backlund transformation

7/26/2019 Backlund Transformation

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Backlund TransformationGraham W. Griffiths1

City University, UK.

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Contents

1 Introduction 1

2 Applications of the Backlund transformation method 2

2.1 Backlund transformation of Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Auto-Backlund transformation of the sine-Gordon equation . . . . . . . . . . . . . . . . . 6

2.3 Miura transformation of the KdV equation . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Backlund transformation of the KdV equation . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Nonlinear superposition 13

3.1 Bianchis Permutability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Nonlinear superposition applied to the KdV equation . . . . . . . . . . . . . . . . . . . . . 143.3 Three-soliton solution to the KdV equation using nonlinear superposition . . . . . . . . . 17

3.4 Nonlinear superposition applied to the sine-Gordon equation . . . . . . . . . . . . . . . . 20

1 Introduction

The Backlund transformation(BT) was introduced in 1880 by A. V. Backlund [Bac-80]as a transformation between 3-dimensional surfaces. By applying Bianchis integrabilitycondition,

2u

x1x2

2u

x2x1= 0, (1)

discussed subsequently in section (3), a Backlund transformation can be used to generatea series of solutions to a partial differential equation from a known trivial solution.

A Backlund transformation transforms a nonlinear partial differential equation into an-other partial differential equation. Thus, a solution to the second partial differentialequation must be compatible to the first partial differential equation. Hence, applicationof the Backlund transformation can provide a powerful method for generating solutionsto nonlinear PDEs.

Examples of well known Backlund transformations are [Miu-68a][Miu-68b] a) theMiuratransformation, which transforms the KdV equation into the modified KdV equation and,b) the Gardner transformation which transforms the KdV equation into the Gardnerequation.

Anauto-Backlund transformationis a transform that leaves a partial differential equationinvariant. It can also provide a method whereby a known solution to a partial differ-ential equation can be transformed into a different second solution to the same partialdifferential equation. This solution can then be used to obtain a third solution, and soon. However, it may not be possible to carry this process on to find an infinite sequence,or even many solutions, as they can repeat at some stage.

There is no known general systematic method for generating Backlund transformations,

and therefore this subject continues to be an active area of research. Detailed discussionon the origin of the Backlund transformation, along with a wide variety of applications,

1email: [email protected]

Graham W Griffiths 1 14 April 2012


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can be found in the excellent books by Rogers and Shadwick [Rog-82], and Dodd, et al.[Dod-82].

2 Applications of the Backlund transformation method

2.1 Backlund transformation of Burgers equation

Burgers equation is given by,ut+uux uxx= 0, (2)

where u = u(x, t) and is the conduction coefficient, and we wish to find a solutionusing the Backlund transformation method.

A Backlund transformation that transforms Burgers equation to the heat equation,

vt = vxx, (3)

is given byvx=

vu

2, vt =

u2v

4

vux2

, (4)

where v =v(x, t) and the first equation represents the x-part of the BT and the secondequation the t-part.

That eqn. (4) is actually the required BT can be easily demonstrated. First we differen-tiate the x-part with respect to t and the t-part with respect to x, then on subtractingwe obtain after some manipulation,

ut+ uux uxx= uvt

v

u2vx2v

+vxux

v . (5)

Now, the left hand side of eqn. (5) is the Burgers equation (2) and is therefore equalto zero. Consequently, for compatibility, the right hand side must also be equal to zero.Therefore, on setting the right hand side to zero and rearranging, we obtain

vt = v

u

u2vx

2v

vxuxv

. (6)

We now subtract multiplied by the derivative of the x-part of eqn. (4) from eqn. (6),to obtain

vt vxx= ux

2u

(2vx+vu) . (7)

But the left hand side of eqn. (7) is the heat equation (3), and is therefore equal to zero.Assuming that u = 0, compatibility considerations require that the bracketed term onthe right hand side is equal to zero. Therefore, on setting the bracketed term to zero andrearranging we arrive at the x-part of the Backlund transformation, eqn. (4), i.e.

vx= vu

2. (8)

Therefore, by definition we have (2vx+ vu) = 0. Thus, we have arrived at the pointwhere, given a solution to the heat eqn (3), we can derive a solution to Burgers equationby employing the Backlund transformation, eqn. (4). A maple program that derives this

result is given in Listing (1).



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Listing 1: Maple program that performs a Backlund transformation of a heat equation solution 1 to aBurgers equation solution

# B a ck l un d t r a ns f o rm a t io n o f H e at e q ua t io n s o lu t io n

# t o B u rg e rs e q ua t io n s o lu t io n

r e s t a rt ; w i t h ( P D E t o ol s ) : w i t h ( P o l y n o m i al T o o l s ) :

w i t h ( p l o t s ) :

a l i as ( u = u ( x , t ) ) : a l i as ( v = v ( x , t ) ) :

p d e _ H e a t E q n : = d i f f ( v , t ) - a l p h a * d i f f ( v , x , x ) = 0 ;

p d e _ B u rg e r s : = d i f f ( u , t ) + u * d i f f ( u , x ) - a l ph a * d i f f ( u , x , x ) = 0 ;B T x : = d i f f ( v , x ) = - 1 / ( 2 * a l p h a ) * u * v ;

B T t : = d i f f ( v , t ) = 1 / (4 * a l p h a ) * u ^ 2 * v - v * d i f f ( u , x ) / 2 ;

e q n 1 : = d i f f ( B T x , t ) - d i f f ( B T t , x ) ;

e q n 2 : = e x p a n d ( e q n 1 * 4 * a l p h a ) ;

# R e ar r a ng e e q n2

e q n 3 : = - r e m ov e ( h a s , r h s ( e q n 2 ) , { d i f f ( v , x ) , d i ff ( v , t ) } ) =

s e l e c t ( h a s , r h s ( e q n 2 ) , { d i f f ( v , x ) , d i f f ( v , t ) } ) ;

# S i mp l if y e qn 3

e q n 4 : = e x p a n d ( e q n 3 / ( 2 * v ) ) ;

# L HS o f e qn 4 i s B ur ge rs e qn a nd , t hu s , e qu al t o z er o ! T he re fo re ,

e q n 5 : = i s o l a t e ( e v a l ( r h s ( e q n 4 ) * ( 2 * v ) = 0 ) , d i f f ( v , t ) ) ;

e q n 6 : = a l p h a * d i f f ( B T x , x ) ;

e q n 7 : = e x p a n d ( e q n 5 - e q n 6 ) ;# L HS e qn 7 is H ea t e qn a nd t hu s e qu al t o z er o .

# T h e r ef o r e , c o m p a t a bi l i t y c o n s i d e r at i o n s r e q u ir e t h a t

# t he R HS of e qn 7 i s a ls o e qu al t o z er o !

e q n 8 : = s i m p li f y ( e v a l ( r h s ( e q n 7 ) = 0 ) , s y m b o li c ) ;

# F or e qn 8 t o b e t ru e , t he b ra ke te d t er m m us t e qu al z er o .

# T hi s i s t ru e a s i t i s e qu al t o B Tx , w hi ch i s z er o b y d ef in it io n , i . e.

e q n 9 : = i s o l at e ( e q n8 , d i f f ( v , x ) ) ;

As an example, we consider the following classical source solution to to the heat equation(3),

v= 1

(4t)exp(x2/4t), t >0, (9)

which, from the x-part of the Backlund transformation, u= 2vx/v, we obtain the thefollowing solution to Burgers equation (2),

u=x

t. (10)

A plot of this solution if given in Fig (1).

A second example is a slightly different source solution to to the heat equation(3),

v= 1 + 1

(4t)exp(x2/4t), t >0, (11)

which, from the x-part of the Backlund transformation, u = 2vx/v, we obtain thefollowing very different solution to Burgers equation (2),

u=x

t

1(4t)

exp(x2/4t)

1 + 1

(4t)exp(x2/4t)

. (12)


A third (variable separable) solution to the heat equation (3) is,

v=A sinx

L

exp

2

L2 t

, (13)



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Figure 1: Initial profile att = 0.1 of Burgers equation solution,u(x, t) =x/t. The solution moves towardsa horizontal profile such that u(x,) = 0.

where A and L are arbitrary constants. Thus, again from u = 2vx/v we obtain thefollowing stationary (non-time-varying) solution to Burgers equation (2),

u= 2

L cot

xL

. (14)


Thus, the above examples demonstrate that the Backlund transformation can produce avariety of very different solutions to the Burgers equation.

