compact on talk 4

8/6/2019 Compact on Talk 4

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Compactons (solitary waves with compact support) inGeneralized Korteweg-DeVries Equations

Talk by Fred Cooper (Santa Fe Institute, LANL) atUniversity of Delhi March 2010

Done in Collaboration with Avinash Khare (Inst. ofPhysics, Bhubaneswar), Bogdan Mihaila and Avadh

Saxena (LANL), and Carl Bender (WashingtonUniversity, St. Louis).

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We discuss generalizations of the K-dV equation by Rosenau and Hyman(RH) and Cooper Shepard and Sodano (CSS) as well as a new PT symmetric

extension by Bender, Cooper, Khare, Mihaila, and Saxena

We find a new two parameter family of compact solitary wave solutions ofthe form AZ((x + ct))), continuous, where we require (Z)2 = 1 Z2qand q is continuous. This family includes all previous solutions

We find an exact relation for all solitary wave solutions that the Height,Width and velocity are related in a simple fashion. To show this one only

needs to know that a solution of the form AF((x + ct)) is a minimum ofthe reduced Hamiltion as a function of for fixed total Momentum.

We find find E = rP c for all the solitary wave solutions, where r is explictilyknown.

Using (Z)2 = 1 Z2q we find explicit expressions for E, and P.

We determine the domain of stability for the new solutions using variousmethods.

We discuss some shocking new results of numerical studies.

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1 Introduction

Solitary waves appear in many nonlinear dynamical systems described by

nonlinear partial differential equations.

If we KNOW there are organized structures (solitary waves, vortices, blobs)can we determine their gross features without solving the equation?

In the 1990s using variational wave functions with time dependent param-eters for width, height and position of center. we found universal relations

between height, position and velocity, as well as Energy, Momentum andstability which seemed independent of the choice of trial function.

We will discuss several equations having compact solitary wave solutions,discuss new solutions and show that previous results can be DERIVED with-

out recourse to explicit trial functions, using related variational methods

(Minimization of ACTION).

We will discuss some interesting new results found numerically about com-pacton collisions.

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Compactons were discovered originally in an extension of the KdV equation by

Rosenau and Hyman [1].

K(m, n) : ut + (um)x + (un)xxx = 0, (1)

For m = n these compactons had the property that the width was independent

of the amplitude. (m = 2, n = 1 is the KdV equation).

To study the solitary waves, one lets u = f(y) where y = x vt in eq. 1 toobtain (f = df/dy)

vf =

d

dy(f

m

) +

d3

dy3 (f

n

). (2)Integrating once we obtain

vf = fm +d2

dy2(fn) + C1 (3)

Muliplying byd

dy(fn

) (4)

and integrating we obtain the equation

n

n + 1vfn+1 =

n

n + mfn+m +

1

2[d(fn)

dy]2 + C1f

n + C2 (5)

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The compactons are solutions with C1 = C2 = 0. f satisfies the differential

equation

(f)2 =2v

n(n + 1)f3n

2

n(n + m)fmn+2. (6)

For m = n 3 the solutions aref = A cos2/(m1)((x vt)); /2 y /2. (7)

and zero elsewhere. In particular for (m, n) = (2, 2)

= 1/4; A =

4

3v (8)

The (2,2) equation and compacton scattering have recently been studied care-

fully numerically as shown in Fig. 1. The compactons are weak solutions be-

ing a combination of a compact object f(x) confined to a region (say initially

x0 < x < x0 and zero elsewhere. At the boundaries x0 the function is as-sumed continuous, f(x0) = 0, but the derivatives most likely are not. For there

to be a weak solution we need that the jump in the quantity second integral ofthe compacton differential equation to be zero: For the RH equation we have

v

n + 1fn+1 f

n+m

n + m n(f)2f2n2 (9)

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must be zero when we go from x0 to x0 + .Disc[(f)2(x)f2n2(x)]x0 = 0. (10)

This is always satisfied if there is no infinite jump in the derivative of the function.

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2 CSS equation

K(l, p) : ut = uxul

2

(11)+

2uxxxup + 4pup1uxuxx + p(p 1)up2(ux)3

(12)

can be derived from

L(l, p) =

1

2xt (x)

l

l(l

1)+ (x)

p(xx)2

dx. (13)

u(x) = x(x). (14)

RH set (m, n) corresponds to the CSS set (l 1, p + 1).)

