k.w. chow and s.y. lou- propagating wave patterns and ‘‘peakons’’ of the davey–stewartson...

8/3/2019 K.W. Chow and S.Y. Lou- Propagating wave patterns and ‘‘peakons’’ of the Davey–Stewartson system

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Propagating wave patterns and ‘‘peakons’’ of theDavey–Stewartson system

K.W. Chow a,*, S.Y. Lou b

a Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong b Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

Accepted 31 March 2005

Abstract

Two exact, doubly periodic, propagating wave patterns of the Davey–Stewartson system are computed analytically

by a special separation of variables procedure. For the first solution there is a cluster of smaller peaks within each per-

iod. The second one consists of a rectangular array of ÔplatesÕ joined together by sharp edges, and is thus a kind of

ÔpeakonsÕ for this system of (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations. A long wave limit will

yield exponentially localized waves different from the conventional dromion. The stability properties and nonlinear

dynamics must await further investigations.

Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction

The Davey–Stewartson model (DS) is an important system of evolution equations, both from the perspectives of

theory and applications. DS can arise in hydrodynamics [1] and plasma physics [2]. Theoretically, many techniques

of the modern theory of nonlinear waves are relevant, e.g., special Hirota bilinear forms [3], Darboux transformations

[4], symmetries [5], rich soliton and related structures [6,7]. We shall take DS as

io A

ot þ 1

2

o2 A

on2þ o

2 A

og2

þ vA2 AÃ ¼ QA;

o2Q

on2

Ào

2Q

og2

¼2v

o2

on2

ð AAÃ

Þ.

ð1:1Þ

In the hydrodynamic context, A is the envelope of the wave packet while Q is the induced mean flow. New coordi-

nates X , Y are defined by

n ¼ X þ Y ffiffiffi2

p ; g ¼ X À Y ffiffiffi2

p ð1:2Þ

0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2005.04.036

* Corresponding author. Tel.: +852 2859 2641; fax: +852 2858 5415.

E-mail addresses: [email protected] (K.W. Chow), [email protected] (S.Y. Lou).

Chaos, Solitons and Fractals 27 (2006) 561–567

www.elsevier.com/locate/chaos

mailto:[email protected]






and the transformed DS considered in the present work will be

io A

ot þ 1

2

o2 A

o X 2þ o

2 A

oY 2

þ vA2 AÃ ¼ QA;

2o

2Q

o X oY ¼ v

o

o X þ o

oY 2

ð AAÃÞ.

ð1:3Þ

Recently a class of novel, exact solutions of DS and other (2 + 1) (2 spatial and 1 temporal) dimensional nonlinear evo-

lution equations can be obtained by a special separation of variables approach [8]. More precisely, one exact solution of

the system (1.3) is

p ¼ p ð xÞ; q ¼ qð y Þ; p 0 ¼ p 0ð xÞ; q0 ¼ q0ð y Þ; r ¼ r ð xÞ; s ¼ sð y Þ;

Q ¼ p 0 þ q0 À p xx þ q yy

a0 þ p þ qþ ð p x þ q y Þ2

ða0 þ p þ qÞ2;

A ¼ ffiffiffi

2

v

r ffiffiffiffiffiffiffiffiffi p xq y

p a0 þ p þ q

expðiðr þ sÞÞ; r x ¼ c1 þ f

p x; s y ¼ c2 À f

q y ;

p 0 ¼ p xxx4 p x

À p 2 xx8 p 2 x

À d

8þ c2

1

2À f2

2 p 2 x; q0 ¼ q yyy

4q y À q2

yy

8q2 y

þ d

8þ c2

2

2À f2

2q2 y

.

ð1:4Þ

x ¼ X À c1t ; y ¼ Y À c2t ð1:5Þare the coordinates translating with the wave pattern. a0, c1, c2, f are constants, and the relevant functions depend on

the indicated variables only. More complicated solutions with terms involving products of p and q in the denominator

can be constructed, but details will be left for future studies.

