We investigate the spectral properties of chaotic quantum graphs. We
demonstrate that the `energy'--average over the spectrum of individual graphs
can be traded for the functional average over a supersymmetric non--linear
$\sigma$--model action. This proves that spectral correlations of individual
quantum graphs behave according to the predictions of Wigner--Dyson random
matrix theory. We explore the stability of the universal random matrix behavior
with regard to perturbations, and discuss the crossover between different types
of symmetries.Comment: 15 pages, Refte

Cooper, Fred

Khare, Avinash

Saxena, Avadh

English

arXiv

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg-De Vries equation K∗ (l,p). Specifically we find that q=r(l,p)H/P where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two-parameter family of exact solutions to these equations, which include elliptic and hyper-elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria

Publications of the IAS Fellows

ar
X
iv
:n
lin
/0
50
80
10
v1
  [
nli
n.P
S]
  4
 A
ug
 20
05
Exact Elliptic Compactons in Generalized Korteweg-DeVries Equations
Fred Cooper∗,†,‡, Avinash Khare∗∗, Avadh Saxena†∗
∗National Science Foundation, Division of Physics, Arlington, VA 22230, USA
†Santa Fe Institute, Santa Fe, NM 87501, USA
‡Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA† and
∗∗Institute of Physics, Bhubaneswar, Orissa 751005, India‡
(Dated: February 8, 2008)
We derive a general theorem relating the energy, momentum and velocity of any solitary wave solution of the
generalized KdV equation which enables us to relate the amplitude, width, and momentum to the velocity of
these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two param-
eter family of exact solutions to these equations which include elliptic and hyper-elliptic compacton solutions.
For this general family we explicitly verify both the theorem and the stability criteria.
In a variety of physical contexts one finds nonlinear field
equations for which a wide class of solitary wave solutions
can exist. However, in many cases one is not able to obtain the
solution in a closed form and thus it is not very easy to study
the stability of such solutions. In this Letter we derive a gen-
eral theorem relating the energy, momentum and velocity of
any solitary wave of a generalized Korteweg-De Vries (KdV)
equation of Cooper, Shephard and Sodano (CSS)1. The im-
portant point is that this particular generalization of the KdV
equation is derivable from an Action Principle. Using the the-
orem, we are able to relate the amplitude, width and velocity
of any of the solitary wave solution even if such a solution is
not known in a closed form and also study its stability. Sec-
ondly, we obtain a two-parameter family of solutions to these
equations which include elliptic and hyper-elliptic compacton
solutions. For this general family we explicitly verify the the-
orem as well as the stability criteria.
Compactons were discovered originally in an extension of
the KdV equation by Rosenau and Hyman in Ref. 2. Com-
pactons are fundamental excitations (i.e., solitary waves with
a compact support) of KdV-like equations that possess a non-
linear dispersion and collide quasi-elastically. They play an
important role in pattern formation and emergence of nonlin-
ear structures in physical systems. Other physical contexts
in which compactons are relevant include fluid dynamics, op-
tical waveguides and the field of intrinsic localized modes3.
Breather compactons4,5 and compactons in other nonlinear
dispersive equations6 are also known. In addition, KdV equa-
tions with higher power of nonlinearity and dispersion may
lead to the phenomema of blow-up and collapse. Therefore,
the study of compactons and their stability in this class of
equations is important in its own right.
Rosenau and Hyman showed that in a particular generaliza-
tion of the KdV equation, defined by parameters (m,n) (with
m,n integers), namely
K(m,n) : ut + (u
m)x + (u
n)xxx = 0 , (1)
a new form of solitary wave with compact support is a solution
of this equation. For the case m = n (m integer) these com-
pactons had the property that the width was independent of the
amplitude. In Ref. 2 it was stated that K(3, 2) had an elliptic
function solution. In a later work, Rosenau7 obtained ellip-
tic function compactons for the case of K(4, 2) and K(5, 3).
Phase compactons have also been investigated8. Because the
equations of Rosenau and Hyman were not equivalent to a
Hamiltonian dynamical system, CSS1 considered instead a re-
lated generalization of the KdV equation
K∗(l, p) : ut + uxul−2 + α[2uxxxup + 4pup−1uxuxx
+p(p− 1)up−2(ux)3] = 0 . (2)
Equation (2) has the property that it is derivable from a La-
grangian. CSS showed that the two equations have the same
class of solitary wave solutions when l = m+1 and p = n−1.
Because of this connection, the two parameter family of new
solutions that we will find here will also be solutions of the
K(m,n) equation with slightly different coefficients. Cooper
and Khare9 later showed that these equations with l = p+ 2,
i.e. m = n ≤ 3 and m continuous, had compacton solutions
of the form
u(x, t) = A[cos(βy)]2/p , y = x− ct , (3)
and for all these compactons also the width was independent
of the amplitude.
