Positon-like Solutions of Nonlinear Evolution Equations in (2+1) Dimensionsfn2

\

Chaos\ Solitons + Fractals Vol[ 8\ No[ 00\ pp[ 0890Ð0801\ 0887 9859Ð9668:87:, ! see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved

Pergamon

PII] S9859Ð9668"87#99017Ð7

Positon!like Solutions of Nonlinear Evolution Equations in "1¦0# Dimensions$ K[ W[ CHOW\ W[ C[ LAI\ C[ K[ SHEK and K[ TSO Department of Mechanical Engineering\ University of Hong Kong\ Pokfulam Road\ Hong Kong\ China "Accepted 7 June 0886#

Abstract*Positons are new exact solutions of classical nonlinear evolution equations in one spatial dimension\ such as the KortewegÐde Vries and sineÐGordon equations[ Recently\ positons have been established as a singular limit of a 1!soliton expression[ Extension to "1¦0#!dimensional phenomena "1 spatial and 0 temporal dimensions# is attempted in this work\ and positon!like solutions are obtained for the well!known KadomtsevÐPetviashvili and DaveyÐStewartson equations\ as well as less familiar examples such as the "1¦0#!dimensional integrable sineÐGordon equation[ Þ 0886 Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

Recently the positons have been introduced as new exact solutions for nonlinear evolution equations "NEEs# ð0\ 1Ł[ Such solutions have been derived for several well!known NEEs] the KortewegÐde Vries "KdV# and the sineÐGordon "sG# equations[ Positons exhibit several novel features which di}er fundamentally from the classical solitons ð2Ł] , Positons are weakly "usually algebraically\ rather than exponentially# localized[ A positon in one spatial dimension possesses a singularity in the near _eld[ , The properties regarding the eigenvalue spectrum and the re~ection coe.cient are special[ , Two positons remain unchanged after mutual interactions[ During a solitonÐpositon collision\ the soliton remains unchanged but the positon is a}ected[ Both the carrier wave and the envelope of the positon experience phase shifts[ These results are originally obtained by the applications of the Darboux transformation[ Recent work has demonstrated that positons can also be derived by a special limiting procedure of a classical 1!soliton solution ð2Ł[ The main goal of the present work is to test whether the same limiting procedure is applicable to "1¦0#!dimensional NEEs "1 spatial and 0 temporal dimen! sions#[ In particular\ by starting with the 1!soliton expression\ new solutions are derived for , , , ,

the KadomtsevÐPetviashvili equation\ the DaveyÐStewartson equations\ the integrable "1¦0#!dimensional sineÐGordon equations\ a special system of coupled NEEs in "1¦0# dimensions[  Author for correspondence[ $ Communicated by Prof[ Hao Bai!Lin[ 0890

0891

K[ W[ CHOW et al[

A remark regarding this limiting process is in order[ One starts from a 1!soliton solution ð2Ł but allows for , complex conjugate wave numbers "a¦ib\ a−ib#\ , a pair of suitably chosen complex phase factors\ , a long wave limit "a:9#[ A related solution of KdV\ called the negaton\ can be derived following a similar procedure\ except one now chooses a¦ib\ −a¦ib and lets b:9 ð2Ł[ A negaton also possesses a singularity in the near _eld but decays exponentially at a large distance[ The associated eigenvalue spectrum is negative[ Collisions of negation with solitons or other negatons in one dimension can display peculiar properties ð3Ł[ To be precise\ the positon and the negaton solutions for the KortewegÐde Vries equation ut ¦5uux ¦uxxx 9\

"0#

fsin"bx¦b2 t#−b"x¦2b1 t#\ u1"log f #xx \

"1#

fsinh"ax−a2 t#¦a"x−2a1 t#\ u1"log f #xx \

"2#

are given by

respectively[ Our contribution can now be explained in a sequence of steps] , The use of complex conjugate wave numbers a2ib and the special limit b:9 will also yield special solutions[ For KdV this process reproduces the negaton solution in eq[ "2#[ , The same procedure is now applied to the KadomtsevÐPetviashvili "KP# equation to generate a new solution of KP[ , To test the universal nature of this technique\ we apply it to the dark 1!soliton of the DaveyÐ Stewartson equations[ , The same limiting process is applicable to the integrable "1¦0#!dimensional sineÐGordon equation\ and a novel solution that is _nite everywhere is generated[ , In an attempt to remove the constraint of integrability\ the identical procedure is attempted on a special system of coupled NEEs[ This system reduces to KdV for xy\ possesses dromion solutions\ but is not entirely understood yet[ 1[ THE KADOMTSEVÐPETVIASHVILI EQUATION