2.1 Exact solitary wave and compacton solutions

Let:u(x, t) = f(y) = f(x + ct), (15)

cf = ffl2 +

2ffp + 4pfp1ff + p(p 1)fp2f3

. (16)

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Integrating we obtain:

cf =fl1

l

1+ [2f

fp + pf2fp1] + C1 (17)

Integrating again:

c

2f2 f

l

l(l 1) f2fp = C1f + C2. (18)

C1 and C2 are zero for compactons Well behaved : : l > 1 and ffp

0, f2fp1 0 at edges where f 0. Then we obtain

f2 = c2

f2p fl

p

l(l 1). (19)For finite f at the edges, we must have p 2, l p. COMPARE WITH RH

(f)2 =2v

n(n + 1)f3n 2

n(n + m)fmn+2 (20)

l = m + 1 and p = n 1 equations identical in form, differing coefficients.

2.2 l = p + 2 (m = n)

Cooper and Khare [4] were able to show that the CSS equation has solutions

u(x, t) = A cos2/p[(x + ct)] (21)

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for noninteger p, the conserved energy and momentum are relatied:

E =2P c

p + 2. (22)

2.3 elliptic and hyperelliptic solutions

Both the RH and CSS equations have elliptic and hyperelliptic solutions: In the

RH equation: For m = 2n 1 with n a continuous variable 1 < n 3 there is acontinuous class of elliptic function compacton solutions of the form (y = x + vt)

f(y) = Acn(y,k2 = 1/2) (23)

with = 2/(n 1).A2n2 =

3n 1n + 1

v; 44 =16v

n2(3n 1)(n + 1). (24)where cn is a Jacobi Elliptic function. The most general compacton solution to

the RH equation is found by letting

f = AZa(y) (25)and requiring that it is a solution of the RH equation which is of the form:

dZ()

d

2

= 1 Z2q() (26)

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Here we allow for m,n,q to be continous variables. We find the conditions for

this are that

m = q(n

1) + 1; a = 2/(n

1)

Am1 = vn + m

n + 1; a22A2 =

2

n(n + m)Amn+2. (27)

So that the width is given by

2 =(n 1)2

2(n + m)

(n + m)v

n + 1

(mn)/(m1)(28)

Similarly for the CSS equation:When l = 2p + 2 we find for p 2

f = Acn(y ; k2 = 1/2) (29)

(f)2 =c

2f2p f

p+2

(2p + 2)(2p + 1). (30)

Matching terms:

=4

l 2;c

2=

Al2

l(l 1)42 =

2c

l(l 1)(l 2

4)2

(31)

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Again letting

f = AZa( = y) (32)

and demanding that

(Z)2 = 1 Z2q (33)we find the conditions:

l = pq + 2; a = 2/p

Al2 = l(l

1)c

2

; a22A2 =c

2

A2p. (34)

From this we get

2 =c

2a2Ap(35)

Apart from different numerical coefficients in and A the solutions to the CSS

and RH equations are identical. For well behaved solutions we need 0 < p 2.Previous solutions are for the cases, q = 1, 2, 3. We will show that all of these

solutions lead to E/P = c/r and are stable for p(q 1) < 4.

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3 Calculus of Variations and the Euler-Lagrange Equation

The advantage of the CSS equation and its generalizations is that the equations

can be obtained from a Variational principle which allows one to study stabilityfrom a Hamiltonian dynamics perspective. Consider a function

F(y,dy

dx, x) (36)

and the integral

I[y(x)] =b

aF(y,

dy

dx, x)dx (37)

I is called a functionalofy(x) since it assigns a number to each function y. Nowwe wish to choose y(x) so that I is stationary with respect to arbitrary small

variations.

y(x) y(x) + (x) (38)To convert this into an ordinary minimization problem consider instead

y(x)

y(x) + (x) (39)

with small and (x) arbitrary

Then ifI[y(x)] is to be stationary then this requires:

dI

d|=0 = 0. (40)

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Expanding I in a Taylor series about = 0 we obtain

I() =b

aF

y + ,

dy

dx+

d

dx, x

dx

= I(0) + b

a

F

y +

F

y d

dx

dx + O(2)

where

y dydx

(41)

We obtain taking the variation

0 =

ba

Fy

+ Fy

ddx

dx

= F

y |ba +

ba

F

y d

dx

F

y

(x)dx

Since (x) is an arbitary function and we will require the variations vanish at the

boundary

(a) = (b) = 0. (42)Since (x) is an arbitrary function we obtain the Euler Lagrange equation:

F

y d

dx

F

y = 0. (43)

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4 Action Principles

4.1 Particle Dynamics

Variables q, q = v. Kinetic Energy = T = 12mv2 Potential Energy V(q).