The CSS equation has three conserved quantities, i.e. the
Hamiltonian H , momentum P and mass M given by
H =
∫ [
αup(ux)
2 − u
l
l(l− 1)
]
dx , (4)
P =
1
2
∫
u2(x, t) dx; M =
∫
u(x, t) dx . (5)
On the other hand, the Rosenau-Hyman equation has in gen-
eral two conserved quantities (except when m = n in which
case there are four conservation laws2,10) given by
M =
∫
u(x, t) dx; Q =
∫
un+1(x, t) dx . (6)
The fact that the CSS equations were derivable from an Ac-
tion Principle allowed CSS to consider time-dependent vari-
ational approximations based on simple time-dependent trial
functions uv(x, t). By using post-gaussian trial wave func-
tions uv(x, t) = A(t) exp [−β(t)|x + q(t)|γ ] it was shown
that these trial wave functions satisfy the relationship
q˙ = r
H
P
, (7)
2with
r = −p+ l + 2
p+ 6− l . (8)
Here q˙ is the velocity of the compacton. However, it was not
known at the time if the result was true in general. In this Let-
ter we will show that this relationship is entirely general and
does not depend on any trial wave function approximation.
We will show explicity that this relationship can be derived for
all solitary wave solutions of the CSS equations of the form
u(x, t) = AZ[β(x + q(t))]. This relationship will also allow
us to relate the amplitude, width and velocity of the solitary
wave. Having a Hamiltonian formulation also simplifies the
discussion of stability, and using general arguments we shall
prove for the CSS equations that the compacton solutions are
stable provided 2 < l < p+ 6.
Energy-momentum relation theorem: To derive the rela-
tionship between the conserved energy and the conserved mo-
mentum, the starting point is the action
Γ =
∫
Ldt , (9)
where L is given by (note φx = u)
L(l, p) =
∫ (
1
2
ϕxϕt +
(ϕx)
l
l(l− 1) − α(ϕx)
p(ϕxx)
2
)
dx .
(10)
If we assume the exact solitary wave solution is of the generic
form
φx = u = AZ(β(x + q(t))) , (11)
then the value of the Hamiltonian for the solitary wave solu-
tion can be shown to be
H = −C1(l) A
l
βl(l − 1) + αβA
p+2C2(p), (12)
where
C1(l) =
∫
Z l(z)dz; C2(p) =
∫
[Z ′(z)]2Zp(z)dz . (13)
SinceH and P are conserved, therefore we can rewrite the pa-
rameter A in terms of the conserved momentum P and obtain
P =
1
2
∫
dxu2 =
A2
2β
C5 , C5 =
∫
dzZ2(z) . (14)
Replacing A by P , we now have
H = −C3(l)P l/2β(l−2)/2 + C4(p)P (p+2)/2β(p+4)/2, (15)
where
C3(l) =
C1(l)
l(l− 1)
[
2
C5
]l/2
; C4 = αC2(p)
[
2
C5
](p+2)/2
.
(16)
The exact solutions have the property that they are the func-
tions of β that minimize the Hamiltonian with respect to β.
Explicit examples showing this for both the CSS equation as
well as a quintic generalization of this equation are found in
the appendix of Ref. 11. On using ∂H/∂β = 0, we obtain
β = P
p−l+2
l−p−6
[
C4
C3
p+ 4
l − 2
]2/(l−p−6)
. (17)
This leads to
H = f(l, p)P−r , (18)
where r is given by Eq. (7), and
f(l, p) = −
(
p− l + 2
p+ 4
)
C3(l)
[
C4(p)
C3(l)
p+ 4
l− 2
](l−2)/(l−p−6)
.
(19)
Hamilton’s equation, q˙ = ∂H/∂P now yields the relationship
as given by Eqs. (7) and (8).
Using Eqs. (7), (14) and (17) to (19) it is easy to show that
the momentum P , amplitude A and the width parameter β
functionally depend on the velocity c (note c = −q˙) by
P ∝ c p+6−l2(l−2) , A ∝ c 1l−2 , β ∝ c l−p−22(l−2) . (20)
Several comments are in order.
1. Notice that when l = p+2 then β is independent of the
velocity c and momentum P and hence the amplitude
A of the solitary wave.
2. Note that the c dependence of the amplitude A solely
depends on the parameter l and is independent of the
parameter p.
3. In the special case when p = 0 and l = d + 2, the
CSS equation reduces to d-th order KdV equation. In
particular, in that case d = 1 corresponds to the KdV
equation while d = 2 corresponds to the modified
Korteweg-De Vries (mKdV) equation. For this case,
a well known exact solution is f(y) = Asech2/d(βy),
where β = (d/2)
√
v and 2Ad = (d+1)(d+2)v which
indeed is consistent with relation (20). For that case we
notice from above that for any d, the width parameter β
varies as c1/2 while the amplitudeA varies as c1/d.
Stability of Solutions. The stability problem at l = p +
2 was studied in Ref. 10, using the results of Karpman12,13.