Throughout this paper\ we shall use the term positon and negaton whenever solutions are obtained from a 1!soliton expression by this special limiting procedure[ This vague terminology should not create any confusion\ provided one understands that more intensive work on the relationship between these new solutions\ the scattering transform and the eigenvalue spectrum of the NEEs still remains to be performed[ A 1!soliton solution for KP with negative dispersion\ "ut ¦5uux ¦uxxx #x −2uyy \

"3#

f0¦exp"f0 #¦exp"f1 #¦M01 exp"f0 ¦f1 #\

"4#

is

ai aj "ai −aj #1 −"aj bi −ai bj #1 :ai aj 2bn1 2 \ f a x¦b y− a ¦ t¦fn"9# \ Mij  n n n n an ai aj "ai ¦aj #1 −"aj bi −ai bj #1 :ai aj

0

1

where i\ j and n are positive integers ð4Ł[ On choosing a0 a1 a¦ib\ b0 b1 p¦iq\

"5#

Positon!like solutions of nonlinear evolution equations

0892

Fig[ 0[ Positon of KP eq[ "7#\ u vs x and y\ b0\ q0\ m9\ t9[

"q−mb#1 ¦b3

exp"f0"9# #ar exp"−ij#\ exp"f1"9# #ar exp"ij#\ r1 

b5

\

"6#

the positon can be obtained by taking the limit a\ pma:9] b2 "x¦my−v9 t# \ u1"log f#xx \ fcos"bx¦qy−V9 t−j#− z"q−mb#1 ¦b3 v9 −2b1 ¦

2"1mqb−q1 # b1

\ V9 −b2 ¦

"7# 2q1 [ b

"8#

A multiplicative factor 1ar is discarded in f as it plays no part in the _nal solution[ One recovers the solution given in eq[ "1# by assuming that mq9\ jp:1[ This is illustrated in Fig[ 0[ For KdV\ one can easily verify that the choice of the complex wavenumbers a2ib in a 1! soliton solution with b:9 reproduces the negaton of eq[ "2#[ This choice is slightly di}erent from that used earlier in the literature ð2Ł[ On repeating the process with eqs[ "4Ð5#\ a negaton for KP is exp"f0"9# #

0

0

fcosh ax¦py− a2 ¦

i i \ exp"f1"9# #− \ sb sb

2p1 x¦ny−ð2a1 ¦2"1pna−p1 #:a1 Łt t − \ a s

11

"09#

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K[ W[ CHOW et al[

Fig[ 1[ Negaton of KP eq[ "09#\ u vs x and y\ a0\ p1[7\ n0\ t9[

"p−na#1 −a3

s1 

a5

[

"00#

In Fig[ 1\ u1"log f#xx is illustrated[ Again\ a multiplicative factor of exp"ax¦py−"a2¦2p1:a#t# has been ignored in f\ since it does not a}ect u[ This kind of simpli_cation allows a more compact form for the dependent variable f\ and will be used repeatedly throughout this paper[ The curve along which u is singular and any given line parallel to the coordinate axes might have zero\ one or two intersections[

1[0[ SolitonÐne`aton interactions The collision of these units is of interest and can be studied via higher!order solutions[ In one dimension "KdV# the soliton remains essentially unchanged while the carrier wave and the envelope of the positon both su}er phase shifts[ In two spatial dimensions we shall see that all units will be a}ected[ For simplicity only the 0!solitonÐ0!negaton collision will be examined[ More complex interactions can be performed in essentially the same manner at the expense of more algebra[ The scattering of a soliton by a negaton is elucidated by a 2!soliton expression[ Two units of solitons will be consumed in the manufacturing of the negaton[ Omitting the algebra the dynamics is governed by