L[q, q] = T V; H = pq L p = Lq

=t2

t1Ldt

Want to show that Newtons laws follow from stationarity of the action . In

what follows we use the short hand

q(t) q(t) + q(t); q(t) = (t)q(t)) q(t) + q(t); q(t) = d(t)

dt

Hamiltons Principle:

= 0 LagrangesEquations = dt Lq ddt Lq q

p = Vq

ddt

L

q=

L

q(44)

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4.2 Classical Field Theory

Variables (x, t), x, xx... t, tt... Lagrangian:

L =

dxL[(x, t), x, xx...t, tt] (45)Action:

S =

dtL (46)

Action Principle

S = 0 ClassicalFieldEquations (47)

5 Conservation laws and canonical structure

Equation (12) can be written in canonical form

ut = xH

u= {u, H} (48)

where H is the Hamiltonian

H =

[() L] dx

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= (x)

l

l(l 1) (x)p(xx)

2

dx,

= u

l

l(l 1) up(ux)2

dx. (49)

(50)

Poisson bracket structure

{u(x), u(y)} = x(x y). (51)

5.1 Conservation of Mass

Integrating

ut = xH

u(52)

over all space, the area under u(x, t) is conserved and is called the Mass:

M =

u(x, t)dx (53)

5.2 Momentum Conservation

Multiplying eq. 52 by u(x, t)

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t(u2

2) = x

u

l

l+ {(p 1)upu2x + 2up+1uxx}

(54)

Integrating over all space

(1/2)

u2(x, t)dx = P (55)

P is the generator of the space translations:

{u(x, t), P} = ux

. (56)

5.3 Energy Conservation

From Lagranges equations we immediately get a

H = u

l

l(l 1) up(ux)

2

dx. (57)

5.4 Conserved Quantities in the RH equation

For the RH equations, in general there are 2 conservation laws.

M =

dxu(x, t); Q =

dxun+1(x, t). (58)

For m = n there are two further conservation laws.

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6 Energy-Momentum relationship

Starting point is the action

=

Ldt, (59)

where L is given by (13).

L(l, p) = 1

2xt (x)

l

l(l 1) + (x)p(xx)

2

dx (60)

If we assume the exact solitary wave solution is of the generic form

x = AZ((x + q(t))) = u, (61)

then we find that

t = xq. (62)

Integrating over the wave function , we can rewrite

1

2

x

tdx = Pq (63)

which shows we have replaced the original problem with a particle mechanics

problem. Here

P =1

2

u2(x, t)dx (64)

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Particle Lagrangian

L = Pq H (65)

H =

dx

u

l

l(l 1) up

(x)(ux)2

(66)Differentiating our ansatz Eq. 61 we have

ux = AZ[(x + q(t))] (67)

The value of the Hamiltonian for the solitary wave solution is just

H = C1(l)Al

l(l 1) Ap+2C2(p) (68)

where

C1(l) =

Zl(z)dz; C2(p) =

[Z(z)]2Zp(z)dz (69)Since H is independent ofq,

P = Hq

= 0, (70)

we obtain again that P is conserved.

We show in the appendix that the exact solutions minimize the Hamiltonian

with respect to with P held fixed We can rewrite A in terms ofP by using

P =1

2

dxu2 =

A2

2C (71)

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C =

dzZ2(z) (72)

So that we have

A2 = 2PC

(73)

H = C3(l)Pl/2(l2)/2 C4(p)P(p+2)/2(p+4)/2 (74)

where

C3(l) =C1(l)

l(l

1)[

2

C]l/2; C4 = C2(p)[

2

C](p+2)/2 (75)

H

|P=constant = 0, (76)

we obtain

= Ppl+2lp6

C4

C3

p + 4

l 2

2/(lp6). (77)

ELIMINATING

H = f(l, p)Pr (78)

r =p + l + 2

p + 6 l (79)

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and

f(l, p) =

p l + 2

p + 4

C3(l)

C4(p)

C3(l)

p + 4

l

2

(l2)/(lp6). (80)

Hamiltons equation now yields

q =H

p= r

H

P(81)

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For the exact solutions satisfying the Hyper-elliptic equation (as well as the

generalized K-DV we can use the fact that the solution obeys the Hyperelliptic

equation to get an exact expression for H and P as well as verify that H = rcP.