A maple program that derives these results and generates accompanying animations isgiven in Listing (2).

Listing 2: Maple program that performs a Backlund transformation of a heat equation solution 2 to aBurgers equation solution

# B a ck l un d t r a ns f o rm a t io n o f H e at e q ua t io n s o lu t io n 2

# t o B u rg e rs e q ua t io n s o lu t io n

r e s t a rt ; w i t h ( P D E t o o l s ) : w i t h ( P o l y n o m i al T o o l s ) :

w i t h ( p l o t s ) :


# H ea t E qn

p d e _ H e a t E q n : = d i f f ( v , t ) - a l p h a * d i f f ( v , x , x ) = 0 ;

# B ur ge rs E qn

p d e _ B u r ge r s E q n : = d i f f ( u , t ) + u * d i ff ( u , x ) - a l p h a * d i f f ( u , x , x ) = 0 ;# K no wn H ea t e qn s ol ut io n ( 1) - p oi nt h ea t s ou rc e

e q n 1 : = v = 1 / s q r t ( 4 * a l p h a * P i * t ) * e x p ( - x ^ 2 / ( 4 * a l p h a * t ) ) ;

# a s s i g n ( e q n 1 ) ;

# C he ck H ea t e qn s ol ut io n ( 1)

p d e t e s t ( e q n 1 , p d e _ H e a t E q n ) ; v 1 : = r h s ( e q n 1 ) ;

# G e ne r at e B u rg e rs e qn s o lu t io n ( 1)

e q n 2 : = u = - 2 * a l p h a * d i f f ( v 1 , x ) / v 1 ;

# C he ck B ur ge rs e qn s ol ut io n ( 1)

p d e t e s t ( e q n 2 , p d e _ B u r g e r s E q n ) ;

# B u rg e rs e q ua t io n p l ot d a ta

dat1:={alpha=1};

# P l ot r e su l ts

# = = = = = = = == = = =

z 1 := s u bs ( d at 1 , r h s ( e qn 2 ) ) ; # S et d a ta v a lu e s# A n im a te s o lu t io n

a n i m a t e ( z 1 , x = - 1 0 . . 1 0 , t = 0 . 1 . . 5 ,



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Figure 2: Initial profile at t = 0 of Burgers equation N-wave solution,

u(x, t) =x

t

1

(4t)

exp(x2/4t)/1 + 1

(4t)

exp(x2/4t) for = 1. The solution flattens out toa horizontal profile such that u(x,) = 0.

Figure 3: Initial profile at t = 0.1 of Burgers equation stationary solution, u(x, t) = 2

L cot

x

L

for

= 1 and L = 100.

n u m p o i nt s = 3 0 0 , f r a m e s = 5 0 , a x e s = f r am e d ,

l a b e l s = [ " x " , " u " ] , t h i c k n e s s = 3 ,

t i t le = " B u r g e r s e q u a t io n s o l u t io n " ,

l a be l f on t = [ T IM ES , R OM AN , 1 6] , a x e s fo n t = [ TI ME S , R OM AN , 1 6] ,

t i t l e f on t = [ T I M ES , R O MA N , 1 6 ] ) ;

# K no wn H ea t e qn s ol ut io n ( 2) - p oi nt h ea t s ou rc e

e q n 3 : = v = 1 + 1 / s q r t ( 4 * a l p h a * P i * t ) * e x p ( - x ^ 2 / ( 4 * a l p h a * t ) ) ;








dat2:={alpha=1};# P l ot r e su l ts

# = = = = = = = == = = =



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z 2 := s u bs ( d at 2 , r h s ( e qn 4 ) ) ; # S et d a ta v a lu e s

# A n im a te s o lu t io n

a n i m a t e ( z 2 , x = - 2 . . 2 , t = 0 . 1 . . 5 ,



t i tl e = " B u rg e rs e q ua t io n - N - w a ve s o lu t io n " ,



# K no wn H ea t e qn s ol ut io n ( 3)e q n 5 : = v = A * s i n ( P i * x / L ) * e x p ( - a l p h a * ( P i / L ) ^ 2 * t ) ;








d a t 3 : = { a l p h a = 1 , A = 1 , L = 1 0 0 } ;


# = = = = = = = == = = =

z 3 := s u bs ( d at 3 , r h s ( e qn 6 ) ) ; # S et d a ta v a lu e s


a n i m a t e ( z 3 , x = 0 . 1 . . 1 0 , t = 0 . . 8 0 ,



t i tl e = " B u rg e rs e q ua t io n - s t at i o na r y s o lu t io n " ,



2.2 Auto-Backlund transformation of the sine-Gordon equation

The sine-Gordonequation is given by

uxt= sin u, (15)

and we wish to find a solution using the Backlund transformation method. An auto-Backlund transformation for eqn. (15) is defined by the following pair of coupled partialdifferential equations.

vx= ux+ 2 sin

v+ u

2 )

,

vt= ut+2

sin

v u

2

,

(16)

where u=u(x, t) and v =v(x, t). We can easily check that this is so by differentiatingthe first equation byt and the second by xto obtain,

vxt= uxt+ 2 sin

v u

2

cos

v+ u

2 )

,

vxt= uxt+ 2 sin

v+u

2

cos

v u

2 )

.

(17)

We see immediately that this is correct by equating eqns. (17) when we obtain theoriginal eqn. (15), whilst adding them yields a second sine-Gordon equation, i.e.

vxt= sin v. (18)

This means that from a known solution to eqn. (15), we should be able to use thetransformation of eqns. (16) to obtain a second solution.



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Clearly, u = 0 is a solution to eqn. (15) and, on substituting this solution 2 is into theauto- Backlund transformation of eqn. (16), we obtain,

vx= 2 sinv

2, vt =

2

sin

v

2, (19)

which can be readily solved for v. Integrating both equations we obtain, dv

sinv

2

= 2

dx, 2x= 2 ln (csc (1/2v) cot(1/2v)) +f(t) , (20)

dv

sinv

2

= 2

dt, 2

t

= 2 ln (csc (1/2v) cot(1/2v)) +g (x) . (21)

Then, after subtracting and rearranging we obtain,

f(t) 2x= g (x) 2t

(22)

from which it follows that, for compatibility, we must have,

f(t) =K2t

, g(x) =K 2x, (23)

where K is an arbitrary constant. Thus, on adding eqns. (20) and (21) and rearranging,we obtain the following new kink solitonsolution to the sine-Gordon equation,

tanv

4=Cexp

x +

t

, (24)

where C= exp(K/2).


Figure 4: Initial profile at t = 20 of the sine-Gordon equation kink soliton solution,

v(x, t) = 4 arctan

Cexp

x +

t

, for C= 1 and = 1. Solution move right to left.

In theory we can now use this solution to obtain a second analytical solution, and the

second to obtain a third, etc. However, the resulting equations are not easily solved, evenwith a CAS such as Maple. Nevertheless, we can use this result, along with nonlinear2This starting point is known as the initiating or seed solution



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superposition, to readily obtain new solutions. This approach is demonstrated in a latersection.

A maple program that derives these results and generates an accompanying animationis given in Listing (3).

Listing 3: Maple program that derives a solution to the Sine-Gordon equation using the Auto-Backlundtransformation

# D e r iv a ti o n o f s o lu t io n t o S in e - G o r do n E q ua t io n u s in g

# A ut o - B a c k lu n d T r a ns f o rm a t io n .


w i t h ( p l o t s ) :


# A ut o - B a c k lu n d t r a ns f o rm a t io n f or s in e - G o r do n e q ua t io n

a B T [ 1 ] : =( u , v ) - > d i f f ( v , x ) = d i ff ( u , x ) + 2 * l a m b d a * s i n ( ( v + u ) / 2 ) ;

a B T [ 2 ] : =( u , v ) - > - d i f f ( v , t ) = d i ff ( u , x ) + ( 2 / l a m b d a ) * s i n ( ( v - u ) / 2 ) ;

# S et u =0 in a BT

o d e 1 : = a BT [ 1 ] ( 0 , v ) ; o d e 2 : = a BT [ 2 ] ( 0 , v ) ;

# N e ga t e a li as

a l i as ( u = u ) : a l i as ( v = v ) :

# I n t eg r a ti n g o d e1 a nd o d e2

e q n 1 : = 2 * l a m b d a * x = i n t ( 1 / s i n ( v / 2 ) , v ) + f ( t ) ;e q n 2 : = ( 2 / l a m b d a ) * t = i n t ( 1 / s i n ( v / 2 ) , v ) + g ( x ) ;