Their analysis is in fact also valid for arbitrary l, p. The result
of detailed analysis is that the criterion for linear stability is
equivalent to the condition,
∂P
∂c
> 0. (21)
Since for all of our solutions P ∝ c(p+6−l)/2(l−2), it imme-
diately follows that the solutions are stable provided 2 < l <
p+6. Analysis of Lyapunov stability following Refs. 10,12,13
also leads to the same restriction on p.
3Exact solitary wave and compacton solutions: If we as-
sume a solution to (2) in the form of a travelling wave:
u(x, t) = f(y) = f(x− ct) , (22)
we then obtain
cf ′ = f ′f l−2+α
(
2f ′′′fp + 4pfp−1f ′f ′′ + p(p− 1)fp−2f ′3) .
(23)
Integrating twice we obtain:
c
2
f2 − f
l
l(l − 1) − αf
′2fp = K1f +K2. (24)
For compactons, the integration constants, K1 and K2 are
zero. The general theorem derived above is valid in the
case that the integration constants K1,K2 are zero. Un-
less stated otherwise, throughout this paper we shall consider
the case when K1,K2 are both zero. On demanding that
f ′′fp → 0, f ′2fp−1 → 0 at edges where f → 0, while
f ′ is finite at edges gives us the following bounds on l and
p : l > 1, 0 < p ≤ 2, p ≤ l.
It is worth noting that Eq. (24) is very similar to the equa-
tion obtained for the Rosenau-Hyman case:
(f ′)2 =
2v
n(n+ 1)
f3−n − 2
n(n+m)
fm−n+2 . (25)
Thus, we see that with l = m+1 and p = n−1 the equations
for finding solutions are identical in form, with only differing
coefficients. Therefore we expect to find similar solutions to
the two sets of equations.
Let us now look at the various different compacton solu-
tions to Eq. (24). For the particular case l = p + 2 (m = n),
Cooper and Khare9 were able to show that the CSS equation
has solutions of the form (3) and that for all these compacton
solutions the width is independent of the amplitude. We now
show that if instead l = 2p + 2 then one gets a one parame-
ter family of elliptic compacton solutions. In particular, it is
easily shown that
f = Acnγ(βy, k2 = 1/2) , (26)
for
−K(k2 = 1/2) ≤ βy ≤ K(k2 = 1/2) , (27)
and zero elsewhere is an exact elliptic compacton solutions to
the field Eq. (23) provided
γ = 2/p , l = 2p+ 2 , A2p = c(p+ 1)(2p+ 1) ,
β4 =
cp4
16α2(p+ 1)(2p+ 1)
. (28)
The (m,n) = (3, 2) and (m,n) = (5, 3) are two special cases
of solution (26) with γ = 1, 2. Here cn(y, k) is a Jacobi ellip-
tic function andK(k) denotes the complete elliptic integral of
the first kind with modulus k.
If instead l = 3p+2, then we obtain a one parameter family
of elliptic solutions of the form
f = A
[
1− cn(2(3)1/4βy, k2)
(1 +
√
3) + (
√
3− 1)cn(2(3)1/4βy, k2)
]γ
, (29)
with the modulus
k2 =
1
2
−
√
3
4
. (30)
Here
γ = 1/p , l = 3p+ 2 , 2A3p = c(3p+ 1)(3p+ 2) ,
β6 =
c2p6
256α3(3p+ 1)(3p+ 2)
. (31)
This is a compacton solution in the range
0 ≤ w = 2(3)1/4ξ ≤ 4K
(
k2 =
1
2
−
√
3
4
)
. (32)
and zero elsewhere, where ξ = βy. The way this solution is
obtained is by starting with the ansatz f = AZa(ξ) and de-
manding that Z satisfies the differential equation (dZ/dξ)2 =
1−Z6. The above integral can be evaluated by converting the
differential equation into the standard differential equation for
the Weierstrass elliptic function14 P(y, k). On simplifying,
we obtain the explicit solution as given by Eq. (28). The
(m,n) = (4, 2) solution of Ref. 7 is a special case of our
general solution (29) with p = 1.
We now show that all the above solutions are in fact special
cases of the two parameter (a, t) family of solutions obtained
by assuming
f = AZa(ξ = βy) , (33)
and demanding that
(Z ′)2 = 1− Z2t . (34)
We find the conditions:
l = pt+ 2; a = 2/p , 2Apt = (pt+ 2)(pt+ 1)c ,
β2t =
ct−1pt
αt23t−1(pt+ 1)(pt+ 2)
. (35)
It may be noted that here l, p, t are all continuous (i.e. real)
parameters. The various solutions discussed above correspond
to t = 1, 2, 3. In fact an explicit compacton solution can also
be found in the case t = 3/2. The solution for t = 3/2 is
essentially the same as that for t = 3, Eq. (29), but with
Eq. (30) replaced by the complementary modulus k2 = 12 +√
3
4 . For t > 3, the compacton solutions are related to hyper-
elliptic functions14.
