Direct method for soliton solution

Hirota’s Direct Method for Soliton Solution
MOHANRAJ K,
M.Phil ,
Department of Theoretical Physics,
University of Madras (Guindy Campus),
Chennai-25.
MOHANRAJ K,
M.Phil ,
Department of Theoretical Physics,
University of Madras (Guindy Campus),
Chennai-25.

Outline
 To present the basics of nonlinear Dynamics
 introduction to solitons
 Solving KdV equation through Hirota’s approach
 Confining soliton behaviour
# Complex burgers equation
# Chengs chemical reaction equation
# General solution of Liouville equation

1.Introduction to nonlinear dynamics
Science : Study of changes
• Engineering : Application of the understanding
of changes
• Evolution : Dynamics

Basic ideas
2
2
d x
F m
dt
=
System Type of dominant force
Pendulum Restoring force
Planetary system Gravitational force
Moving charges Electromagnetic
Stationary charges Electrostatic force
Atomic nucleus Nuclear force

Linear system
For a linear harmonic oscillator the restoring force F is given by
where K is spring constantF kx= −
K = 1

Force is directly proportional to the displacement x, that is F is a linear function of
x.
Equation of motion:
This is Linear differential equation
2
2
d x
m kx
dt
= −
2
2
02
0
d x
x
dt
ω+ =

Nonlinear system
Force: 3
F kx xλ= − −
k-1and α=1

Equation of motion
Nonlinear differential equation !!!
Generally, we call physical systems subjected to linear forces as linear
dynamical systems while systems driven by nonlinear forces as nonlinear
dynamical systems.
2
3
2
d x
m kx x
dt
λ= − −

Differential equation
• Linear
If one of the term has a total
degree either 1 or 0 in the
dependent variable, then it is
called linear differential equation.
Examples
• The linear superposition principle is
valid
• Nonlinear
If one of the term has a degree
different from 1 or 0 in the
dependent variable, then it is
called nonlinear differential
equation.
Examples
• The linear superposition principle is
not valid
2
0
2
2
2
0
0
dx
x
dt
d x dx
t
dt dt
ω+ =
+ + =
3
22
2
2
0
dx
x x
dt
d x dx
t
dt dt
− +
 
+ + = ÷
 
1 2 0x ax bx= + =
1 2 0x ax bx= + ≠

2.Effects of nonlinearities
Young's modulus cantilever
Load vs bending

Human heart beats
For normal body & bP patient

Chemical reaction
2NO + O2→ 2NO2
Rate equation
Where k is a constant
α & β are concentration of NO & O2
2
( ) ( )
dx
k x x
dt
α β= − −

3.Introduction to solitons
What is solitary wave ?
• It is a wave which travels very large distance without
changing its shape.
• Amplitude is constant
• Velocity is not change

John Scott Russell (1808-1882)
Union Canal at Hermiston, Scotland
- Scottish engineer at Edinburgh
First observation of Solitary Waves (1834)

“I was observing the motion of a boat which was rapidly drawn
along a narrow channel by a pair of horses, when the boat
suddenly stopped - not so the mass of water in the channel which
it had put in motion; it accumulated round the prow of the vessel
in a state of violent agitation, then suddenly leaving it behind,
rolled forward with great velocity, assuming the form of a large
solitary elevation, a rounded, smooth and well-defined heap of
water, which continued its course along the channel apparently
without change of form or diminution of speed…”
- J. Scott Russell
Great Wave of Translation

“…I followed it on horseback, and overtook it still rolling on at a
rate of some eight or nine miles an hour, preserving its original
figure some thirty feet long and a foot to a foot and a half in height.
Its height gradually diminished, and after a chase of one or two
miles I lost it in the windings of the channel. Such, in the month of
August 1834, was my first chance interview with that singular and
beautiful phenomenon which I have called the Wave of
Translation.”
“Report on Waves” - Report of the fourteenth meeting of the British Association
for the Advancement of Science, York, September 1844 (London 1845), pp 311-390,
Plates XLVII-LVII.

Scott Russells Experiment

From the Scott Russell experiment , He noticed some key properties.
• The waves are stable, and can travel over very large distances
• The speed depends on the size of the wave, and its width on the
depth of water.
• the velocity of the wave is given by the relation.
Where a is the amplitude of the wave, h is the depth of water and g is
the acceleration due to gravity.
( )c g h a= + ×

distant pacific storms produce nearly perfect
KdV soliton waves that travel from a reef
about 1 mi off the coast of Molokai, Hawaii
interaction of soliton-like surface waves in
very shallow water on Lake Peipsi, Estonia
in July 2003
interaction of two soliton waves in
shallow ocean water off the coast of
Oregon