# r e a rr a n gi n g : e qn 3 : = eq n1 - e q n 2 ;

e q n 3 : = e q n 1 - e q n 2 ;

# e q n 3 : = 2 * l a m b da * x + f ( x ) = g ( t ) + 2 * x / l a m b da ;

# T he re f or e w e m us t h av e :

f ( t ) : = K - 2 * t / l a m b da ; g ( x ) : = K - 2 * l a m b da * x ;

e q n 4 : = e q n 1 + e q n 2 ;

e q n 5 : = i s o l at e ( e q n4 , l n ( c s c ( 1 / 2 * v ) - c o t ( 1 / 2 * v ) ) ) ;

e q n 6 : = e x p ( l h s ( e q n 5 ) ) = e x p ( r h s ( e q n 5 ) ) ;

e q n 7 : = s i m p li f y ( s u b s ( K = - 2 * l n ( C ) , e q n 6 ) , p o w e r ) ;

e q n 8 : = s i m p l i f y ( c o n v e r t ( e q n 7 , t a n ) , s y m b o l i c ) ;

e q n 9 : = c o n v e r t ( e q n 8 , t a n ) ;

# S in e - G o r do n e q ua t io n p l ot d a tad a t 1 : = { C = 1 , l a m b d a = 1 } ;


# = = = = = = = == = = =

z 1 := s u bs ( d at 1 , r h s ( i so l at e ( e qn 9 , v ) ) ) ; # S et d a ta v a lu e s


a n i m a t e ( z 1 , x = - 3 0 . . 3 0 , t = - 2 0 . . 2 0 ,



t i t le = " S i n e - G o r d o n e q u a t io n s o l u t io n " ,



2.3 Miura transformation of the KdV equation

The Korteweg-deVries(KdV) equation is defined as,

ut 6uux+ uxxx= 0, (25)

and we desire to obtain a solution to this equation by application of the Miura transfor-mation [Miu-68a], given by

u= v2 + vx, (26)

where u = u(x, t) and v = v(x, t). This equation, when substituted directly into eqn.

(25), yields 2v+

x

vt 6v

2vx+ vxxx

= 0. (27)



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Thus, ifv is a solution to,vt 6v

2vx+vxxx= 0, (28)

which is known as the modified KdV (mKdV) equation, then eqn. (4) will yield a solutionto eqn. (25). However, a solution to eqn. (25) does not imply a solution to equation(28)owing to the additional operator of eqn. (27).

Thus, equations (4) and (28) can be regarded as a Backlund transformation as theyprovide a derivation route to obtain solutions to the KdV equation. For example, a kinksoliton solution to eqn. (28) is,

v= 2k tanh

k

x x0 2k2t

; (29)

and, on substituting eqn. (29) into eqn. (4), we obtain the following hump solitonsolution to eqn. (25)

u= k2

2tanh

k

x +x0+ 2k2t2

1

. (30)

Plots of these solutions are given in Figs (5) and (6).

Figure 5: Initial profile at t = 50 of mKdV equation anti-kink soliton solution,v(x, t) = 2k tanh

kx x0 2k

2t

for k = 1 and x0 = 0. Solution move right to left.

A Maple program that generate a new KdV solution from a known mKdV solution byimplementing the Miura transformation, and that also generates animations for each, is

given in Listing (4).Listing 4: Maple program that transforms a mKDV equation solutionn to a KdV equation solutionn

# M i ur a t r a ns f o rm a t io n o f m K dV e q ua t io n t o K dV e q ua t io n

w i t h ( p l o t s ) :


# m K dV e q ua t io n

p d e _ m Kd V : = d i f f ( v , t ) - 6 * v ^ 2 * d i f f ( v , x ) + d i f f ( v , x , x , x ) = 0;

# K dV e q ua t io n

p d e _ K dV : = d i f f ( u , t ) - 6 * u * d i f f ( u , x ) + d i f f ( u , x , x , x ) = 0;

# K n ow n m K dv s o lu t io n

s o l 1 : = v = - k * t a n h ( k * ( x + x 0 + 2 * k ^ 2 * t ) ) ;

# m Kd V p lo t d at ad a t 1 : = { k = 0 . 5 , x 0 = 0 } ;


# = = = = = = = == = = =



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Figure 6: Initial profile at t = 20 of KdV equation hump soliton solution,

u= k2

2tanh

k

x + x0+ 2k

2t

2

1

for k = 1 and x0 = 0. Solution move right to left.

z 1 := s u bs ( d at 1 , r h s ( s ol 1 ) ) ; # S et d a ta v a lu e s


a n i m a t e ( z 1 , x = - 5 0 . . 1 0 , t = 0 . . 8 0 ,


l a b e l s = [ " x " , " v " ] , t h i c k n e s s = 3 ,

t i tl e = " m Kd V e q ua t io n - k in k s o li t on " ,



# C h ec k m K dV s o lu t io n

p d e t e s t ( s o l 1 , p d e _ m K d V ) ;

# D e fi n e M i ur a t r a ns f o rm a t io n

M i u r a T r a n s : = u = v ^ 2 + d i f f ( v , x ) ;

# G e ne r at e K dV s o lu t io n u s in g M i ur a t r a ns f o rm a t io ns o l 2 : = s i m p l i f y ( e v a l ( s u b s ( s o l 1 , M i u r a T r a n s ) ) , s i z e ) ;

# C h ec k K dV s o lu t io n

p d e t e s t ( s o l 2 , p d e _ K d V ) ;

# K dV p lo t d at a

d a t 2 : = { k = 0 . 5 , x 0 = 0 } ;


# = = = = = = = == = = =

z : = s ub s ( d at 2 , r h s ( s ol 2 ) ) ; # S et d a ta v a lu e s


a n i m a t e ( z , x = - 5 0 . . 1 0 , t = 0 . . 8 0 ,



t i tl e = " K dV e q ua t io n - h u mp s o li t on " ,l a be l f on t = [ T IM ES , R OM AN , 1 6] , a x e s fo n t = [ TI ME S , R OM AN , 1 6] ,


2.4 Backlund transformation of the KdV equation

We will now discuss a more convenient transformation than the Miura transformationdue to Wahlquist and Estabrook [Wah-73].

Because the KdV equation is Galilean invariant and, following Drazin and Johnson[Dra-92], we decide to work with u rather than u. This has the effect of modifying

the Miura transformation of eqn. (4) to

u= + v2 +vx, (31)



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where u= u(x, t), v= v(x, t) and is a real parameter. Thus, the mKdV equation (28)becomes

vt 6(v2 + )vx+ vxxx= 0. (32)

Now eqns (31) and (32) are an alternative Backlund transformation as they imply KdVequation (25) foru. Clearly, we observe that ifv is a solution to eqn. (32) then so is v.From this observation, Wahlquist and Estabrook were led to introduce the two functions

u1= + v2 + vx, u2= + v

2 vx (33)

where and v are given. By adding and subtracting these equations we obtain

u1 u2= 2vx, u1+ u2= 2

+v2

. (34)

We now introduce an additional useful transformation,

ui=wi

x, w= w(x, t), (i= 1, 2), (35)

from which we obtain, using eqns (33) and (34),

w1 w2= 2v (36)

and

(w1+ w2)x= 2 +1

2(w1 w2)

2 . (37)

Equation (36) is obtained by a single integration that requires the addition of an arbitraryfunction in f(t). However, this arbitrary function can be absorbed into the definitionofwi without changing ui and, therefore, there is no loss of generality by setting thisfunction to zero. Equation (37) constitutes the x-part of the Backlund transformationfor w1 and w2 which, along with eqn (35), enable solutions to the KdV equation to be

generated. From eqns (32)-(36) and some algebraic manipulation we obtain the followingt-part of the Backlund transformation,

(w1 w2)t 3

w21x w22x

+ (w1 w2)xxx= 0. (38)

A Maple program that generates eqns (37) and (38) from the KdV equation by applicationof the Wahlquist and Estabrook transformation, is given in Listing (5).