It is interesting to note that even though none of the hyper-
elliptic solutions can be obtained in an analytic form (t > 3),
still their momentum and energy can be obtained analytically.
In particular, on using Eq. (34) in the expressions for H,P as
given by Eqs. (4) and (5) it is easily shown that
H = −A
2c
2βt
(6 + p− l)
(l + p+ 2)
B
(
p+ 4
2pt
,
1
2
)
, (36)
P =
A2
2βt
B
(
p+ 4
2pt
,
1
2
)
, (37)
4where B denotes the Beta function14. For all these solu-
tions with p, l continuous (i.e. real) variables, the relationship
H/P = c/r is always satisfied.
For the hyper-elliptic compacton solutions we have that l,
p and t are related by l = pt + 2 so that the general stability
criterion can be rewritten as 0 < pt < p+ 4, or for t > 1
0 < p(t− 1) < 4. (38)
The requirement for non-singular solutions is that 0 < p ≤ 2.
This means for t ≤ 3, compactons with arbitrary p in the
allowed range are linearly stable while when t > 3 , the com-
pactons are stable only for 0 < p < 4/(t − 1). Analysis of
Lyapunov stability following Refs. 10,12,13 also leads to the
same restrictions on p.
From the stability analysis we also recover the well known
result that the higher order KdV equations, characterized by
p = 0, l = d+2, have stable soliton solutions provided d < 4
while one has unstable soliton solutions in the case15,16,17 d >
4.
General remarks. In conclusion, we have obtained explicit
exact compacton solutions in terms of Jacobi elliptic functions
which exist at particular values of the elliptic modulus k. Note
that at k = 0 these solutions do not exist. In addition, we de-
rived a quite general theorem that relates the energy momen-
tum and velocity of solitary wave solutions without knowing
the explicit form of the solution. We note that the above analy-
sis should also hold for the generalized quintic KdV equation.
We also notice that the radial part of the generalized nonlinear
Schro¨dinger equation with nonlinear power κ
i
∂ψ
∂t
+∇2ψ + g|ψ ∗ ψ|κ = 0 (39)
obeys the same equation as (24) with p = 0 and l = 2κ + 2,
so that the stability analysis of these two problems are related.
We believe that it should be possible to derive a similar theo-
rem in other nonlinear systems.
Finally, as a byproduct of our results, we are able to obtain
analytic expressions for K(k) and the complete elliptic inte-
gral of the second kind, E(k), at k2 = 1/2 − √3/4 (see Eq.
30) or at k = sin(pi/12). Specifically,
K
(
k = sin
( pi
12
)
=
√
3− 1
2
√
2
)
=
31/4
√
piΓ(1/6)
6Γ(2/3)
, (40)
where Γ(a) denotes the Gamma function14. In addition, at
k = sin(pi/12) using the relations K ′ =
√
3K and
E =
pi
√
3
12K
+
√
2
3
k′K, E′ =
pi
√
3
4K ′
+
√
2
3
k′K ′, (41)
we also have the explicit analytic expressions for E and the
two complete elliptic integrals with complementary modulus
k′ =
√
1− k2 = cos(pi/12), namely K ′ and E′.
F.C. would like to thank the Santa Fe Institute and A.K.
would like to thank the Center for Nonlinear Studies and
Theoretical Division at LANL for their hospitality during the
completion of this work. We would also like to thank the U.S.
DOE and the NSF for their partial support of this work.
∗ Electronic address: fcooper@nsf.gov
† Electronic address: khare@iopb.res.in
‡ Electronic address: avadh@lanl.gov
1 F. Cooper, H. Shepard, and P. Sodano, Phys. Rev. E 48, 4027
(1993).
2 P. Rosenau and J.M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
3 Y. S. Kivshar, Phys. Rev. E 48, R43 (1993).
4 P. Rosenau and S. Schochet, Phys. Rev. Lett. 94, 045503 (2005).
5 P. T. Dinda and M. Remoissenet, Phys. Rev. E 60, 6218 (1999).
6 A. M. Wazwaz, Chaos, Solitons and Fractals 13, 321 (2002).
7 P. Rosenau, Phys. Lett. A 275, 193 (2000).
8 P. Rosenau and A. Pikovsky, Phys. Rev. Lett. 94, 174102 (2005).
9 A. Khare and F. Cooper, Phys Rev. E 48, 4843 (1993).
10 B. Dey and A. Khare, Phys. Rev. E 58, R2741 (1998).
11 F. Cooper, J. M. Hyman, and A. Khare, Phys Rev. E 64, 026608.
(2001).
12 V. I. Karpman, Phys. Lett. A. 210, 77 (1996).
13 V. I. Karpman, Phys. Lett. A. 215, 254 (1996).
14 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and
Products, Fifth ed. (Academic, San Diego, 1994).
15 M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering
Transform (SIAM, Philadelphia, PA, 1981).
16 E.A. Kuznetsov, Phys. Lett. A101, 314 (1984).
17 E.A. Kuznetsov, A.M. Rubenchik, and V.E. Zakharov, Phys. Rep.
142, 103 (1986).