6 0t x xxxu uu u+ + =
Korteweg de Vries (KdV) Equation
KdV Equation
t
x
u
u
t
u
u
x
∂
=
∂
∂
=
∂

4. Soliton solutions of KdV equation
2
( ), ( )
( )
sec ( )
2 2
u f x ct x ct
u f
c c
u h x ct
ξ
ξ
= − = −
=
 
= − 
 
A. Travelling wave method
u (x,t)
c = 1.4
t = 0

5.Solving KdV equation by Hirota’s method
Procedure
• Identify suitable bilinear transformation
• convert the bilinear form by using Hirota D operator
More general , Hirota defined
( ) ( ) ( ) ( )
'
' ' ' '
' ' '
, . , , . ,
.
n m
n m
x t
x x
D D F x t G x t F x t G x t
x x t t t t
=∂ ∂ ∂ ∂   
= − − ÷  ÷
∂ ∂ ∂ ∂ =   
( ) ( )' '
'
. .xD G F G x F x x x
x x
∂ ∂ 
= − = ÷
∂ ∂ 

( )
( )
2
2 2
4 2
2
2( )
. 2( )
D . 2 4 3
t t t
x x x
x xx x x xx
x t xt x t t x xt
x t xt x t
x xx x
x xxxx xxxx x xx
D G F G F FG
D G F G G FG
D G F G F G F FG
D D G F G F G F G F GF
D D F F F F F F
D F F F F F
F F F F F F F
× = −
× = −
× = − +
× = − − −
× = −
= −
= − +
Some of Hirota operators

• After making the bilinearizing transformation the next step is to decouple the
resulting equation suitably into a bilinear form, of which there are many
possibilities.
• Then in order to obtain soliton solutions, first the functions F and G are
expressed as a formal power series expansion in terms of a small parameter.
• This power series is terminated suitably and then the resulting linear PDEs at
various powers of “ε “ are solved recursively.
• Solve recursive those of system of linear PDE’s

Step-1 bilinearization
KdV equation is
Or
Dependent variable is given by
Substituting above and integrate with respect ‘ x’ , we have
6 0 (1)t x xxxu uu u+ + = →
( ) 2
, 2 logxu x t F= ∂
( ) ( )2
2 2 4 3 0 (2)xt x t xxxx xxx x xxF F F F F F F F F− + − + = →
( )2
3 0t xx x
u u u+ + =

Step 2. Application of the Hirota’s perturbation
Hirota’s ‘D’ operator is given by
By using Hirota’s operator the equation (2) becomes
Expand F interms of power series
(1) 2 (2)
1 .......... (4)F f fε ε= + + + →
( )
.
.
x x x
x t xt x t t x xt
D G F G F GF
D D G F G F G F G F GF
= −
= − − +
3
( ) . 0 (3)x t xD D D F F+ = →

Introducing F value and Equating each power of separately to zero, we get
system of linear PDE’s
'
ε
( )
0
1 1 1
22 2 2 1 1 1 1 1
: 0
: 0 (5 )
: 4 3 (5 )
xt xxxx
xt xxxx x t xxx x xx
f f a
f f f f f f f b
ε
ε
ε
+ = →
+ = + − →
( ) ( )2
2 2 4 3 0 (2)xt x t xxxx xxx x xxF F F F F F F F F− + − + = →
(1) 2 (2)
1F f fε ε= + +

From equation 5b, we have
The solution is given by
(1)
1
(0)
(6)
.
N
i
i
i i i i
f e
Where k x k t
η
η η
=
= →
= − +
∑
1 1
0xt xxxxf f+ =
3
i ikω =

Step-3 Construction of soliton solutions
One soliton Solution
For one soliton construction, we choose N=1 in equation (6)
Substitute above equation in (4)
Substitute F in Bilinear variable, we have
1(1) 3 (0)
1 1 1 1,f e k x k tη
η η= = − +
1
1F eη
= +
2
2 3 (0)1 1
1 1 1 1( , ) sec ;
2 2
k
u x t h k x k t
η
η η= = − +

Properties: one soliton
It confirms the John Scott Russell original relation . The one –
soliton solution is illustrated for the parametric value k1=1.2 at t = 0. The
figure shows that, solitary wave symmetrically situated at x = 0.The one
soliton has a peculiar property that is it travels with constant shape and
velocity.
( )2
c g h a= +

Two soliton solution
Proceeding in a similar way, Put N = 2 in (6), we have
Where
Substituting in 5b and solving it, we obtain
where
1(1) 2
,f e eη η
= +
(0)
1 1 3 1k x k tη η= − +
(0)
2 2 3 2k x k tη η= − +
( )
22 2 1 1 1 1 1
4 3 (5 )xt xxxx x t xxx x xxf f f f f f f b+ = + − →
(1)
f
1 1 12(2) A
f eη η+ +
= 12
2
1 2
1 2
( )
( )
A k k
e
k k
 −
=  
+ 
1 2 1 2 12
1 A
F e e eη η η η+ +
= + + +

