded real constant on the whole coordinate axes (−∞,∞), where α, k are freeparameters, and c(y) is a free function of space y.

For the special case σ = Gα2, Eq. (11) is just the normalization condition for the wave function g(y), i.e.∫ ∞−∞ |g(y)|2 dy = 1. There, in fact, exist many types of functions for g(y) to solve Eq. (11) (e.g. the Hermite–Gaussian

function, the hyperbolic secant function, etc.). Therefore for the chosen function g(y) given by Eq. (11), we can obtain therogon-like solutions of Eq. (5) in terms of the similarity transformation (6) with Eq. (10) and the rogon (rational) solutionsof the NLS equation (9).

To illustrate the dynamical behaviors of the obtained rogon-like solutions of Eq. (5) we choose the wave function g(y)

as the Hermite (also called Hermite–Gaussian) function in the form [50]

g(y) = 1√n!2n

√π

Hn(ωy)e−ω2 y2/2, (12)

where Hn(ωy) is the Hermite polynomial [50] and ω �= 0 is a constant, which is substituted into Eq. (11) to yield thefollowing condition for the parameters α, ω and σ :

∞∫−∞

∣∣g(y)∣∣2

dy = |ω|−1 = α2/σ (G = 1), i.e. σ = α2|ω| > 0. (13)

692 Z. Yan / J. Math. Anal. Appl. 380 (2011) 689–696

Fig. 2. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A1|2 given in Eq. (15) forα = ω = 1.0, n = 0 with k = 1. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

Fig. 3. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A1|2 given in Eq. (15) forα = ω = 1.0, n = 1 with k = 0. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

3.1. One-rogon-like solutions

Thus based on the obtained similarity transformation (6) with Eqs. (10)–(12) and the one-rogon solutions of the NLSequation (9), we arrive at the one-rogon-like (first-order rational-like) solutions of Eq. (5) in the form

A1(x, y, t) = 1√n!2n

√π

Hn(ωy)e−ω2 y2/2[

1 − 4 + 8iα2t

1 + 2α2(x − 2kt)2 + 4α4t2

]ei[kx+(α2−k2)t+c(y)], (14)

whose intensity can be written as

∣∣A1(x, y, t)∣∣2 = 1

n!2n√

πH2

n(ωy)e−ω2 y2 [2α2(x − 2kt)2 + 4α4t2 − 3]2 + 64α4t2

[1 + 2α2(x − 2kt)2 + 4α4t2]2, (15)

which involves four free parameters n, ω, α, k to control the different types of rogue-like wave propagations.Figs. 1 and 2 display the cross-sections of the density distribution (15) for the parameters α = ω = 1, n = 0 and k = 0,1

in (x, t, y = 1)-, (x, y, t = 0)-, and (t, y, x = 0)-space, respectively. Figs. 3 and 4 exhibit the cross-sections of the densitydistribution (15) for the other parameters α = ω = 1, k = 0 and n = 1,2 in (x, t, y = 1)-, (x, y, t = 0)-, and (t, y, x = 0)-space, respectively.

Z. Yan / J. Math. Anal. Appl. 380 (2011) 689–696 693

Fig. 4. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A1|2 given in Eq. (15) forα = ω = 1.0, n = 2 with k = 0. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

Fig. 5. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A2|2 given in Eq. (18) forα = ω = 1.0, n = 0 with k = 0. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

3.2. Two-rogon-like solutions

By means of the similarity transformation (6) with Eqs. (10)–(12) and two-rogon solutions of the NLS equation (9), weobtain the following two-rogon-like (second-order rational-like) solutions of Eq. (5)

A2(x, y, t) = 1√n!2n

√π

Hn(ωy)e−ω2 y2/2[

1 + P2(x, t) + i Q 2(x, t)

R2(x, t)

]ei[kx+(α2−k2)t+c(y)], (16)

with these functions P2(x, t), Q 2(x, t) and R2(x, t) being of the polynomial forms of variables x, t

P2(x, t) = −1

2α4(x − 2kt)4 − 6α6t2(x − 2kt)2 − 10α8t4 − 3

2α2(x − 2kt)2 − 9α4t2 + 3

8,

Q 2(x, t) = −α2t

[α4(x − 2kt)4 + 4α6t2(x − 2kt)2 + 4α8t4 − 3α2(x − 2kt)2 + 2α4t2 − 15

],

4

694 Z. Yan / J. Math. Anal. Appl. 380 (2011) 689–696

Fig. 6. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A2|2 given in Eq. (18) forα = ω = 1.0, n = 0 with k = 1. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

Fig. 7. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A2|2 given in Eq. (18) forα = ω = 1.0, n = 1 with k = 0. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

R2(x, t) = 1

12α6(x − 2kt)6 + 1

2α8t2(x − 2kt)4 + α10t4(x − 2kt)2 + 2

3α12t6 + 1

8α4(x − 2kt)4

+ 9

2α8t4 − 3

2α6t2(x − 2kt)2 + 9

16α2(x − 2kt)2 + 33

8α4t2 + 3

32, (17)

whose intensity can be written as

∣∣A2(x, y, t)∣∣2 = 1

n!2n√

πH2

n(ωy)e−ω2 y2 [R2(x, t) + P2(x, t)]2 + Q 22 (x, t)

R22(x, t)

, (18)

which involves four free parameters n, ω, α, k to manege the different types of rogue-like wave propagations in Eq. (5).Figs. 5 and 6 depict the cross-sections of the density distribution (18) for the parameters α = ω = 1, k = 0,1 and n = 0

in (x, t, y = 1)-, (x, y, t = 0)-, and (t, y, x = 0)-space, respectively. Figs. 7 and 8 exhibit the cross-sections of the densitydistribution (18) for the parameters α = ω = 1, k = 0 and n = 1,2 in (x, t, y = 1)-, (x, y, t = 0)-, and (t, y, x = 0)-space,respectively. In addition, we can also obtain the multi-rogon-like solution of Eq. (5) in terms of the similarity transforma-tion (6) with Eqs. (10)–(11) and multi-rogon solutions of Eq. (9) [19], which are omitted here.

Z. Yan / J. Math. Anal. Appl. 380 (2011) 689–696 695

Fig. 8. (Color online.) Cross-sections of wave propagations (top row) and contour plots (bottom row) for the density distribution |A2|2 given in Eq. (18) forα = ω = 1.0, n = 2 with k = 0. (a) and (d): (x, t)-plane with y = 1, (b) and (e): (x, y)-space with t = 0, (c) and (f): (y, t)-plane with x = 0.

4. Conclusion

In conclusion, based on the two-dimensional similarity transformation, we have reported the analytical one- and two-rogon-like solutions of the two-dimensional nonlocal nonlinear Schrödinger equation (5). These obtained solutions containa free function of space y, which leads to the abundant structures of wave propagation. The Hermite–Gaussian function ischosen to exhibit the dynamical behaviors of the obtained rogon-like solutions of Eq. (5). This may also further excite thepossibility of relative experiments and potential applications for the higher-dimensional rogon-like solutions in the field ofnonlinear science.

Acknowledgment

This work was partially supported by the NSFC60821002F02 and 11071242.

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