Exact elliptic compactons in generalized Korteweg-De Vries equations

Indian Academy of Sciences

ar
X
iv
:n
lin
/0
50
80
10
v1
  [
nl
in
.P
S
]  
4 
A
ug
 2
00
5
Exact Elliptic Compactons in Generalized Korteweg-DeVries Equations
Fred Cooper∗,†,‡, Avinash Khare∗∗, Avadh Saxena†∗
∗National Science Foundation, Division of Physics, Arlington, VA 22230, USA
†Santa Fe Institute, Santa Fe, NM 87501, USA
‡Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA† and
∗∗Institute of Physics, Bhubaneswar, Orissa 751005, India‡
(Dated: February 8, 2008)
We derive a general theorem relating the energy, momentum and velocity of any solitary wave solution of the
generalized KdV equation which enables us to relate the amplitude, width, and momentum to the velocity of
these solutions. We obtain the general condition for linearand Lyapunov stability. We then obtain a two param-
eter family of exact solutions to these equations which include elliptic and hyper-elliptic compacton solutions.
For this general family we explicitly verify both the theorem and the stability criteria.
In a variety of physical contexts one finds nonlinear field
equations for which a wide class of solitary wave solutions
can exist. However, in many cases one is not able to obtain the
solution in a closed form and thus it is not very easy to study
the stability of such solutions. In this Letter we derive a gen-
eral theorem relating the energy, momentum and velocity of
any solitary wave of a generalized Korteweg-De Vries (KdV)
equation of Cooper, Shephard and Sodano (CSS)1. The im-
portant point is that this particular generalization of theKdV
equation is derivable from an Action Principle. Using the th-
orem, we are able to relate the amplitude, width and velocity
of any of the solitary wave solution even if such a solution is
not known in a closed form and also study its stability. Sec-
ondly, we obtain a two-parameter family of solutions to these
equations which include elliptic and hyper-elliptic compacton
solutions. For this general family we explicitly verify thethe-
orem as well as the stability criteria.
Compactons were discovered originally in an extension of
the KdV equation by Rosenau and Hyman in Ref. 2. Com-
pactons are fundamental excitations (i.e., solitary waveswith
a compact support) of KdV-like equations that possess a non-
linear dispersion and collide quasi-elastically. They play an
important role in pattern formation and emergence of nonlin-
ear structures in physical systems. Other physical contexts
in which compactons are relevant include fluid dynamics, op-
tical waveguides and the field of intrinsic localized modes3.
Breather compactons4,5 and compactons in other nonlinear
dispersive equations6 are also known. In addition, KdV equa-
tions with higher power of nonlinearity and dispersion may
lead to the phenomema of blow-up and collapse. Therefore,
the study of compactons and their stability in this class of
equations is important in its own right.
Rosenau and Hyman showed that in a particular generaliza-
tion of the KdV equation, defined by parameters(m,n) (with
m,n integers), namely
K(m,n) : ut + (u
m)x + (u
n)xxx = 0 , (1)
a new form of solitary wave with compact support is a solution
of this equation. For the casem = n (m integer) these com-
pactons had the property that the width was independent of the
amplitude. In Ref. 2 it was stated thatK(3, 2) had an elliptic
function solution. In a later work, Rosenau7 obtained ellip-
tic function compactons for the case ofK(4, 2) andK(5, 3).
Phase compactons have also been investigated8. Because the
equations of Rosenau and Hyman were not equivalent to a
Hamiltonian dynamical system, CSS1 considered instead a re-
lated generalization of the KdV equation
K∗(l, p) : ut + uxu
l−2 + α[2uxxxu
p + 4pup−1uxuxx
+p(p− 1)up−2(ux)3] = 0 . (2)
Equation (2) has the property that it is derivable from a La-
grangian. CSS showed that the two equations have the same
class of solitary wave solutions whenl = m+1 andp = n−1.
Because of this connection, the two parameter family of new
solutions that we will find here will also be solutions of the
K(m,n) equation with slightly different coefficients. Cooper
and Khare9 later showed that these equations withl = p+ 2,
i.e. m = n ≤ 3 andm continuous, had compacton solutions
of the form
u(x, t) = A[cos(βy)]2/p , y = x− ct , (3)
and for all these compactons also the width was independent
of the amplitude.