Properties of two soliton
• When two solitary waves get closer, they gradually deform finally merge into a
single wave packet. This packet soon splits into two solitary waves with the same
shape and velocity before collision.
( )
2 2 2 2
2 2 2 2 1 2
2 1 2
2 2 1 1
cos ( / 2) sec ( / 2)1
( , )
2 ( cot( / 2)) tanh( / 2)
k ech k h
u x t k k
k k
η η
η η
 +
= −  
− 

The above equation has been studied by Brushchi and Ragnisco [22], who
derived an infinite number of constants of motion to prove the integrability of the
equation.
( )4 0 2.1t x xxu uu iu+ + = →
u
u Nonlinear term
x
∂
⇒
∂
2
2
u
i diffusionterm
x
∂
⇒
∂
Complex Burgers equation

( )
( )
( )
,
, (2.2)
,
G x t
u x t
F x t
= →
Let us introduce the following dependent variable transformation
Substituting (2.2) in (2.1), we get
[ ] [ ] [ ]2 3 2 3
4 2 2 0 (2.3)t t
x x xx x x xx xx x x xx
FG GF G i iG
FG GF G F G F GF F F F F FF
F F F F
−
+ − + − + − − + = →

Step 2. Application of the Hirota’s perturbation
By using Hirota’s operator the equation (2.3) becomes
( )
( )
.
.
.
x x x
x t xt x t t x xt
x t xt x t t x xt
D a b a b ab
D D a b a b a b a b ab
= −
= − − −
= − − +
( )
2
2
2 3 2 3
. .
4 . . 0 2.4t x
x x
D G F D G FG G
D G F i i D F F
F F F F
+ + − = →

Now decoupling above equation into the coupled form for F and G are chosen as
( )2
4 . . 0 (2.5 )x xD G F iD F F b− = →
( )2
. 0 (2.5 )t xD iD G F a+ = →

To obtain the exact solution of (2.1), we now expand F and G as a formal power series in
the parameter ε :
One solitary wave solution
To obtain one solitary solution, we substitute n = 1, in (2.6) and using the equation G
and F in (2.4) and equate the coefficients of ε to zero. We obtain following linear
equations
1 0
1 (2.6)n
n n
n n
F f G gε
∞ ∞
= =
= + = →∑ ∑
( ) ( )1 1 0 (2.8)t xx
g i g+ = →
( ) ( )1 14 2 0 (2.9)x xx
g i f+ = →

Let us start with solution
where and are an arbitrary complex parameters. Using (2.10) in
(2.8), we have
Similarly we rewritten equation (2.9) as follows
1 1exp (2.10)g η= →
( )1 1 2 ,p x ip tη = + 1p 2p
( )2
1 1 1exp (2.11)g p x ip t= − →
( ) ( )1 12 (2.12)xx x
f i g=− →

( )2
1 1 1
1
2
exp (2.13)
i
f p x ip
p
= − − →
Therefore, the one soliton solution of (2.1) is
( )
( )
( )
( )
1
1
1
exp
, (2.14)
2
1 exp
u x t
i
p
η
η
= →
−

One soliton solution of complex burgers equation is plotted for the value at
t = 0. This figure confirms that one soliton solution of complex burgers
equation is kink type. This kink soliton also behaves like usual K-dV soliton
solution.
1 1p =
u (x,t)
t = 0

Two solitary wave solution
To obtain two-solitary wave solution of complex burgers equation, we put n = 2 in
(2.6) and using it in (2.5) and equating the coefficients of . We have the following
equation,
By solving above equations and simplifying, we get
and
2
ε
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 1 1 1 1 1 1 1 1 22 0t t t xx x x xx xx
g f g f g i f g i f g i g f i g+ − + − + + =
( ) ( ) ( ) ( ) ( ) ( )
2
1 1 2 1 1 1 1 2 14 4 4 2 2 2 0.x x x xx xx x
g f g g f i f f i f i f+ − − − − =
( ) ( )1 2exp expG η η= + ( ) ( )1 2
1 2
2 2
1 exp exp .
i i
F
p p
η η
   
= − − ÷  ÷
   

( )
( ) ( )
( ) ( )
1 2
1 2
1 2
exp exp
, .
2 2
1 exp exp
u x t
i i
p p
η η
η η
+
=
   
− − ÷  ÷
   
Substituting the expression of G and F in (4.2), we get two soliton solution of
complex burgers equation.