Listing 5: Maple program that performs a Wahlquist and Estabrook transformation on the KDV equation

# W a hl q u is t a nd E s ta b r oo k t r a ns f o rm a t io n o f m K dV e q ua t io n

# t o K dV e qu at io n


w i t h ( p l o t s ) :a l i as ( u = u ( x , t ) , v = v ( x , t ) ) : a l i as ( w [ 1 ] = w [ 1 ] ( x , t ) , w [ 2 ] = w [ 2 ] ( x , t ) ) :

# K dV a nd m Kd V e q ua ti on s



# W a hl q u is t a nd E s ta b r oo k m K dV e q ua t io n

p d e _ m K dV _ W E : = d i f f ( v , t ) - 6 *( v ^ 2 + l a m b d a ) * ( d i f f ( v , x ) ) +

d if f (v , x , x , x ) =0 ;

# D e fi n e t r a ns f o rm a t io n s

# M i ur a

M i u r a T r a n s : = u = v ^ 2 + d i f f ( v , x ) ;

# W a h lq u is t a nd E s t ab r oo k

WE_Trans:=isolate(subs(u=u-lambda ,MiuraTrans),u);

# D e ri v e n ew m K dV s o lu t io n a nd c h ec k c o rr e ctmKdV_WE := ev al ( su bs ( WE _T ran s , pde _KdV )) ;

mKdV_CK :=2 * v * pde _mKdV _WE + dif f ( pd e_m KdV_ WE , x) ;

t e s t e q ( s i m p l i f y ( l h s ( m K d V _ W E ) - l h s ( m K d V _ C K ) , s i z e ) ) ;



12/23

# N ot e: u a nd - u a re b ot h s ol ut io ns of m Kd V e qn

e q n 1 : = s u b s ( u = u [ 1 ] , W E _ T r a n s ) ;

e q n 2 : = s u b s ( { u = u [ 2 ] , v = - v } , W E _ T r a n s ) ; # u n a s s i g n ( u ) ;

# A dd a nd s u bt r ac t W E t r a ns f o rm a t io n s

e q n 3 : = e qn 1 - e q n 2 ; e q n 4 : = e q n1 + e q n 2 ;

# s u b st i tu t e u = d if f ( w , x)

e q n 5 : = s u b s ( u [ 1 ] = d i f f ( w [ 1 ] , x ) , u [ 2 ] = d i f f ( w [ 2 ] , x ) , e q n 3 ) ;

e q n 6 : = s u b s ( u [ 1 ] = d i f f ( w [ 1 ] , x ) , u [ 2 ] = d i f f ( w [ 2 ] , x ) , e q n 4 ) ;

e q n 7 : = i n t ( l h s ( e q n 5 ) , x ) = i n t ( r h s ( e q n 5 ) , x ) ;e q n 8 : = s u b s ( i s o l a t e ( e q n 7 , v ) , e q n 6 ) ;

e q n 9 : = s u b s ( i s o l a t e ( e q n 7 , v ) , p d e _ m K d V _ W E ) ;

e q n 1 0 : = r h s ( e q n 8 ) / 2 = l h s ( e q n 8 ) / 2 ;

# F i na l r e su l t

e q n 1 1 : = s i m p l i f y ( s u b s ( e q n 1 0 , e q n 9 ) , s i z e ) ;

The Wahlquist and Estabrook transformation is relatively straight forward to use; forexample, if we set w2= 0, eqn (37) becomes

w1x= 2 +1

2w21, (39)

which on integration yieldsw1= 2k tanh [kx +f(t)] (40)

where we have set = k2 and f is an arbitrary function. Similarly, eqn (38) becomes

w1t 3w21x+w1xxx= 0 (41)

and, using eqn (39) twice to give w1xxx= w21x+w

21w1x we obtain

w1t 2w1x

w1x+

1

2w21

= 0. (42)

Then, using eqn (39) once more for simplification, we can write eqn (41) as

w1t+ 4k2w1x= 0. (43)

This represents an advection equation with velocity 4k2, and which has the followinggeneral solution

w1=g(x, t) =g

x 4k2t

. (44)

Now, for consistency with eqn. (40), the function f(t) must take the form

f(t) = 4k3t kx0, (45)

where x0 is an arbitrary constant. The Backlund transformation therefore yields,

w1= 2k tanh [k x x0 4k2t . (46)Finally, we obtain from eqn (35)

u1= 2k2sech2

k

x x0 4k2t

, (47)

which is the same single soliton solution to the KdV equation, eqn (30), that we obtainedusing the Miura transformation and that is plotted in Fig( (6). This solution is valid for|w1| 2k we obtain the singular solution,

w1 = 2k2 coth

k

x x0 4k2t

, (48)

andu1= 2k

2 1 coth2 k x x0 4k2t . (49)A Maple program that derives the above example by application of the Wahlquist andEstabrook transformation, is given in Listing (6).



13/23

Listing 6: Maple program that performs a Wahlquist and Estabrook transformation on a KdV equationsolution to obtain a mKdV equation solution

# W a hl q u is t a nd E s ta b r oo k t r a ns f o rm a t io n o f K dV s o lu t io n

# t o m Kd V s ol ut io n


w i t h ( p l o t s ) :

a l i as ( u = u ( x , t ) , v = v ( x , t ) ) : a l i as ( w [ 1 ] = w [ 1 ] ( x , t ) , w [ 2 ] = w [ 2 ] ( x , t ) ) :

# K dV a nd m Kd V e q ua ti on s

p d e _ K dV : = d i f f ( u , t ) - 6 * u * d i f f ( u , x ) + d i f f ( u , x , x , x ) = 0;p d e _ m Kd V : = d i f f ( v , t ) - 6 * v ^ 2 * d i f f ( v , x ) + d i f f ( v , x , x , x ) = 0;

# W a hl q u is t a nd E s ta b r oo k m K dV e q ua t io n

p d e_ m K dV _ W E := d i ff ( v , t ) - 6 *( v ^ 2 + l am b da ) * ( d if f ( v , x ) ) + d if f ( v , x , x , x ) = 0;

# x - pa rt of B T

e q n1 : = d i ff ( w [ 1] , x ) + d if f ( w [2 ] , x ) = 2 * (( 1 /2 ) * w [ 1] - ( 1 /2 ) * w [ 2] ) ^ 2+

2 * l a m b d a ;

# t - pa rt o f B T

e q n2 : = ( 1 /2 ) * ( d if f ( w [1 ] , t ) ) - (1 /2 ) * ( d if f ( w [2 ] , t ) ) -

3 *( d i f f ( w [1 ] , x ) ) ^ 2 *( 1 /2 ) +

3 *( d i f f ( w [2 ] , x ) ) ^ 2 *( 1 /2 ) +

( 1/ 2) * ( d if f (w [ 1] , x , x , x ) ) -

( 1/ 2) * ( d if f (w [ 2] , x , x , x ) ) = 0 ;

# S et w [ 2] =0 i n x - p ar t o f BTe q n 3 : = e v a l ( s u b s ( { w [ 2 ] = 0 } , e q n 1 ) ) ;

# S ol ve O DE a nd r ea r ra ng e

e q n 4 : = e v a l ( i n t ( 1 / ( ( w 1 ^ 2 + 4 * l a m b d a ) / 2 ) , w 1 ) = i n t ( 1 , x ) + f ( t ) ) ;

e q n 5 : = i s o l a t e ( s i m p l i f y ( s u b s ( l a m b d a = - k ^ 2 , e q n 4 ) , s y m b o l i c ) , w 1 ) ;

# S et w [ 2] =0 i n t - p a rt o f B T a nd c ro ss m ul ti pl y

e q n 6 : = e v a l ( s u b s ( { w [ 2 ] = 0 } , e q n 2 ) ) ;

e q n 6 a : = n u m e r ( l h s ( e q n 6 ) ) * d e n o m ( r h s ( e q n 6 ) ) = n u m e r ( r h s ( e q n 6 ) ) *

d e n o m ( l h s ( e q n 6 ) ) ;

# S i mp l if y a g ai n b y s u bs t i tu t i on

e q n 7 : = e v a l ( d i f f ( e q n 3 , x ) ) ;

e q n 7 a : = l h s ( e q n 7 ) = s u b s ( e q n 3 , r h s ( e q n 7 ) ) ;

e q n 7 b : = e v a l ( d i f f ( e q n 7 a , x ) ) ;


e q n 8 : = i s o l a t e ( s u b s ( e q n 7 b , e q n 6 ) , d i f f ( w [ 1 ] , t ) ) ;e q n 8 a : = s i m p l i f y ( e q n 8 , s i z e ) ;


e q n 9 : = s u b s ( e q n 3 , e q n 8 a ) ;

e q n 9 a : = s u b s ( ( 1 / 2 ) * w [ 1 ] ^ 2 + 2 * l a m b d a = d i f f ( w [ 1 ] , x ) , e q n 9 ) ;

e q n 9 b : = s u b s ( l a m b d a = - k ^ 2 , e q n 9 a ) ;

e q n9 c : = l hs ( e q n 9b ) - r hs ( e q n9 b ) = 0 ;