H =

dx

2f

l

l(l 1) c

2f2

(82)

Letting f = AZa( = y), with Al2 = l(l 1)c/2We can write H as

H =

A2c

2

d[2Z

a(pq+2)

() Z2a

()] (83)

P =A2

2

dZ2a() (84)

We now use the equation for Z to change variables from to Z since

dZ/d =

1 Z2q (85)

H =A2c

10

dZ1 Z2q [2Z

a(pq+2) Z2a] (86)

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This can be evlauated in terms of the Beta function B(, ) by substituting

t = Z2q. We then obtain

H =A2c

2q

(6 + p

l)

(l + p + 2) Bp + 4

2pq ,1

2

. (87)

Similarly evaluating P we obtain

P =A2

2qB

p + 4

2pq,

1

2

. (88)

where we have used the previous result that a = 2/p, and a(l

2) = 2q Thus we

find for all these solutions with p, l continuous varables that

H/P = c/r (89)

7 Stability of Solutions

The stability problem at q = 1 was studied in [10], using the results of Karpman

[11][12]. The result of detailed analysis is that the criteria for Linear Stability is

equivalent to the condition,P

c> 0. (90)

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We have

P =A2

2qB

p + 4

2pq,

1

2

. (91)

where the dependence of A and on c can be obtained from:

Al2 = l(l 1)c2

; a22A2 =c

2A2p. (92)

From this we deduce that stability requires: (q = (l 2)/p)p(q 1) < 4 (93)

The requirement for non-singular solutions is that 0 < p

2. This means for

q 3, compactons with arbitary p in the allowed range are linearly stable. Whenq > 3 , the compactons are stable only for

0 < p < 4/(q 1). (94)Analysis of Lyapunov stability following [10] [11] [12] leads to the same restric-

tions on p.

8 PT extensions of the CSS equation

For the ordinary KdV equation there has been recent interest in extensions of

the equations that are invariant under PT symmetry. These extensions exist in

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the complex plane but also lead to new equations that are entirely real. The first

extension of the KdV equation by Bender et. al. [14] was a study of the equation:

ut iu(iux)

+ uxxx = 0. (95)which reduced when = 1 to the usual KdV equation and which was analyzed for

= 3 in [14]. This extension of the KdV equation was however not a Hamiltonian

dynamical system at arbitrary . To find extensions of the generalized KdV

equation which are invariant under the joint operation of Mirror inversion (P) and

time inversion (T), we use the following facts: Under the operation P, x x. Since u is a velocity, applying the operation P , u u . Under the timereversal operation T, we have i i , u u , and t t. Thereforeiux is P T even. We find that an appropriate P T symmetric Lagrangian which

generalizes Eq. (13) is [13]:

LP T =

dx

1

2xt +

(x)l

l(l

1)

+ (x)p(ixx)

m

(96)

For the Lagrangian we will need to find the correct P T symmetric contour which

is on the real axis for m = 2. For P T to be a good symmetry, branch cuts

need to be taken along the positive imaginary axis in the complex x plane. The

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Hamiltonian resulting from the above Lagrangian is given by

H =

dx

u

l

l(l

1) up(iux)m

(97)

When m is an even integer, a convenient choice for which allows for solitary

wave solutions and a real equation for the generalized KdV system is

im = 1m 1. (98)

9 Traveling wave solutions

Starting with:

ut +ul2ux p

m 1x

up1(ux)m

+m

m 12

x2

upum1x

= 0. (99)

We assume thatu(x, t) = f(x ct) f(y) (100)

Then

cf = fl2f +

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1

m 1p d

dx

fp1(f)m

m d

2

dx2

fp(f)m1

.