The CSS equation has three conserved quantities, i.e. the
HamiltonianH , momentumP and massM given by
H =
∫
[
αup(ux)
2 − u
l
l(l− 1)
]
dx , (4)
P =
1
2
∫
u2(x, t) dx; M =
∫
u(x, t) dx . (5)
On the other hand, the Rosenau-Hyman equation has in gen-
eral two conserved quantities (except whenm = n in which
case there are four conservation laws2,10) given by
M =
∫
u(x, t) dx; Q =
∫
un+1(x, t) dx . (6)
The fact that the CSS equations were derivable from an Ac-
tion Principle allowed CSS to consider time-dependent vari-
ational approximations based on simple time-dependent trial
functionsuv(x, t). By using post-gaussian trial wave func-
tions uv(x, t) = A(t) exp [−β(t)|x + q(t)|γ ] it was shown
that these trial wave functions satisfy the relationship
q̇ = r
H
P
, (7)
2
with
r = −p+ l + 2
p+ 6 − l . (8)
Hereq̇ is the velocity of the compacton. However, it was not
known at the time if the result was true in general. In this Let-
ter we will show that this relationship is entirely general and
does not depend on any trial wave function approximation.
We will show explicity that this relationship can be derivedfor
all solitary wave solutions of the CSS equations of the form
u(x, t) = AZ[β(x + q(t))]. This relationship will also allow
us to relate the amplitude, width and velocity of the solitary
wave. Having a Hamiltonian formulation also simplifies the
discussion of stability, and using general arguments we shall
prove for the CSS equations that the compacton solutions are
stable provided2 < l < p+ 6.
Energy-momentum relation theorem: To derive the rela-
tionship between the conserved energy and the conserved mo-
mentum, the starting point is the action
Γ =
∫
Ldt , (9)
whereL is given by (noteφx = u)
L(l, p) =
∫
(
1
2
ϕxϕt +
(ϕx)
l
l(l− 1) − α(ϕx)
p(ϕxx)
2
)
dx .
(10)
If we assume the exact solitary wave solution is of the generic
form
φx = u = AZ(β(x + q(t))) , (11)
then the value of the Hamiltonian for the solitary wave solu-
tion can be shown to be
H = −C1(l)
Al
βl(l − 1) + αβA
p+2C2(p), (12)
where
C1(l) =
∫
Z l(z)dz; C2(p) =
∫
[Z ′(z)]2Zp(z)dz . (13)
SinceH andP are conserved, therefore we can rewrite the pa-
rameterA in terms of the conserved momentumP and obtain
P =
1
2
∫
dxu2 =
A2
2β
C5 , C5 =
∫
dzZ2(z) . (14)
ReplacingA byP , we now have
H = −C3(l)P l/2β(l−2)/2 + C4(p)P (p+2)/2β(p+4)/2, (15)
where
C3(l) =
C1(l)
l(l− 1)
[
2
C5
]l/2
; C4 = αC2(p)
[
2
C5
](p+2)/2
.
(16)
The exact solutions have the property that they are the func-
tions of β that minimize the Hamiltonian with respect toβ.
Explicit examples showing this for both the CSS equation as
well as a quintic generalization of this equation are found in
the appendix of Ref. 11. On using∂H/∂β = 0, we obtain
β = P
p−l+2
l−p−6
[
C4
C3
p+ 4
l − 2
]2/(l−p−6)
. (17)
This leads to
H = f(l, p)P−r , (18)
wherer is given by Eq. (7), and
f(l, p) = −
(
p− l + 2
p+ 4
)
C3(l)
[
C4(p)
C3(l)
p+ 4
l− 2
](l−2)/(l−p−6)
.
(19)
Hamilton’s equation,̇q = ∂H/∂P now yields the relationship
as given by Eqs. (7) and (8).
Using Eqs. (7), (14) and (17) to (19) it is easy to show that
the momentumP , amplitudeA and the width parameterβ
functionally depend on the velocityc (notec = −q̇) by
P ∝ c
p+6−l
2(l−2) , A ∝ c 1l−2 , β ∝ c
l−p−2
2(l−2) . (20)
Several comments are in order.
1. Notice that whenl = p+2 thenβ is independent of the
velocity c and momentumP and hence the amplitude
A of the solitary wave.
2. Note that thec dependence of the amplitudeA solely
depends on the parameterl and is independent of the
parameterp.
3. In the special case whenp = 0 and l = d + 2, the
CSS equation reduces tod-th order KdV equation. In
particular, in that cased = 1 corresponds to the KdV
equation whiled = 2 corresponds to the modified
Korteweg-De Vries (mKdV) equation. For this case,
a well known exact solution isf(y) = Asech2/d(βy),
whereβ = (d/2)
√
v and2Ad = (d+1)(d+2)v which
indeed is consistent with relation (20). For that case we
notice from above that for anyd, the width parameterβ
varies asc1/2 while the amplitudeA varies asc1/d.
Stability of Solutions. The stability problem atl = p +
2 was studied in Ref. 10, using the results of Karpman12,13.
Their analysis is in fact also valid for arbitraryl, p. The result
of detailed analysis is that the criterion for linear stability is
equivalent to the condition,
∂P
∂c
> 0. (21)
Since for all of our solutionsP ∝ c(p+6−l)/2(l−2), it imme-
diately follows that the solutions are stable provided2 < l <
p+6. Analysis of Lyapunov stability following Refs. 10,12,13
also leads to the same restriction onp.