# S ol ve a dv ec ti o n p de a nd t es t

s o l : = p d s o l v e ( e q n 9 c ) ; p d e t e s t ( s o l , e q n 9 c ) ;

# S ol ve f or f ( t)

e q n 1 0 : = k * x + k * f ( t ) = c * ( - x + 4 * k ^ 2 * t ) / ( 4 * k ^ 2 ) ;

f 1 : = s o l v e ( s u b s ( c = - 4 * k ^ 3 , e q n 1 0 ) , f ( t ) ) ;

# F i na l s o lu t io n f or m Kd V e q ua t io n

e q n 1 1 : = s u b s ( f ( t ) = f 1 , e q n 5 ) ;# F i na l s o lu t io n f or K dV e q ua t io n

e q n 1 2 : = u 1 = s i m p l i f y ( d i f f ( r h s ( e q n 1 1 ) , x ) , t r i g ) ;

3 Nonlinear superposition

3.1 Bianchis Permutability Theorem

We need to first introduce the Bianchi Permutability Theoremthat provides a simplifiedmethod of obtaining solutions to evolutionary equations by means ofnonlinear superpo-sition.

Theorem:[Con-08, 101]



14/23

Given a nonlinear PDE F(u,x,t) = 0 and its auto-Backlund transformation BT(u, U; ) =0, if one applies this BT to a given solution un1 with two different spectral parameters,BT(un1, un; n) = 0 and BT(un1,un; n+1) = 0, then there exists a fourth solutionun+1 which can be obtained by either BT(un, un+1; n+1) = 0 or BT(un, un+1; n) = 0,i.e. permuting the two spectral parameters.

[Put simply: from three distinct separate solutions to a given PDE, a fourth solution may

be constructed by purely algebraic means using a suitable auto-Backlund transformation,if one exists].

This theorem is represented diagrammatically in figure 7.

Figure 7: Diagrammatic representation of Bianchis theorem of permutability [Dra-92, p115].

No general proof of this theorem is known to exist and therefore it has to be proved foreach PDE admitting an auto-Backlund transformation. Therefore, we will illustrate itsuse by examples.

3.2 Nonlinear superposition applied to the KdV equation

We have now developed the tools to enable a more straightforward method of obtainingsolutions to the KdV equation using nonlinear superposition. From the above B acklund

transformation we generate two solutions from the same seed solution but using twodifferent values of. Thus, thex-part of the Backlund transformation, i.e. eqn. (37),can be written as

(w1+w0)x= 21+ 12(w1 w0)

2 ,

(w2+w0)x= 22+ 12(w2 w0)

2 . (50)

Similarly, we construct two additional solutions: w12 from w1 and 1; and w21 from w2and 2

(w12+w1)x= 21+ 12(w12 w1)

2 ,

(w21+w2)x= 22+ 12(w21 w2)

2 . (51)

Now, the above Bianchi theorem of permutability states that w12 and w21 in eqns (51)

will be equal, i.e.w12=w21. (52)



15/23

This provides a way of obtaining additional solutions to the KdV equation in a straight-forward manner. First, subtract the difference of eqns (50) from the difference of eqns(51) to give

0 = 4 (2 1) +1

2

(w12 w1)

2 (w21 w2)2 (w1 w0) + (w2 w0)

2

. (53)

Then, using eqn (52), we obtain after some rearranging,

w12 = w0 4 (1 2)

w1 w2. (54)

Thus, we have arrived at the point where, given three solutions to the mKdV equation(28): w0, w1 and w2, we can derive a fourth solution to the mKdV equation by purelyalgebraic means. Consequently, the mKdV solutions can be used to obtain correspondingKdV solutions by application of eqn (35).

A Maple program that derives the above result is given in Listing (7).

Listing 7: Maple program for the nonlinear superposition of KdV equation

# K dV N o n li n ea r S u p er p os i t io n

restart;

a l i as ( w [ 0 ] = w [ 0 ] ( x , t ) ) : a l i a s ( w [ 1 ]= w [ 1 ] ( x , t ) ) :

a l i a s ( w [ 2 ] = w [ 2 ] ( x , t ) ) :

a l i a s ( w [ 1 2 ] = w [ 1 2 ] ( x , t ) ) : a l i a s ( w [ 2 1 ] = w [ 2 1 ] ( x , t ) ) :

# D e fi n e t wo d i ff e re n t x - p a rt a ut o - B a c kl u nd t r a ns f o rm a t io n s

e q n 1 : = d i ff ( ( w [ 1 ] + w [ 0 ] ) , x ) = 2 * l a m b da [ 1 ] + ( 1 / 2) * ( w [ 1 ] - w [ 0 ] ) ^ 2 ;

e q n 2 : = d i ff ( ( w [ 2 ] + w [ 0 ] ) , x ) = 2 * l a m b da [ 2 ] + ( 1 / 2) * ( w [ 2 ] - w [ 0 ] ) ^ 2 ;

# D e fi n e t wo a d di t i on a l d i ff e re n t x - p a rt a ut o - B a c k lu n d

# t r a n s f o r m at i o n s

e q n 3 : = d i ff ( ( w [ 1 2 ] + w [ 1 ] ) , x ) = 2 * l a m b da [ 2 ] + ( 1 / 2) * ( w [ 1 2 ] - w [ 1 ] ) ^ 2 ;

e q n 4 : = d i ff ( ( w [ 2 1 ] + w [ 2 ] ) , x ) = 2 * l a m b da [ 1 ] + ( 1 / 2) * ( w [ 2 1 ] - w [ 2 ] ) ^ 2 ;

# S ub tr ac t d if f er ec e o f s ec on d B Ts f ro m f ir st B Ts

e q n 5 : = ( e q n 1 - e q n 2 ) - ( e q n 3 - e q n 4 ) ;# A p pl y " i n t e gr a b il i t y c o nd i t io n s "

w [ 2 1 ] : = w [ 1 2 ] ;

# R e ar r a ng e t o o b ta i n f i na l s o lu t io n

e q n 6 : = c o l l e c t ( i s o l a t e ( e q n 5 , w [ 1 2 ] ) , w [ 0 ] ) ;

As an example application of the above, let w0= 0 and use solutions from eqns (46) and(48), i.e. w1 = 2k tanh([k (x x0 4k

2t)], w2 = 2k2 coth[k (x x0 4k

2t)]; theneqn (54) yields,

w12 = 2

k1

2 k22

k1tanh

k1

x 4k12

t k2coth

k2

x 4k22

t , (55)

where we have set the arbitrary constant x0 = 0 (recall that = k2). Then finally from

eqn (35) we obtain,

u12 =2

k12 k2

2

k12sech2

k1

x 4k12t

+ k22csch2

k2

x 4k22t

k1tanh

k1

x 4k12t k2coth

k2

x 4k22t2 , (56)

which represents a two-soliton solution to the KdV equation. A plot of this solution isgiven in Fig (8).

A Maple program that derives the above example and includes an animation, is given inListing (8).



16/23

Figure 8: Initial profile at t = 4 of KdV equation 2-soliton solution,

u12 = 2

k1

2 k2

2

k1

2sech2

k1

x 4k1

2t

+ k2

2csch2

k2

x 4k2

2t

k1

tanhk1

x

4k12

t

k2

cothk2

x

4k22

t2 for k1 = 1/2 and k2 = 1.

The solitons move left to right with the taller faster soliton overtaking the shorter slower soliton.