The choice of exponential functions as the basis functions in (1.4) leads to generalized solutions of localized solitons

or dromions. The purpose of the present note is to demonstrate that the choice of the Jacobi elliptic functions as build-

ing blocks (or p 0(x), q0( y) in (1.4)) is feasible too, and will result in doubly periodic, propagating wave patterns for DS.

Two constraints will dictate the choice of elliptic functions. Firstly, the building block functions need to be nonnegative

as a square root is taken in the process. Simple choices like the functions sn and cn, which oscillate in a sinusoidal man-

ner, must be rejected. Secondly, for analytical convenience, we restrict the attention to simple cases where both A and Q

can be evaluated in simple, closed forms which do not involve the elliptic integrals in this paper. The selections of the

Jacobi elliptic function dn [9,10] and its reciprocal will satisfy these requirements, and will now lead to these two newwave patterns for DS (Section 2). Further properties like the long wave limit and the boundary conditions will also be

investigated (Section 3).

2. Doubly periodic wave patterns

2.1. First solution

By choosing both basis functions p 0(x), q0( y) in (1.4) in the x, y directions as ÔdnÕ, we obtain

A ¼ ffiffiffi

2

v

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidnða x; k Þdnðb y ; k 1Þ

p R1

expðiU 1Þ; ð2:1Þ

U 1 ¼ c1 x À fa ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 À k 2p sinÀ1 cnða x; k Þ

dnða x; k Þ þ c2 y þ f

b

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 À k 21

q sinÀ1 cnðb y ; k 1Þdnðb y ; k 1Þ

; ð2:2Þ

R1 ¼ a0 þ 1

asinÀ1½snða x; k Þ þ 1

bsinÀ1½snðb y ; k 1Þ; ð2:3Þ

Q ¼ k 2a2

4sn2ða x; k Þ À cn2ða x; k Þ À k 2sn2ða x; k Þcn2ða x; k Þ

2dn2ða x; k Þ

!þ c2

1

2À f2

2dn2ða x; k Þ

þ k 21b2

4sn2ðb y ; k 1Þ À cn2ðb y ; k 1Þ À k 21sn2ðb y ; k 1Þcn2ðb y ; k 1Þ

2dn2ðb y ; k 1Þ

!þ c2

2

2À f2

2dn2ðb y ; k 1Þ

þ ½k 2asnða x; k Þcnða x; k Þ þ k 21bsnðb y ; k 1Þcnðb y ; k 1Þ R1

þ ½dnða x; k Þ þ dnðb y ; k 1Þ2

R2

1

. ð2:4Þ

562 K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567



a, b are the wave numbers in the x and y directions respectively. k and k 1 are the distinct, independent moduli of the

Jacobi elliptic functions. The parameters c1, c2 measure the speeds of propagation in the x, y directions. The arbitrary

constant a0 must be sufficiently large to avoid any singularity. To ensure the accuracy of the solutions, direct substitu-

tion in (1.3) and differentiation for solutions (2.1)–(2.4) by this separation of variables procedure are performed inde-

pendently with the computer software Mathematica.

As expected from the formulation (2.1)–(2.4), the solution is nonsingular, doubly periodic, and translating steadily in

the x, y directions. There are clusters of smaller peaks within each period (Fig. 1).

2.2. Second solution

The second solution is obtained by choosing the reciprocal of the Jacobi elliptic function dn as the building block for

the formulation [8]. The profile for the intensity jAj2 resembles a sequence or a rectangular array of ÔplatesÕ, and is a kind

of (2 + 1) (2 spatial and 1 temporal) dimensional ÔpeakonÕ solutions of nonlinear evolution equations. As the intensity

profile of the wave pattern is continuous, but has a jump in the slope (or a sharp edge), we use the term ÔpeakonÕ [11–15]

here as well. This usage is loose in the sense that no strict analytical comparison has been performed nor implied. A

(1 + 1) dimensional ÔpeakonÕ will have a sharp corner.