3
Exact solitary wave and compacton solutions: If we as-
sume a solution to (2) in the form of a travelling wave:
u(x, t) = f(y) = f(x− ct) , (22)
we then obtain
cf ′ = f ′f l−2+α
(
2f ′′′fp + 4pfp−1f ′f ′′ + p(p− 1)fp−2f ′3
)
.
(23)
Integrating twice we obtain:
c
2
f2 − f
l
l(l − 1) − αf
′2fp = K1f +K2. (24)
For compactons, the integration constants,K1 andK2 are
zero. The general theorem derived above is valid in the
case that the integration constantsK1,K2 are zero. Un-
less stated otherwise, throughout this paper we shall consider
the case whenK1,K2 are both zero. On demanding that
f ′′fp → 0, f ′2fp−1 → 0 at edges wheref → 0, while
f ′ is finite at edges gives us the following bounds onl and
p : l > 1, 0 < p ≤ 2, p ≤ l.
It is worth noting that Eq. (24) is very similar to the equa-
tion obtained for the Rosenau-Hyman case:
(f ′)2 =
2v
n(n+ 1)
f3−n − 2
n(n+m)
fm−n+2 . (25)
Thus, we see that withl = m+1 andp = n−1 the equations
for finding solutions are identical in form, with only differing
coefficients. Therefore we expect to find similar solutions to
the two sets of equations.
Let us now look at the various different compacton solu-
tions to Eq. (24). For the particular casel = p + 2 (m = n),
Cooper and Khare9 were able to show that the CSS equation
has solutions of the form (3) and that for all these compacton
solutions the width is independent of the amplitude. We now
show that if insteadl = 2p + 2 then one gets a one parame-
ter family of elliptic compacton solutions. In particular,it is
easily shown that
f = Acnγ(βy, k2 = 1/2) , (26)
for
−K(k2 = 1/2) ≤ βy ≤ K(k2 = 1/2) , (27)
and zero elsewhere is an exact elliptic compacton solutionso
the field Eq. (23) provided
γ = 2/p , l = 2p+ 2 , A2p = c(p+ 1)(2p+ 1) ,
β4 =
cp4
16α2(p+ 1)(2p+ 1)
. (28)
The(m,n) = (3, 2) and(m,n) = (5, 3) are two special cases
of solution (26) withγ = 1, 2. Herecn(y, k) is a Jacobi ellip-
tic function andK(k) denotes the complete elliptic integral of
the first kind with modulusk.
If insteadl = 3p+2, then we obtain a one parameter family
of elliptic solutions of the form
f = A
[
1 − cn(2(3)1/4βy, k2)
(1 +
√
3) + (
√
3 − 1)cn(2(3)1/4βy, k2)
]γ
, (29)
with the modulus
k2 =
1
2
−
√
3
4
. (30)
Here
γ = 1/p , l = 3p+ 2 , 2A3p = c(3p+ 1)(3p+ 2) ,
β6 =
c2p6
256α3(3p+ 1)(3p+ 2)
. (31)
This is a compacton solution in the range
0 ≤ w = 2(3)1/4ξ ≤ 4K
(
k2 =
1
2
−
√
3
4
)
. (32)
and zero elsewhere, whereξ = βy. The way this solution is
obtained is by starting with the ansatzf = AZa(ξ) and de-
manding thatZ satisfies the differential equation(dZ/dξ)2 =
1−Z6. The above integral can be evaluated by converting the
differential equation into the standard differential equation for
the Weierstrass elliptic function14 P(y, k). On simplifying,
we obtain the explicit solution as given by Eq. (28). The
(m,n) = (4, 2) solution of Ref. 7 is a special case of our
general solution (29) withp = 1.
We now show that all the above solutions are in fact special
cases of the two parameter (a, t) family of solutions obtained
by assuming
f = AZa(ξ = βy) , (33)
and demanding that
(Z ′)2 = 1 − Z2t . (34)
We find the conditions:
l = pt+ 2; a = 2/p , 2Apt = (pt+ 2)(pt+ 1)c ,
β2t =
ct−1pt
αt23t−1(pt+ 1)(pt+ 2)
. (35)
It may be noted that herel, p, t are all continuous (i.e. real)
parameters. The various solutions discussed above correspnd
to t = 1, 2, 3. In fact an explicit compacton solution can also
be found in the caset = 3/2. The solution fort = 3/2 is
essentially the same as that fort = 3, Eq. (29), but with
Eq. (30) replaced by the complementary modulusk2 = 12 +√
3
4 . For t > 3, the compacton solutions are related to hyper-
elliptic functions14.
It is interesting to note that even though none of the hyper-
elliptic solutions can be obtained in an analytic form(t > 3),
still their momentum and energy can be obtained analytically.
In particular, on using Eq. (34) in the expressions forH,P as
given by Eqs. (4) and (5) it is easily shown that
H = −A
2c
2βt
(6 + p− l)
(l + p+ 2)
B
(
p+ 4
2pt
,
1
2
)
, (36)
P =
A2
2βt
B
(
p+ 4
2pt
,
1
2
)
, (37)
4
whereB denotes the Beta function14. For all these solu-
tions withp, l continuous (i.e. real) variables, the relationship
H/P = c/r is always satisfied.