Positon!like solutions of nonlinear evolution equations

0

0

fcosh ax¦py− a2 ¦

0

0894

2p1 x¦ny−ð2a1 ¦2"1pna−p1 #:a1 Łt t − a s

11

0

¦A exp a2 x¦b2 y− a22 ¦

2b21 t a2

1 10 0

Ah¦0:"Ah# x¦ny−ð2a1 ¦2"1pna−p1 #:a1 Łt − \ 1 s

1

"01#

2p1 t \ a

11

"02#

a21 a1 "a2 −a#1 −"ab2 −a2 p#1 \ A 1 1 a2 a "a2 ¦a#1 −"ab2 −a2 p#1

"03#

0

hexp ax¦py− a2 ¦

with s given by eq[ "00#[ Figure 2a and Fig[ 2b illustrate the interaction process[

1[1[ Phase shift The two {arms| of the soliton on the two sides of the negaton are not on the same straight line\ and this phase shift can be determined from eq[ "01#[ For the sake of brevity\ only one case in the parameters space will be examined closely] a2×9\ b2³9\ a×9\ p×9[ On the wave crest a2x¦b2yconstant and time tO"0#\ x\y: implies that hð0 and

0

0

f½0¦A1 exp a2 x¦b2 y− a22 ¦

2b21 t \ a2

11

"04#

since exponential functions always dominate algebraic growth for large arguments[ Scalars and linear functions in the exponential as overall multiplicative factors are ignored since these quantities will not a}ect u[ Conversely the limit x\y:− will mean that hð0 and

0

0

f½0¦exp a2 x¦b2 y− a22 ¦

2b21 t [ a2

11

"05#

The phase shift of 1 log A can thus be deduced from this information[

2[ THE DAVEYÐSTEWARTSON EQUATIONS

Complex NEEs typically display dark as well as bright solitons[ The dynamics\ methods of solution and properties of the two regimes are quite di}erent[ Hence\ as a test for the universal nature of the technique\ we examine this dark soliton regime[ We shall see that positon!like solutions continue to exist\ but a modi_cation of the limiting procedure is necessary[ The DaveyÐStewartson equations "DS# iAt ¦Axx ¦Ayy −1A1 A−1QA\ −Qxx ¦Qyy −1"AA#xx \

"06#

admit dark 1!soliton in the bilinear format "see ð4Ł# r exp"−1ir1 t#` \ Q1"log f#xx \ A f

"07#

"iDt ¦Dx1 ¦Dy1 #`[f9\ "−D x1 ¦Dy1 #f[f1r1 " f 1 −``#\

"08#

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Fig[ 2[ Collision of a soliton and a negaton of KP eq[ "01#\ u1"log f#xx vs x and y\ a0\ p1[7\ n0\ a21\ b2−0\ "a# t9\ "b# t0[

Positon!like solutions of nonlinear evolution equations

0896

`0¦a0 exp"f0 #¦a1 exp"f1 #¦f01 a0 a1 exp"f0 ¦f1 #\

"19#

f0¦exp"f0 #¦exp"f1 #¦f01 exp"f0 ¦f1 #\

"10#

fn pn x¦qn y−Vn t¦fn"9# \

"11#

3r1 pn1 ¦qn1 ¦iVn 1 1 1 1 an −exp"1ijn #− 1 \ V "p ¦q # −0 \ n n n pn ¦qn1 −iVn qn1 −pn1

0

−3r1 cos ji cos jj cos"ji −jj #¦qi qj −pi pj

fij 

3r1 cos ji cos jj cos"ji ¦jj #¦qi qj −pi pj

1

[

"12#

"13#

Here D is the Hirota operator[ We now test whether there are positon!like solutions[ An inspection on the derivation shows that one cannot use the technique of complex wavenumbers here[ Instead the only available alternative is the choice p0 p−p1 \ q0 q1 q[

"14# 1 1

3

3

The limit p:9 implies that f01½3r p :q ¦O"p #[ Hence one now chooses q1 q1 \ exp"f1"9# #− \ exp"f0"9# # 1rp 1rp