More precisely, the wave packet is now:

A ¼ ffiffiffi

2

v

r 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dnða x; k Þdnðb y ; k 1Þp expðiU 2Þ

R2

; ð2:5Þ

U 2 ¼ c1 x þ f

asinÀ1½snða x; k Þ þ c2 y À f

bsinÀ1½snðb y ; k 1Þ; ð2:6Þ

R2 ¼ a0 À 1

a ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 À k 2p sinÀ1 cnða x; k Þ

dnða x; k Þ

À 1

b

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 À k 21

q sinÀ1 cnðb y ; k 1Þdnðb y ; k 1Þ

; ð2:7Þ

Q ¼ a2k 2

4dn2ða x; k Þ 1 þ ðk 2 À 2Þsn2ða x; k Þ À k 2cn2ða x; k Þsn2ða x; k Þ2

þ c2

1

2À f2dn2ða x; k Þ

2

þ b2k 21

4dn2

ðb y ; k 1

Þ1 þ ðk 21 À 2Þsn2ðb y ; k 1Þ À k 21cn2ðb y ; k 1Þsn2ðb y ; k 1Þ

2 þ c22

2À f2dn2ðb y ; k 1Þ

2

Fig. 1. Intensity jAj2 versus x, y for the first solution (2.1)–(2.4) of the Davey–Stewartson system, a = b = 1, a0 = 8, k 2 = 0.5, k 21 ¼ 0.6,

v = 2.

K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567 563



À ak 2cnða x; k Þsnða x; k Þdn2ðb y ; k 1Þ þ bk 21cnðb y ; k 1Þsnðb y ; k 1Þdn2ða x; k Þdn2ða x; k Þdn2ðb y ; k 1Þ R2

þ ½dnða x; k Þ þ dnðb y ; k 1Þ2

dn2ða x; k Þdn2ðb y ; k 1Þ R22

; ð2:8Þ

wherex

,y

are defined by Eq. (1.5),c1

,c2

are again the speeds of propagation. Fig. 2 shows that (2.5)–(2.8) represent a

kind of ÔpeakonsÕ for DS.

3. Further investigations

3.1. Long wave limit

To demonstrate that the present family of solutions is truly different from ones found earlier in the literature, an

instructive perspective is to study the long wave limits (k , k 1 going to 1) of (2.1)–(2.4). As sn, cn, dn tend to tanh, sech,

sech respectively, one nonsingular limit is

A ¼ ffiffiffi2

vr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisecha x sechb y

p R10 ! expðic1 x þ ic2 y Þ; ð3:1Þ

R10 ¼ a0 þ sinÀ1ðtanh a xÞa

þ sinÀ1ðtanh b y Þb

; ð3:2Þ

Q ¼ a2

8ð1 À 3sech2a xÞ þ c2

1

2þ b2

8ð1 À 3sech2b y Þ þ c2

2

2

þ ða tanh a x secha x þ b tanhb y sechb y Þ R10

þ ðsecha x þ sechb y Þ2

R210

. ð3:3Þ

While jAj2 described by (3.1) is still exponentially localized, it differs from the known dromion of the Davey–Stewartson

system [16,17] and deserves further studies.

Fig. 2. Intensity jAj2 versus x, y for the second solution (2.5)–(2.8) of the Davey–Stewartson system, a = b = 1, a0 = 8, k 2 = 0.2,

k

2

1 ¼ 0.3, v = 2.




3.2. Boundary conditions for the mean flow Q

Another very important aspect to consider is the boundary condition for the mean flow Q, as the conventional local-

ized dromion jAj2 is typically situated at the intersection(s) of the ÔtracksÕ provided by Q [16–19].