Listing 8: Maple program that generates a 2-soliton solution to the KdV equation by nonlinear superpo-sition of mKdV equation solutions

# K dV / m K dV E q ua t io n s , 2 - s o li t on s o lu t io n - B a ck l un d T r a ns f o rm a t io n


w i t h ( p l o t s ) :

a l i as ( u = u ( x , t ) ) : a l i as ( w = w ( x , t ) ) : a l i a s ( v = v ( x , t ) ) :

# D ef in e K dV a nd m Kd V e qu at i on s



# W a hl q ui s t a n d E s ta b r oo k t r a n s fo r m at i o n

p d e 1 : = ( w ) - > d i f f ( w , x ) = 2 * l a m b d a + w ^ 2 / 2 ;# D e fi n e n o nl i ne a r s u p er p o si t i on

w [ 1 2 ] : = w [ 0 ] - 4 * ( l a m b d a [ 1 ] - l a m b d a [ 2 ] ) / ( w [ 1 ] - w [ 2 ] ) ;

# D e fi n e t h re e m Kd V s o l ut i on s

w[0]:=0;

w [ 1 ] : = - 2 * k [ 1 ] * t a n h ( k [ 1 ] * ( x - 4 * k [ 1 ] ^ 2 * t ) ) ;

w [ 2 ] : = - 2 * k [ 2 ] * c o t h ( k [ 2 ] * ( x - 4 * k [ 2 ] ^ 2 * t ) ) ;

# C a lc u l at e c o r re s p on d in g K dV s o l ut i on s a nd t e st t h ey a r e c o rr e ct

s o l 1 : = u = s i m p l if y ( d i f f ( w [ 1 ] , x ) , t r i g ) ;

s o l 2 : = u = s i m p l i f y ( d i f f ( w [ 2 ] , x ) , t r i g ) ;



# S o lv e f or l a mb d a u s in g W a h lq u is t a nd E s ta b ro o k t r a ns f o rm a t io n

e q n 1 : = p d e 1 ( w [ 1 ] ) ;e q n 2 : = p d e 1 ( w [ 2 ] ) ;

l a m b d a [ 1 ] : = s o l v e ( e q n 1 , l a m b d a ) ;


# F i na l s o lu t io n f r om n o nl i n ea r s u p er p o si t io n

e v a l ( w [ 1 2 ] ) ;

s o l 3 : = u = d i f f ( w [ 1 2 ] , x ) ;

# C h ec k s o lu t io n s t a ti s fi e s K dV e q ua t io n


# A p pl y m a nu a l s i m pl i f ic a t io n a nd c h ec k i t i s c o rr e ct

s o l 3a : = u = ( 2 * ( k [ 1 ]^ 2 - k [ 2 ] ^ 2 ) ) * ( k [ 1 ] ^ 2 * s e c h ( k [ 1 ] * ( x - 4 * k [ 1 ] ^2 * t ) ) ^ 2 +

k [ 2 ] ^ 2 * c s c h ( k [ 2 ] * ( x - 4 * k [ 2 ] ^ 2 * t ) ) ^ 2 ) /

( k [ 1 ] * t a n h ( k [ 1 ] * ( x - 4 * k [ 1 ] ^ 2 * t ) ) -

k [ 2 ] * c o t h ( k [ 2 ] * ( x - 4 * k [ 2 ] ^ 2 * t ) ) ) ^ 2 ;

t e s t e q ( r h s ( s o l 3 ) = r h s ( s o l 3 a ) ) ;# P l ot r e su l ts

# = = = = = = = == = = =



17/23

d a t : = { k [ 1 ] = 1 / 2 , k [ 2 ] = 1 } ;

u 12 : = s u bs ( d at , r h s ( s o l3 ) ) ; # S et d a ta v a lu e s


a n i m a t e ( u 1 2 , x = - 3 0 . . 3 0 , t = - 4 . . 4 ,


l a b e l s = [ " x " , " u [ 1 2 ] " ] , t h i c k n e s s = 3 ,

t i t le = " K d V e q u a t io n \ n 2 - s o l i t o n s o l u t io n " ,



3.3 Three-soliton solution to the KdV equation using nonlinear superposi-

tion

The two-soliton solution presented above can be extended to generate multi-soliton so-lutions where, for a three-soliton solution, eqn. (54) becomes.

w123 = w1 4 (2 3)

w12 w13. (57)

Or, on expansion,

w123=1w1(w2 w3) +2w2(w3 w1) +3w3(w1 w2)

1(w2 w3) +2(w3 w1) +3(w1 w2) . (58)

For example, if we define the following solutions to the mKdV equation,

w0 =0, (59)

w1 = 2tanh

k1

x 4k21t

, (60)

w2 = 2coth k2 x 4k22t , (61)

w3 = 2tanh

k3

x 4k23t

, (62)

then, following the same procedure as for the two soliton solution, we obtain

w12= 2

k1

2 k22

k1tanh

k1

x 4k12t k2coth

k2

x 4k22t , (63)

w13= 2

k1

2 k32

k1tanh

k1

x 4k12t k3tanh

k3

x 4k32t . (64)

and from eqn (57) ,

w123= N1D1, (65)

where,

N1=2tanh

4tk13 xk1

k1

3 coth

4tk23 xk2

k2

2tanh

4tk13 xk1

k1coth

4tk2

3 xk2

k23

2tanh

4tk13 xk1

k1

3 tanh

4tk33 xk3

k3

+ 2 tanh

4tk13 xk1

k1tanh

4tk3

3 xk3

k33

+ 2k23 coth

4tk2

3 xk2

tanh

4tk33 xk3

k3

2k2coth

4tk23

xk2

tanh

4tk33

xk3

k33

(66)



18/23

and

D1= tanh

4tk13 xk1

k1k2

2 + tanh

4tk13 xk1

k1k3

2

+ coth

4tk23 xk2

k2k1

2 coth

4tk23 xk2

k2k3

2

tanh

4tk33 xk3

k3k1

2 + tanh

4tk33 xk3

k3k2

2. (67)

The solution to the KdV equation is obtained by application of eqn (35) yielding,

u123 =N2D2

, (68)

where,

N2 =2k22

k12 (k1 k2)

2 (k1+ k2)2 tanh2

4tk1

3 xk1

+ 2tanh

4tk33 xk3

k1k3(k2 k3) (k2+ k3) (k1 k2) (k1+ k2)

tanh

4tk13 xk1

+k3

2 (k2 k3)2 (k2+ k3)

2 tanh2

4tk33 xk3

(k1 k3)

2 (k1+ k3)2 k12 + k32 k22 coth2 4tk23 xk2

4k1(k1+k3) (k1 k3) k2k2

2k1+k13

tanh

4tk33 xk3

tanh2

4tk1

3 xk1

+ k3

tanh

4tk33 xk3

1

tanh

4tk33 xk3

+ 1

(k2 k3) (k2+ k3)tanh

4tk1

3 xk1

+k1

3 + k22k1

tanh

4tk3

3 xk3

k3coth

4tk23 xk2

+ 2k1

2

k32 (k1 k3)

2 (k1+ k3)2 tanh2

4tk3

3 xk3

+ (k2 k3)2 (k2+ k3)

2 k22 +k12 k32 tanh2 4tk13 xk1 4k3tanh

4tk3

3 xk3

k22k1(k2 k3) (k2+k3) (k1 k2) (k1+ k2)

tanh 4tk13 xk1 2k32 tanh2 4tk33 xk3 (k1 k2)

2 (k1+k2)2

k12 +k2

2 k32

(69)

and

D2=k3k1

2 +k3k22

tanh

4tk33 xk3

+k2

2 +k32

tanh

4tk13 xk1

k1+ k2coth

4tk2

3 xk2

(k1 k3) (k1+ k3)2

. (70)

This represents a three-soliton solution to the KdV equation and a plot of this solutionis given in Fig (9).

A Maple program that derives the above result and includes an animation, is given in

Listing (9).Listing 9: Maple program that generates a 3-soliton solution to the KdV equation by nonlinear superpo-sition of mKdV equation solutions

# K dV / m K dV E q ua t io n s , 3 - s o li t on s o lu t io n - B a ck l un d T r a ns f o rm a t io n


w i t h ( p l o t s ) :

a l i as ( u = u ( x , t ) ) : a l i as ( w = w ( x , t ) ) : a l i a s ( v = v ( x , t ) ) :

# D ef in e K dV a nd m Kd V e qu at i on s



# W a hl q ui s t a n d E s ta b r oo k t r a n s fo r m at i o n

p d e 1 : = ( w ) - > d i f f ( w , x ) = 2 * l a m b d a + w ^ 2 / 2 ;# D e fi n e n o nl i ne a r s u p er p o si t i on

w [ 1 2 3 ] : = w [ 1 ] - 4 * ( l a m b d a [ 2 ] - l a m b d a [ 3 ] ) / ( w [ 1 2 ] - w [ 1 3 ] ) ;




19/23

Figure 9: Initial profile at t = 40 of KdV equation 3-soliton solution, eqn. (68). Solitons move left toright with the taller faster solitons overtaking the shorter slower solitons.