For the hyper-elliptic compacton solutions we have thatl,
p andt are related byl = pt + 2 so that the general stability
criterion can be rewritten as0 < pt < p+ 4, or for t > 1
0 < p(t− 1) < 4. (38)
The requirement for non-singular solutions is that0 < p ≤ 2.
This means fort ≤ 3, compactons with arbitraryp in the
allowed range are linearly stable while whent > 3 , the com-
pactons are stable only for0 < p < 4/(t − 1). Analysis of
Lyapunov stability following Refs. 10,12,13 also leads to the
same restrictions onp.
From the stability analysis we also recover the well known
result that the higher order KdV equations, characterized by
p = 0, l = d+2, have stable soliton solutions providedd < 4
while one has unstable soliton solutions in the case15,16,17d >
4.
General remarks. In conclusion, we have obtained explicit
exact compacton solutions in terms of Jacobi elliptic functions
which exist at particular values of the elliptic modulusk. Note
that atk = 0 these solutions do not exist. In addition, we de-
rived a quite general theorem that relates the energy momen-
tum and velocity of solitary wave solutions without knowing
the explicit form of the solution. We note that the above analy-
sis should also hold for the generalized quintic KdV equation.
We also notice that the radial part of the generalized nonlinear
Schrödinger equation with nonlinear powerκ
i
∂ψ
∂t
+ ∇2ψ + g|ψ ∗ ψ|κ = 0 (39)
obeys the same equation as (24) withp = 0 andl = 2κ + 2,
so that the stability analysis of these two problems are related.
We believe that it should be possible to derive a similar theo-
rem in other nonlinear systems.
Finally, as a byproduct of our results, we are able to obtain
analytic expressions forK(k) and the complete elliptic inte-
gral of the second kind,E(k), atk2 = 1/2 −
√
3/4 (see Eq.
30) or atk = sin(π/12). Specifically,
K
(
k = sin
( π
12
)
=
√
3 − 1
2
√
2
)
=
31/4
√
πΓ(1/6)
6Γ(2/3)
, (40)
whereΓ(a) denotes the Gamma function14. In addition, at
k = sin(π/12) using the relationsK ′ =
√
3K and
E =
π
√
3
12K
+
√
2
3
k′K, E′ =
π
√
3
4K ′
+
√
2
3
k′K ′, (41)
we also have the explicit analytic expressions forE and the
two complete elliptic integrals with complementary modulus
k′ =
√
1 − k2 = cos(π/12), namelyK ′ andE′.
F.C. would like to thank the Santa Fe Institute and A.K.
would like to thank the Center for Nonlinear Studies and
Theoretical Division at LANL for their hospitality during the
completion of this work. We would also like to thank the U.S.
DOE and the NSF for their partial support of this work.
∗ Electronic address: fcooper@nsf.gov
† Electronic address: khare@iopb.res.in
‡ Electronic address: avadh@lanl.gov
1 F. Cooper, H. Shepard, and P. Sodano, Phys. Rev. E48, 4027
(1993).
2 P. Rosenau and J.M. Hyman, Phys. Rev. Lett.70, 564 (1993).
3 Y. S. Kivshar, Phys. Rev. E48, R43 (1993).
4 P. Rosenau and S. Schochet, Phys. Rev. Lett.94, 045503 (2005).
5 P. T. Dinda and M. Remoissenet, Phys. Rev. E60, 6218 (1999).
6 A. M. Wazwaz, Chaos, Solitons and Fractals13, 321 (2002).
7 P. Rosenau, Phys. Lett. A275, 193 (2000).
8 P. Rosenau and A. Pikovsky, Phys. Rev. Lett.94, 174102 (2005).
9 A. Khare and F. Cooper, Phys Rev. E48, 4843 (1993).
10 B. Dey and A. Khare, Phys. Rev. E58, R2741 (1998).
11 F. Cooper, J. M. Hyman, and A. Khare, Phys Rev. E64, 026608.
(2001).
12 V. I. Karpman, Phys. Lett. A.210, 77 (1996).
13 V. I. Karpman, Phys. Lett. A.215, 254 (1996).
14 I. S. Gradshteyn and I. M. Ryzhik,Tables of Integrals, Series, and
Products, Fifth ed. (Academic, San Diego, 1994).
15 M.J. Ablowitz and H. Segur,Solitons and the Inverse Scattering
Transform (SIAM, Philadelphia, PA, 1981).
16 E.A. Kuznetsov, Phys. Lett.A101, 314 (1984).
17 E.A. Kuznetsov, A.M. Rubenchik, and V.E. Zakharov, Phys. Rep.
142, 103 (1986).


Spectral correlations of individual quantum graphs

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows

Indian Academy of Sciences