(101)

Integrating once we obtain:

cf =fl1

l 1 +mfp(f)m2f + p(f)mfp1 + K1

. (102)

For compact solutions K1 is zero. If we set the integration constant to zero

and multiply this equation by f and integrate over y we obtain:

cI2 =1

l 1Il p + m

m 1Jm,p (103)where

In = f

l(y)dy; Jm,p =(f

)m fp(y)dy. (104)

Integrating twice as before and setting the integration constants to zero we get

:c

2f2

fl

l(l 1) (f)m

fp

= 0.. (105)For p = 1, l = 3 one has : For p = 1, l = 3 the equation for the solitary wave is :

c

2f f

2

6

= (f)m (106)

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So we are led to (for the positive branch of the solution) the differential equation

x ct =f

0

du

[ c2

u

u2

6]1/m

(107)

Integrating this equation one obtains:

x ct = 2 1m3m1m cm2m B f3c

m 1

m,

m 1m

(108)

For m = 2 this simplifies to

x ct = 2

6sin1

f

3c

(109)

Which leads to the previous compacton result:

f = 3c sin2 1

2

6(x ct)

(110)

For m = 4 we get instead

x ct = 42 33/4 c B f3c

34, 34 B 34, 34

(111)

In Fig 4 we plot [B f3c

34,

34

B

34,

34

as a function off /3c and its mirror image.

Here y = (x ct)/( 4233/4c)28


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Appendix- Alternative Generating Function and stability

The equation for the solitary waves of the form

f(y = x ct)can be derived by considering the following function:

[f(y), f(y)] =

dy (H[f, f] + P[f]c)

dx [f, f] (112)

(We notice that is the negative of the Lagrangian density). That is the original

equation for the solitary wave can be written as

x

f= 0. (113)

The once integrated equation (with no constants) Eq. (102) is obtained from

f= 0. (114)

or equivalently the Euler Lagrange equation:

f=

d

dx

f. (115)

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Explicitly we have:

=

dy

f

l

l(l

1)

+ fp(f)m +1

2cf2

(116)

The first variation after an integration by parts can be written as

=

dy

f

l1

l 1 + cfpfp1(f)m mfp(f)m2f

f (117)

The second variation which is important for the Linear Stability analysis can be

written as

2 =

dyf L f, (118)

Where L is the operator

L = c fl2 p(p 1)fp2(f)m mpfp1(f)m2f

mpfp1(f)m1 ddy

m(m 2)fp(f)m3f ddy

mfp(f)m2 d2

dy2. (119)

When m = 2 this reduces to the result given in Dey and Khare [18]. One can

write in terms ofIl and Jm,p as follows:

[f] =1

m 1Jm,p 1

l(l 1)Il +c

2I2 (120)

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Following Derrick [15], we consider the scale transformation x x. Under thistransformation we find:

Il[f(y)] =

1

Il; Jm,p[f(y)] =

m

1

Jm,p, (121)so that

[f(y)] =1

m 1m1Jm,p Il

l(l 1) +c

2I2. (122)

If we assume that taking the derivative of with respect to and setting = 1

gives a solution, then we find:

d()

d |=1 = 0 = Jm,p +I

ll(l 1)

c

2I2 (123)

This is indeed the equation of motion integrated over space which we found

before (Eq.105). Derrick then considered the second derivative and looked at

whether this became negative, indicating that the solitary wave was unstable. If

we determine the second derivative we obtain:

d2()

d2 = cI2(l

2)

2 (m 1)ml2 m2 7m 2p + 4 l + 2(3m + p 2)

l(l 1)[l(1 + m) + m + p](124)

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This does not factorize to give a simple criteria for stability. However another

choice will lead to a simple stability criteria.

Suppose we instead make the scaling

f(y) f(y) (125)This again will lead to the equations of motion plus a boundary term since:

d

d|=1 =

dy

f d

dx

f

(f + xf)

+

f

(f + xf) |ymax

ymin = 0. (126)

Assuming the boundary term vanishes at the edges of the compacton, one recovers

the equation of motion. We have

[f(y)] =1

m 1(m1+(m+p)Jm,p Il

l(l 1)l1

+

c

2

2

1

I2. (127)

The condition for a minimum is

d()

d|=1 = 0 =

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(m 1 + (m + p)m 1 Jm,p +

(l 1)Ill(l 1)

c

2(2 1)I2.