We consider the important physical system proposed by Cheng
corresponding to dynamics of photo sensitive molecule when the light
beam pass through them. Consider a uniformly distributed
photosensitive molecule in liquid or film irradiated by a light beam. As
the beam penetrates the emulsion, it is absorbed by the molecules,
which decompose and become transparent to light. This can be
represented by
where a, b are the absorption constant and proportionality constant
respectively. Also the variable U and V are the light intensity and the
density of molecule respectively.
Cheng chemical reaction equation
,xU aUV= −
,t xV bU=

The equations (3.1) and (3.2) can be transformed into Hirota’s bilinearization form
using following variable transformations
And
1
(3.3)xF
V
a F
= →
( )
( )
,
(3.4)
,
G x t
U
F x t
= →

And
2 2
0x x xG F GF GF
F F
−
+ =
2 2
1
0 (3.6)xt x t x xFF F F G F GF
b
a F F
− −   
− = →   
   
2
0 (3.5)xG F
F
= →
2 2
1
0 (3.7)xt x t xFF F F GF
b
a F F
−   
+ = →   
   

we make the transformations
and
where , P(x) and Q (t) are arbitrary functions.
Using above transformation in equation (3.7), we get the
( )11 expF θ= + ( )
_
1expG θ=
( ) ( )P x Q tθ = +
1
1
exp tG
ab
θ θ
− −  
= = ÷  ÷
   

( ) ( )( )
1
,
1 exp
tQ
U
ab P x Q t
= −
 + + 
( ) ( )( )
( ) ( )( )
1 exp1
1 exp
xP P x Q t
V
a P x Q t
 + +   =  ÷
 + +   
Therefore, the solutions of the equation (3.1) and (3.2) are written as

Similarly for Two – solitary wave solution
And
Substituting these in equation (3.7), we get
And
( ) ( )1 21 exp expF θ θ= + +
( ) ( )
_ _
1 2exp exp .G θ θ= +
( )1 1
1
exp t
ab
θ θ
− −  
= ÷  ÷
   
( )2 2
1
exp .
ab
θ θ
− −  
= ÷  ÷
   
( ) ( )
( ) ( )
1 2
1 2
exp exp1
1 exp exp
U
ab
θ θ
θ θ
+ 
= − ÷
+ + 
( ) ( ) ( ) ( )
( ) ( )
1 1 2 2
1 2
exp exp1
,
1 exp exp
x x
V
a
θ θ θ θ
θ θ
 +
=  
+ +  

4.Liouville equation
Liouville equation arises in the context of interactions of relativistic particles and fields.
Under the transformation
Introducing this in equation (4.1), we get
Now introducing dependent variable transformation
0 (4.1)U
xtU e− = →
logU V=
3
0 (4.2)xt x tVV V V V− − = →
2
(4.3)
G
V
F
= →

2 2 3
2 3 4 4
2 2 0 (4.4)xt x t xt x tGG G G G F G F F G
F F F F
−
− − − = →
( )
( )
.
.
.
x x x
x t xt x t t x xt
x t xt x t t x xt
D a b a b ab
= −
= − − −
= − − +
. 0 (4.5)x tD D G G = →
0 (4.6)xtF = →
2 (4.7)x tF F G= →

Solving equation (4.6), we obtain
where A(x) and B(t) are the arbitrary functions and substituting this in equation (4.7),
we get
Substituting above two expressions in (4.3), we get
introducing this in equation (4.2), we get the general solution of Liouville equation.
( ) ( ) (4.8)F A x B t= + →
2 (4.9)x tG A B= →
( )
2
2 .x tA B
V
A B
=
+ ( )
2
log 2 .x tA B
U
A B
 
=  ÷
 ÷+ 

Conclusion
Soliton theory is a subject which began to attract the attention of many scientists
with the study of nonlinearity in dynamical systems. I studied and
summarized following
• The analogy between linear and nonlinear systems which are solvable
or not, can be identified by using linear superposition principle.
• John Scott Russel’s observation of solitary wave and remarkable
stability properties in the year 1834.
• In 1895, Kortweg de Vries formulating nonlinear equation for solitary
wave and find single soliton solution by travelling wave phenomena.
• In 1971, Hirota’s technique used to solve KdV equation. This method
only applicable to simple PDE’s.

Future of Solitons
"Any where you find waves you find solitons."

1. Nonlinear dynamics -M.lakshmanan
2. Methods for exact solution, Comm. Pure and Appl. Math. 27 (1974),
pp. 97-133
3. R. Miura, The Korteweg-de Vries equation: a survey of results,
SIAM Review 18 (1976), No. 3, 412-459.
4. Solitons Home Page: http://www.ma.hw.ac.uk/solitons/
5. Light Bullet Home Page:
http://people.deas.harvard.edu/~jones/solitons/solitons.html
References

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