"15#

and this will yield the solution Ara exp"−1ir1 t#

q1 x:r−"aE−0:"aE# q1 x:r−"E−0:"E#

\ Eexp"qy−Vt#\

q¦iz3r1 −q1 \ Vqz3r1 −q1 \ a−exp"1ij#− q−iz3r1 −q1 q1 x 0 − E− [ r E

Q1"log f#xx \ f

0 1

"16# "17# "18#

A multiplicative factor of E has been ignored in both g and f\ since it does not a}ect the _nal solutions Q and A\ given in eq[ "07#[ In an independent calculation we verify that eqs[ "16Ð18# solve eq[ "06# by direct di}erentiation with the computer algebra software Mathematica[ Several features reveal the uniqueness of this solution[ The envelope A and the mean _eld Q are singular along the curve sinh"qy−Vt#

q1 x [ 1r

"29#

In one dimension\ positons basically only have point singularities but the additional spatial dimension does indeed provide a richer variety of solutions[ Furthermore\ eq[ "16# tends to the uniform plane wave in the far _eld algebraically in the x!direction but exponentially in the y! direction[ This is in marked contrast with the other known solutions\ e[g[ solitons\ dromions ð5Ł and algebraic lumps ð4Ł where the decay rate does not show this mixed character[

2[0[ SolitonÐpositon interactions The collision of a dark soliton and the positon!like solution in eqs[ "16Ð18# can again be studied by a 2!soliton expression\ with two units being diverted to create the positon]

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Ara exp"−1ir1 t#

q1 x:r−"aE−0:aE#¦a2 exp"p2 x¦q2 y−V2 t#x0 q1 x:r−"E−0:E#¦exp"p2 x¦q2 y−V2 t#x1

\

f12 q1 x f12 q1 x 0 0 1 1 aE\ x1  E[ ¦ −f 12 ¦ −f 12 r aE r E

x0 

"20#

"21#

The limit p:9 is taken and the parameters are −3r1 cos j cos j2 cos"j−j2 #¦qq2

f02 ½f12 

3r1 cos j cos j2 cos"j¦j2 #¦qq2

\

with p2\ q2\ V2\ j2 satisfying the dispersion relation in eq[ "12#[ As a concrete example\ consider the case p2\q2×9[ The wave trough of the dark soliton is along the line p2x¦q2yconstant[ In the second quadrant "x:−\ y:¦\ E××0#\ 1 a2 exp"p2 x¦q2 y−V2 t# 0¦f12

A½ra1 exp"−1ir1 t#

1 0¦f12 exp"p2 x¦q2 y−V2 t#

[

In the fourth quadrant "x:¦\ y:−\ Eð0#\ 0¦a2 exp"p2 x¦q2 y−V2 t# [ 0¦exp"p2 x¦q2 y−V2 t#

A½r exp"−1ir1 t#

Thus the two arms of the dark soliton have a relative phase shift of 1 log f12[

3[ THE INTEGRABLE "1¦0#!DIMENSIONAL SINEÐGORDON EQUATION

The straightforward higher!dimensional extension of the sineÐGordon equation cxx ¦cyy −ctt sin c\

"22#

does not appear to be integrable ð6Ł even though it has a 2!soliton solution under certain conditions ð7Ł[ A more reasonable generalization is the system ð8\ 09Ł

0 1 0 1 0 1 0 1 fx sin u cx sin u

−

x

−

x

fy sin u cy sin u

¦

fy ux −fx uy sin1 u

y

¦

cx uy −cy ux

y

sin1 u

9\

"23#

9\

"24#

ut f¦c[

"25#

An alternative version ð8\ 09Ł of this integrable "1¦0#!dimensional system of sineÐGordon equations is "exp"if#"f¦c#xt #x −"exp"if#"f¦c#yt #y 9\

"26#

"exp"−if#"f−c#xt #x −"exp"−if#"f−c#yt #y 9[

"27#

In terms of characteristic coordinates jy−x\ hy¦x\ the system can be rewritten as

Positon!like solutions of nonlinear evolution equations

0898

uxyt ¦ux vyt ¦uy vxt 9\ vxy ux uy \

"28#

after renaming "j\h# as "x\y#[ Equation "28# has been investigated earlier in the literature in an entirely di}erent context ð00\ 01Ł[ It can be shown ð09Ł that eq[ "28# is equivalent to eqs[ "23Ð24#[ The bilinear forms of eq[ "28# are ui log