In deriving the localized solutions of the Davey–Stewartson equations in terms of the calculus of variations, a Ham-

iltonian integral will work smoothly for the wave packet A [16]. However, additional constraints must apply for the

mean flow Q. More precisely, the cross sections of the mean flow Q in the far field positions must match.For (3.1)–(3.3), the asymptotic forms,

Qj y !À1 ¼ a2

8ð1 À 3sech2a xÞ þ c2

1

2þ c2

2

2þ b2

8þ a tanh a x secha x

RÀþ sech2 a x

R2À

;

RÀ ¼ a0 þ sinÀ1ðtanh a xÞa

À p

2b;

Qj y !þ1 ¼ a2

8ð1 À 3sech2 a xÞ þ c2

1

2þ c2

2

2þ b2

8þ a tanh a x secha x

Rþþ sech2 a x

R2þ

;

Rþ ¼ a0 þ sinÀ1ðtanh a xÞa

þ p

2b

ð3:4Þ

are different (Figs. 3 and 4), and hence a Hamiltonian will not exist in this case. Stability of the localized solution (3.1)probably needs to be studied numerically.

3.3. Semi-localized solutions

An additional degree of freedom is to take the long wave limit in one of the moduli only (say k 1 tends to one but still

0 < k < 1):

A ¼ ffiffiffi

2

v

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidnða x; k Þsechb y

p R11

expðic1 x þ ic2 y Þ;

R11 ¼ a0 þ 1

asinÀ1ðsnða x; k ÞÞ þ sinÀ1ðtanhb y Þ

b;

Q ¼k 2a2

4 sn2

ða x; k Þ À cn2

ða x; k Þ Àk 2sn2

ða x; k

Þcn2

ða x; k

Þ2dn2ða x; k Þ !þc2

1

2 þc2

2

2

þ b2

8ð1 À 3sech2 b y Þ þ ½k 2asnða x; k Þcnða x; k Þ þ b tanh b y sechb y

R11

þ ½dnða x; k Þ þ sechb y 2

R211

.

ð3:

5Þ

Fig. 3. The mean flow Q versus x for the limit y going to negative infinity, Eq. (3.4), a0 = 4, a = 1, b = 2, v = 2.




Physically this corresponds to wave patterns localized in one direction but periodic in the other direction (Fig. 5). TheÔpeakonÕ behavior is again evident as there are sharp edges where the derivatives will be discontinuous. Depending of

the value of a0 the wave form generally is not symmetric for a cross section in a direction the pattern is localized ( y in the

present case).

4. Conclusions

In conclusions two new exact solutions of the DS system are derived, and differ from some earlier works in the lit-

erature as these new solutions are doubly periodic (versus growing and decaying, or localized, or periodic in one direc-

tion [20,21]), steadily propagating (versus standing [22,23]) waves. Indeed the Jacobi elliptic functions have been

employed earlier to obtain solutions for some (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations related

to the DS system [24]. The difference in terms of the analytic structure is that rational functions are employed in the

Fig. 5. The intensity jAj2 versus x, y for a Ôsemi-localizedÕ solution (periodic in x, localized in y), Eq. (3.5), a0 = 4, a = 1, b = 2, k = 0.5,

v = 2.

Fig. 4. The mean flow Q versus x for the limit y going to positive infinity, Eq. (3.4), a0 = 4, a = 1, b = 2, v = 2.




present separation of variables approach, as opposed to a polynomial in Jacobi elliptic functions. Furthermore, inverse

trigonometric functions of the elliptic functions are involved in the present formulation, and hence we believe that the

present approach is novel. Solutions in the form of a ÔpeakonÕ, where derivatives might be discontinuous, are possible.

In terms of applications, previously ignored physics, or higher order nonlinear effects, might be needed in the vicinity of

the sharp edges.

The long wave limits for these doubly periodic wave patterns yield an exponentially localized structure that differs

from the conventional dromion. This aspect deserves further studies. As the mismatch in far field conditions shows thata Hamiltonian probably does not exist, the stability of these structures must be investigated numerically.

Strictly speaking, the DS system considered in the present paper is the DS I system, as the mean flow Eq. (1.1) for Q

is hyperbolic. We anticipate that similar solutions will also exist for DS II and other related evolution equations [24,25],

but details will be left for future studies.

Acknowledgement

Partial financial support has been provided by the Research Grants Council contracts HKU7184/04E and

HKU7006/02E.

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k.w. chow and s.y. lou- propagating wave patterns and ‘‘peakons’’ of the davey–stewartson...

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