w [ 1 2 3 ] : = c o l l e c t ( n u m e r ( w [ 1 2 3 ] ) , { l a m b d a [ 1 ] , l a m b d a [ 2 ] , l a m b d a [ 3 ] } ) /

c o l l e c t ( d e n o m ( w [ 1 2 3 ] ) , { l a m b d a [ 1 ] , l a m b d a [ 2 ] , l a m b d a [ 3 ] } ) ;

# D e fi n e f o ur m K dV s o lu t io n s

w[0]:=0;

w [ 1 ] : = - 2 * k [ 1 ] * t a n h ( k [ 1 ] * ( x - 4 * k [ 1 ] ^ 2 * t ) ) ;

w [ 2 ] : = - 2 * k [ 2 ] * c o t h ( k [ 2 ] * ( x - 4 * k [ 2 ] ^ 2 * t ) ) ;

w [ 3 ] : = - 2 * k [ 3 ] * t a n h ( k [ 3 ] * ( x - 4 * k [ 3 ] ^ 2 * t ) ) ;

w [ 1 2 ] : = e v a l ( w [ 1 2 ] ) ;

w [ 1 3 ] : = e v a l ( w [ 1 3 ] ) ;

w [ 1 2 3 ] : = s i m p l i f y ( w [ 1 2 3 ] , s i z e ) ;

# C a lc u l at e c o r re s p on d in g K dV s o l ut i on s a nd t e st t h ey a r e c o rr e ct

s o l 1 : = u = s i m p l i f y ( d i f f ( w [ 1 ] , x ) , t r i g ) ;s o l 2 : = u = s i m p l i f y ( d i f f ( w [ 2 ] , x ) , t r i g ) ;

s o l 3 : = u = s i m p l i f y ( d i f f ( w [ 3 ] , x ) , t r i g ) ;




# S o lv e f or l a mb d a u s in g W a h lq u is t a nd E s ta b ro o k t r a ns f o rm a t io n

e q n 1 : = p d e 1 ( w [ 1 ] ) ;

e q n 2 : = p d e 1 ( w [ 2 ] ) ;

e q n 3 : = p d e 1 ( w [ 3 ] ) ;




w [ 1 2 3 ] : = e v a l ( w [ 1 2 3 ] ) ;

s o l 1 2 3 : = u = s i m p l i f y ( d i f f ( w [ 1 2 3 ] , x ) , s i z e ) ;# P l ot r e su l ts

# = = = = = = = == = = =

d a t : = { k [ 1 ] = 1 / 2 , k [ 2 ] = 3 / 4 , k [ 3 ] = 1 } ;

u 1 2 3 : = s i m p li f y ( s u b s ( d at , r h s ( s o l 1 23 ) ) , s i z e ) ; # S e t d a t a v a l u es


a n i m a t e ( u 1 2 3 , x = - 4 0 . . 4 0 , t = - 8 . . 8 ,


l a b e l s = [ " x " , " u [ 1 2 3 ] " ] , t h i c k n e s s = 3 ,

t i t le = " K d V e q u a t io n \ n 3 - s o l i t o n s o l u t io n " ,





20/23

3.4 Nonlinear superposition applied to the sine-Gordon equation

We take a similar approach to that used for the KdV equation and start by rewriting thex-part of the auto-Backlund transformation given in eqns (16) as

(v1 v0)x= 21sinv1+v02

,(v2 v0)x= 22sin

v2+v0

2

,

(71)

where v =v(x, t). Similarly, we construct two additional solutions: v12 from v1 and 1;and v21 fromv2 and 2

(v12 v1)x= 21sin

v12+ v1

2

,

(v21 v2)x= 22sin

v21+ v2

2

.

(72)

This provides a way of obtaining additional solutions to the S-G equation in a straight-forward manner. First, subtract the difference of eqns (71) from the difference of eqns(72) to give

0 = 2

sin

v0 v1

2

+ sin

v2 v12

2

1+

2

sin

v0 v2

2

+ sin

v1 v21

2

2. (73)

Again, the above Bianchi theorem of permutability states that v12 and v21 in eqns (72)will be equal, i.e.

v12=v21. (74)

Then, using eqn (74) and some standard trigonometrical identities 3, we obtain aftersome rearranging,

v12 = v0 4 arctan

1+ 21 2

tan

v1 v2

4

. (75)

Thus, as for the KdV equation, we have arrived at the point where, given three solutionsto the S-G equation: v0, v1 and v2, we can derive a fourth by purely algebraic means.

A Maple program that derives the above result is given in listing (10).

Listing 10: Maple program that generates a fourth solution to the sine-Gordon equation by nonlinearsuperposition of three known solutions

# S in e - G o r do n N o nl i ne a r S u p er p o si t i on

restart;

a l i a s ( v [ 0 ] = v [ 0 ] ( x , t ) , v [ 1 ] = v [ 1 ] ( x , t ) , v [ 2 ] = v [ 2 ] ( x , t ) , u = u ( x , t ) ) :

a l i a s ( v [ 1 2 ] = v [ 1 2 ] ( x , t ) ) : a l i a s ( v [ 2 1 ] = v [ 2 1 ] ( x , t ) ) :

# D e fi n e s in e - G o r do n e q ua t io n

p d e _ S G : = d i f f ( u , x , t ) = s i n ( u ) ;

# D e fi n e t wo d i ff e re n t x - p a rt a ut o - B a c kl u nd t r a ns f o rm a t io n s

e q n 1 : = d i ff ( ( v [ 1 ] + v [ 0 ] ) , x ) = 2 * l a m b da [ 1 ] * s i n ( ( v [ 1] - v [ 0 ] ) / 2 ) ;

e q n 2 : = d i ff ( ( v [ 2 ] + v [ 0 ] ) , x ) = 2 * l a m b da [ 2 ] * s i n ( ( v [ 2] - v [ 0 ] ) / 2 ) ;

# D e fi n e t wo a d di t i on a l d i ff e re n t x - p a rt a ut o - B a c k lu n d t r a ns f o rm a t io n s

e q n 3 : = d i ff ( ( v [ 1 2 ] + v [ 1 ] ) , x ) = 2 * l a m b da [ 2 ] * s i n ( ( v [ 12 ] - v [ 1 ] ) / 2 ) ;

e q n 4 : = d i ff ( ( v [ 2 1 ] + v [ 2 ] ) , x ) = 2 * l a m b da [ 1 ] * s i n ( ( v [ 21 ] - v [ 2 ] ) / 2 ) ;

3Identities used: A) sin + sin= 2sin +

2 cos

+

2 , B) sin ( + ) = sin cos+ cos sin .



21/23

# S ub tr ac t d if f er ec e o f s ec on d B Ts f ro m f ir st B Ts

e q n 5 : = ( e q n 1 - e q n 2 ) - ( e q n 3 - e q n 4 ) ;

# A p pl y " i n t e gr a b il i t y c o nd i ti o n "

v [ 2 1 ] : = v [ 1 2 ] ;

# R e ar r a ng e

eqn6:=collect(eqn5 ,{lambda[1],lambda[2]});

# A p p ly t r i g i d e n t i t y : s i n ( A + B ) + s i n ( A - B ) = 2 * s i n ( ( A + B ) / 2 ) ) * c os ( ( A - B ) / 2 )

e q n 7 : = a l g s ub s ( s i n ( - ( 1 /2 ) * v [ 1 ] + ( 1 / 2 ) * v [ 0 ]) + s i n ( - ( 1 /2 ) * v [ 1 2 ] + ( 1 / 2 ) * v [ 2 ] ) =

2 * s i n ( ( - ( 1 / 2 ) * v [ 1 ] + ( 1 / 2 ) * v [ 0 ] - ( 1 / 2 ) * v [ 1 2 ] + ( 1 / 2 ) * v [ 2 ] ) / 2 ) *c o s ( ( - ( 1 / 2 ) * v [ 1 ] + ( 1 / 2 ) * v [ 0 ] + ( 1 / 2 ) * v [ 1 2 ] - ( 1 / 2 ) * v [ 2 ] ) / 2 ) , e q n 6 ) ;

# A p p ly t r i g i d e n t i t y : s i n ( A + B ) + s i n ( A - B ) = 2 * s i n ( ( A + B ) / 2 ) ) * c os ( ( A - B ) / 2 )

e q n 8 : = a l g s ub s ( s i n ( - ( 1 /2 ) * v [ 2 ] + ( 1 / 2 ) * v [ 0 ]) + s i n ( - ( 1 / 2) * v [ 1 2 ] + ( 1 / 2 ) * v [ 1 ] ) =