01

01

G ` −1 tan−0 \ GFf¦i`\ F f

"39#

vlxt¦myt¦log"GF#lxt¦myt¦log"`1 ¦f1 #\

"30#

"Dx Dy Dt ¦lDy ¦mDx #G[F9\ Dx Dy G[F9[

"31#

f0¦m01 exp"f0 ¦f1 #\ `exp"f0 #¦exp"f1 #\

"32#

The 1!soliton solution

fn pn x¦qn y−

0

1

l m "p0 −p1 #"q0 −q1 # \ ¦ t¦fn"9# \ m01 − pn qn "p0 ¦p1 #"q0 ¦q1 #

"33#

with the choice of complex wavenumbers and phase factors p0 p1 a¦ib\ q0 q1 p¦iq\ exp"fn"9# #3

i n "for n0\1#\ qnb\ s1  \ sb ap

"34#

3

"35#

and the same limiting procedure "b\ qnb:9# yield u−1 tan−0

2X

ap x¦ny¦"l:a1 ¦mn:p1 #t [ n cosh"ax¦py−"l:a¦m:p#t#

The negaton given by eq[ "35# is nonsingular for all _nite x\y\t and hence is drastically di}erent from the positon!like solution of KP and DS "Figure 3#[

3[0[ SolitonÐne`aton interactions The scattering of a kink soliton by the negaton in eq[ "35# can be studied via a 2!soliton expression[ The reasoning follows closely that of KP and DS[ The dynamics is now governed by u−1 tan−0 z0 ax¦py−

X

Z0 1

0 1

Z0 \ Z1

"36#

0 1

0

0

1

l m l mn ¦ t\ z1 x¦ny¦ 1 ¦ 1 t\ a p a p

0

"37#

11

ap l m exp"z0 #z1 ¦exp p2 x¦q2 y− ¦ t n p2 q2 ¦

Z1 0¦exp"1z0 #¦1

"a−p2 #1 "p−q2 #1 1

1

"a¦p2 # "p¦q2 #

X

0

0

exp 1z0 ¦p2 x¦q2 y−

0

11

0

11

m l ¦ t \ p2 q2

ap l m ¦ t exp"z0 #x\ exp p2 x¦q2 y− n p2 q2

"38#

"49#

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K[ W[ CHOW et al[

Fig[ 3[ Globally smooth solution eq[ "35#\ −u vs x and y\ l0\ m0\ a0\ p1[7\ n0\ t9[

"a−p2 #"p−q2 #z1 1p2 "p1 −q21 #¦1nq2 "a1 −p21 # − [ "a¦p2 #"p¦q2 # "a¦p2 #1 "p¦q2 #1

x−

"40#

This interaction can be quite complex and is illustrated in Fig[ 4[ 4[ A SPECIAL COUPLED SYSTEM

The system ð02Ł ut ¦uxxx ¦2"uv#x 9\ ux vy \

"41#

u1"log f#xy \ v1"log f#xx \ "Dy Dt ¦Dy D x2 #f[f9\

"42#

admits bilinear form

and a special exact solution ð00Ł f0¦exp"f#¦exp"F#¦exp"c#¦m exp"f¦F#¦k"exp"f¦c#¦exp"F¦c## ¦m exp"f¦F¦c#\ fpx−p2 t¦f"9# \ Fqx−q2 t¦F"9# \ cby[ The choice

"43#

Positon!like solutions of nonlinear evolution equations

0800

Fig[ 4[ An exact solution of the "1¦0#!dimensional sG\ "Equations "36Ð40##\ −u vs x and y\ l0\ m0\ a0\ p1[7\ n0\ p29[4\ q2−9[6\ t9[

pqa¦ib\ exp"f"9# #

i i \ exp"F"9# #− \ ms1 b1 \ sb sb

yields x−2a1 t k"x−2a1 t# f cosh"ax−a2 t#− ¦exp"by# cosh"ax−a2 t#− [ s s

0

1

0

1

"44#

Again a multiplicative exponential factor is ignored since it does not a}ect u and v "see eq[ "42##[ Equation "44# is singular for t³t9 but regular for t×t9\ where t9 will depend on the values of a\ k and s[ This can easily be con_rmed by elementary graphical considerations[ The equation cosh xcx¦d might have 9\ 0\ or 1 roots depending on the values of c and d[ In an independent calculation we verify by direct substitution that eq[ "44# solves eq[ "42#[ The situation here is again quite di}erent from the positons and negatons of KP and DS\ where the solutions remain singular for all time[ 5[ DISCUSSION