(128)

The particular case = 1/2 is special in that the conserved momentum P is

invariant under this transformation. That is

P[1/2f(y)] = P[f(y)]. (129)

So if we choose this value we find that when we vary , we are varying the

Hamiltonian with the constraint that P is held fixed. This particular variation

was first considered by Kuznetzov [17] and then further elaborated on by Karpman

[19] and Dey and Khare [18].

For = 1/2 we obtain that

Jmp =(l 2)l(l 1)

(m 1)(3m + p 2)Il (130)

which can be derived directly from the equations of motion. For arbitrary wewould get a linear combination of the equations Eq. 102 and Eq. 105. For

arbitrary the second derivative does not factorize into a simple form which

allows one to say when it changes sign. However for = 1/2 the answer does

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factorize and the second derivative yields

d2()

d2|=1 = P c(l 2)

4

(3m + p l)(3m + p 2)[l(

1 + m) + m + p

(131)

The conditions for the existence of a compacton solution that

p 2;p l (132)leads to the statement that solitary waves will be unstable under this type of

deformation when

l > p + 3m. (133)

References

[1] P. Rosenau and J.M. Hyman, Phys. Rev. Lett. 70, 564 (1993).

[2] P. Rosenau Phys. Lett. A 275, 193 (2000).

[3] F. Cooper, H. Shepard, and P. Sodano Phys. Rev. E 48, 4027 (1993).

[4] A. Khare and F. Cooper, Phys Rev. E 48, 4843 (1993).

[5] F. Cooper, J. Hyman, and A. Khare, Phys Rev. E 64, 026608. (2001).

[6] A. Das, Integrable Models (World Scientific Lecture Notes in Physics, Singapore, 1989) Vol.30.

[7] F.Cooper, H. Shepard, C. Lucheroni, and P. Sodano, Physica D68 (1993), 344. hep-ph/9210234

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[8] F. Cooper, C. Lucheroni, H. Shepard, and P. Sodano, Phys. Lett. A 173, 33 (1993).

[9] P.F. Byrd and M.D. Friedman Handbook of Elliptic Integrals for Engineers and Scientists 2nd Edition(Springer, Berlin, 1971).

[10] B. Dey and A. Khare, Phys. Rev. E 58, R2741 (1998).

[11] V. I. Karpman, Phys. Lett. A. 210, 77, 1996.

[12] V. I. Karpman, Phys. Lett. A. 215, 254, 1996.

[13] Compactons In PT symmetric generalized K-dV Equations Carl Bender, Fred Cooper, Avinash Khare,Bogdan Mihaila and Avadh Saxena , arXiv:0810.3460v1 [math-ph] Pramana, vol. 73 (2009) No. 2, 375-385.

[14] C. Bender, D. Brody, J. Chen, and E. Furlan arXiv: math-ph/06010003 (2006).

[15] J. Math. Phys. 5, 1252 (1964)

[16] O. Holder Nachr . Ges. Wiss. Gottingen 38 (1889). See also G.H. Hardy, J. E. Littlewood and G. Polya,Inequalities , Cambridge Univ. Press, ISBN 0521358809 (1934).

[17] Kuznetzov, Phys. Lett. 101, 314 (1984)

[18] B. Dey and A. Khare, Phys. Rev. E58, R2741 (1998).

[19] V.I. Karpman, Phys. Lett. A210, (1996) 77.

[20] V.I. Karpman, Phys. Lett. A215, (1996) 254 and references therein.

[21] B.Dey, A. Khare and C.N. Kumar, Phys. Lett. A223, 449 (1996).

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Figure 1: (Color online) Collision of two m = n = 2 compactons with c=1 and 2. The simulation is performedin the comoving frame of reference of the first compacton. The collision is shown to be inelastic, see Fig. 3.

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Figure 2: (Color online) Dynamics of the ripple created as a results of the collision depicted in Fig. 1. Thisripple features two shock components when the ripple switches from negative to positive values.

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Figure 3: (Color online) Dynamics of a blob decomposition into two compactons and ripple featuring aset of compacton-anticompacton pairs. Similarly to the collision problem, the ripple has positive- andnegative-value components, separated by a shock front. This simulation was performed using the (6,4,4)scheme described in text.

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0.2 0.4 0.6 0.8 1fy3c

-1.5

-1

-0.5

0.5

1

1.5

y

Figure 4: f/3c vs. y for m = 4, l = 3 p = 1

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compact on talk 4

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