2 * s i n ( ( - ( 1 / 2 ) * v [ 2 ] + ( 1 / 2 ) * v [ 0 ] - ( 1 / 2 ) * v [ 1 2 ] + ( 1 / 2 ) * v [ 1 ] ) / 2 ) *

c o s ( ( - ( 1 / 2 ) * v [ 2 ] + ( 1 / 2 ) * v [ 0 ] + ( 1 / 2 ) * v [ 1 2 ] - ( 1 / 2 ) * v [ 1 ] ) / 2 ) , e q n 7 ) ;

# R e mo v e c o mm o n f a ct o r c os ( . . )

e q n 9 : = 0 = o p ( 3 , r h s ( f a c t o r ( e q n 8 ) ) ) ;

# A p pl y t r ig i d en t it y : s in ( A + B )= s i n (A ) c os ( B ) + co s ( A ) si n ( B )

e q n 10 : = a l g s u b s ( s i n ( - ( 1/ 4 ) * v [ 2 ] + ( 1 / 4) * v [ 0 ] - ( 1 /4 ) * v [ 1 2 ] + ( 1 / 4 ) * v [ 1 ] ) =

s i n ( ( 1 / 4 ) * v [ 0 ] - ( 1 / 4 ) * v [ 1 2 ] ) * c o s ( - ( 1 / 4 ) * v [ 2 ] + ( 1 / 4 ) * v [ 1 ] ) +

c o s ( ( 1 / 4 ) * v [ 0 ] - ( 1 / 4 ) * v [ 1 2 ] ) * s i n ( - ( 1 / 4 ) * v [ 2 ] + ( 1 / 4 ) * v [ 1 ] ) , e q n 9 ) ;

# A p pl y t r ig i d en t it y : s in ( A + B )= s i n (A ) c os ( B ) + co s ( A ) si n ( B )

e q n 11 : = a l g s u b s ( s i n ( - ( 1/ 4 ) * v [ 1 ] + ( 1 / 4) * v [ 0 ] - ( 1 /4 ) * v [ 1 2 ] + ( 1 / 4 ) * v [ 2 ] ) =

s i n ( ( 1 / 4 ) * v [ 0 ] - ( 1 / 4 ) * v [ 1 2 ] ) * c o s ( - ( 1 / 4 ) * v [ 1 ] + ( 1 / 4 ) * v [ 2 ] ) +

c o s ( ( 1 / 4 ) * v [ 0 ] - ( 1 / 4 ) * v [ 1 2 ] ) * s i n ( - ( 1 / 4 ) * v [ 1 ] + ( 1 / 4 ) * v [ 2 ] ) , e q n 1 0 ) ;

# C o ll e ct l ik e t e rm s

e q n 12 : = s i m p l i f y ( c o l l e c t ( e q n1 1 , { s i n ( ( 1 / 4) * v [ 0 ] - ( 1 /4 ) * v [ 1 2 ] ) ,

c o s ( - ( 1 / 4 ) * v [ 2 ] + ( 1 / 4 ) * v [ 1 ] ) } ) , s i z e ) ;

# R e ar r a ng e

e q n 1 3 : = o p ( 1 , r h s ( e q n 1 2 ) ) = - o p ( 2 , r h s ( e q n 1 2 ) ) ;

# D e fi n e c o mm o n f a ct o r

d i v i s or : = s e l e c t ( h a s , l h s ( e q n 1 3 ) , c o s ) * s e l e ct ( h a s , r h s ( e q n 1 3 ) , c o s ) ;

# R e mo v e c o mm o n f a ct o r

eqn14:=convert(eqn13/divisor ,tan);

# F i na l s o lu t io n

i s o l a te ( e q n 1 4 , v [ 1 2 ] ) ;

As an example application of the above, letv0= 0,v1 = arccos

2tanh2 (a1x2 + t/a1) 1

(peakonsoliton) and v2 = 4 arctan [exp (a2x + t/a2)] (anti-kink soliton); then eqn. (75)yields the following new solution to the sine-Gordon equation,

v12= 4 arctan

1+21 2

tan

1

4arccos

2tanh2 (a2x + t/a2) 1

arctan [exp (a1x + t/a1)]

. (76)

A plot of this solution is given in Fig (10).

A Maple program that derives the above example and includes an animation, is given inListing (11).

Listing 11: Maple program that generates a 2-soliton solution to the sine-Gordon equation by nonlinearsuperposition

# S in e - G o r do n e q ua t io n - n o n li n ea r s u pe r p os i t io n

r e s t a rt ; w i t h ( p l o t t oo l s ) : w i t h ( p l o ts ) :


# D e fi n e s in e - G o r do n e q ua t io n

p d e _ S G : = d i f f ( u , x , t ) - s i n ( u ) = 0 ;

# D e fi n e S - G n o nl i ne a r s u p er p o si t i on e q ua t io nv [ 1 2 ] : = v [ 0 ] - 4 * a r c t a n ( t a n ( - ( 1 / 4) * v [ 2 ] + ( 1 / 4 ) * v [ 1 ] ) *

( l a m b d a [ 2 ] + l a m b d a [ 1 ] ) / ( l a m b d a [ 1 ] - l a m b d a [ 2 ] ) ) ;

# D e fi n e t h re e S - G s o lu t io n s



22/23

Figure 10: Initial profile at t = 200 of sine-Gordon equation2-soliton solution,

v12 = 4 arctan1+ 21 2

tan1

4arccos 2 tanh

2 (a2x + t/a2) 1 arctan [exp (a1x + t/a1)] for a1 = 5,a2 = 4, 1 = 1 and 2 = 2. The solution moves from right to left with the peakonovertaking the anti-kinksoliton.

v[0]:=0;

v [ 1 ] : = a r c c o s ( 2 * t a n h ( a 2 * x + t / a 2 ) ^ 2 - 1 ) ;

v [ 2 ] : = 4 * a r c t a n ( e x p ( a 1 * x + t / a 1 ) ) ;

# C h ec k s o lu t io n s a re c o rr e ct

p d e t e s t ( u = v [ 1 ] , p d e _ S G ) ; p d e t e s t ( u = v [ 2 ] , p d e _ S G ) ;

# E v al u at e s o lu t io n

v [ 1 2 ] : = e v a l ( v [ 1 2 ] ) ;


# = = = = = = = == = = =

d a t : = { a 1 = 5 , a 2 = 4 , l a m b da [ 1 ] = 1 , l a m b da [ 2 ] = 2 } ;z z : = s u b s ( d a t , v [ 1 2 ] ) ;

a n i m a t e ( z z , x = - 1 5 . . 1 5 , t = - 2 0 0 . . 2 0 0 , n u m p o i n t s = 1 0 0 0 , f r a m e s = 5 0 ,


l a b e l s = [ " x " , " u [ 1 2 ] " ] , t h i c k n e s s = 3 ,

t i t le = " S i n - G o r d o n 2 - s o l i to n s o l u t io n \ n P e a k o n o v e r t ak e s a n ti - K i n k " ,



References

[Bac-80] Backlund, A. V. (1880), Zur Theorie der Partiellen Differentialgleichungenerster Ordnung,Math. Ann., XVII, 285-328.

[Con-08] Conte, R. and M. Musette (2008), The Painleve Handbook, Springer, Dor-drecht, Netherlands.

[Dra-92] Drazin, P. G. and R. S. Johnson (1992). Solitons: an introduction, Cam-bridge University Press, Cambridge.

[Dod-82] Dodd, R. K., J. C. Eilbeck. J. D. Gibbon and H. C. Morris (1982),Solitons and nonlinear wave equations, Academic Press.

[Miu-68a] Miura, R. A.

(1968). Korteweg-de Vries Equation and Generalizations I. Aremarkable explicit transformation. J. Math. Phys., 9, pp1202-4.



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[Miu-68b] Miura, R. A., C. S. Gardner and M. D. Kruskal (1968). Korteweg-deVries Equation and generalizations II. Existence of conservation laws and constantsof motion. J. Math. Phys., 9, pp1204-9.

[Rog-82] Rogers, C. and W. F. Shadwick(1982), Backlund transformations and theirapplications, Academic Press, New York.

[Wah-73] Wahlquist, H. D. and F. B. Estabrook (1973). Backlund transformationfor solutions to the Korteweg-de Vries equation. Phys. Rev. Lett., 31,1386-90.

[Waz-09] Wazwaz, Abdul-Majid (2009). Partial Differential Equations and SolitaryWaves Theory, Springer-Verlag, Berlin.

backlund transformation

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