Positon!like solutions have been derived for intensively studied NEEs "KP\ DS# as well as for less well!known examples in "1¦0# dimensions[ They are obtained from a special limiting procedure applied to a 1!soliton solution[ Provided that a higher!order soliton expression exists\

0801

K[ W[ CHOW et al[

M!positonÐN!soliton solutions can theoretically be computed[ The procedure is robust and the phenomenon is likely to be universal[ As supporting evidence\ the theory is applicable to the dark soliton regime as well[ However\ the relation between these new solutions and the eigenvalue spectrum of the NEEs remains to be determined[ The main features of these new solutions are the following] , These solutions are generally singular along curves in the x\y plane for KP and DS\ but can be globally smooth for the "1¦0#!dimensional sG[ , Unlike the one!dimensional case\ the soliton su}ers a phase shift during interactions with these solutions[ It is not inconceivable that solutions might possess "only# point singularities in "1¦0# dimen! sions\ although we have not yet found any example[ An instructive example is provided by a special coupled KdV system[ Positon!like solutions are nonsingular beyond a certain threshold time[ We have every reason to suspect that regular "or partially regular# solutions of this nature will likely exist for other NEEs as well[ The valuable point here is that "1¦0#!dimensional NEEs display an extremely rich variety of behaviour\ and further research will certainly be fruitful[ Acknowled`ements*Partial _nancial support has been provided by the Hung Hing Ying Physical Sciences Research Fund "Grant 262:953:5587# and the Committee of Research and Conference Grants of the University of Hong Kong "Grant 262:953:9966#[

REFERENCES 0[ Matveev\ V[ B[\ Asymptotics of the multi!positon!soliton t function of the KortewegÐde Vries equation and the supertransparency[ J[ Math[ Phys[\ 0883\ 24\ 1844Ð1869[ 1[ Beutler\ R[\ Positon solutions of the sineÐGordon equation[ J[ Math[ Phys[\ 0882\ 23\ 2987Ð2098[ 2[ Jaworski\ M[\ Zagrodzinski\ J[\ Positon and positon!like solutions of the KortewegÐde Vries and sineÐGordon equations[ Chaos\ Solitons + Fractals\ 0884\ 4\ 1118Ð1123[ 3[ Rasinariu\ C[\ Sukhatme\ U[\ Khare\ A[\ Negaton and positon solutions of the KdV and mKdV hierarchy[ J[ Phys[ A\ 0885\ 18\ 0792Ð0712[ 4[ Satsuma\ J[\ Ablowitz\ M[ J[\ Two dimensional lumps in nonlinear dispersive systems[ J[ Math[ Phys[\ 0868\ 19\ 0385Ð0492[ 5[ Hietarinta\ J[\ Hirota\ R[\ Multi!dromion solutions of the DaveyÐStewartson equation[ Phys[ Lett[ A\ 0889\ 034\ 126Ð133[ 6[ Ablowitz\ M[ J[ and Clarkson\ P[ A[\ Solitons\ Nonlinear Evolution Equations and Inverse Scatterin`[ London Mathematical Society\ 0880[ 7[ Hirota\ R[\ Exact three!soliton solution of the two dimensional sineÐGordon equation[ J[ Phys[ Soc[ Japan\ 0862\ 24\ 0455[ 8[ Konopelchenko\ B[\ Solitons in Multi!Dimensions[ World Scienti_c\ 0882[ 09[ Radha\ R[\ Lakshmanan\ M[\ The "1¦0# dimensional sineÐGordon equation\ integrability and localized solutions[ J[ Phys[ A\ 0885\ 18\ 0440Ð0451[ 00[ Chow\ K[ W[\ Solito}s solutions of nonlinear evolution equations[ J[ Phys[ Soc[ Japan\ 0885\ 54\ 0860Ð0865[ 01[ Nimmo\ J[ J[ C[\ in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations\ ed[ P[ A[ Clarkson[ Kluwer Academic Press\ 0882\ pp[ 072Ð081[ 02[ Radha\ R[\ Lakshmanan\ M[\ Singularity analysis and localized coherent structures in "1¦0# dimensional generalized Korteweg!de Vries equations[ J[ Math[ Phys[\ 0883\ 24\ 3635Ð3645[