We obtain exact periodic solutions of the positive and negative modified
Kortweg-de Vries (mKdV) equations. We examine the dynamical stability of these
solitary wave lattices through direct numerical simulations. While the positive
mKdV breather lattice solutions are found to be unstable, the two-soliton
lattice solution of the same equation is found to be stable. Similarly, a
negative mKdV lattice solution is found to be stable. We also touch upon the
implications of these results for the KdV equation.Comment: 8 pages, 3 figures, to appear in J. Phys. 

A Saxena

Ablowitz M J

Agop M

Avinash Khare

Chou K S

Dodd R K

Drazin P G

G Herring

Infeld E

Kevrekidis P G

Khater A H

Komatsu T S

Matsutani S

Ohta Y

Ono H

P G Kevrekidis

Ralph E A

Schief W K

Scott A

English

arXiv

Crossref

On some classes of mKdV periodic solutions

We obtain exact periodic solutions of the positive and negative modified Kortweg-de Vries (mKdV) equations. We examine the dynamical stability of these solitary wave lattices through direct numerical simulations. While the positive mKdV breather lattice solutions are found to be unstable, the two-soliton lattice solution of the same equation is found to be stable. Similarly, a negative mKdV lattice solution is found to be stable. We also touch upon the implications of these results for the KdV equation

Kevrekidis, P. G.

Khare, Avinash

Saxena, A.

Herring, G.

Publications of the IAS Fellows

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On Some Classes of mKdV Periodic Solutions
P.G. Kevrekidis1, Avinash Khare2, A. Saxena3 and G.
Herring1
1 Department of Mathematics and Statistics, University of Massachusetts,
Amherst, MA 01003-4515, USA.
2 Institute of Physics, Bhubaneswar, Orissa 751005, India.
3 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA.
Abstract. We obtain exact periodic solutions of the positive and negative
modified Kortweg-de Vries (mKdV) equations. We examine the dynamical
stability of these solitary wave lattices through direct numerical simulations.
While the positive mKdV breather lattice solutions are found to be unstable, the
two-soliton lattice solution of the same equation is found to be stable. Similarly,
a negative mKdV lattice solution is found to be stable. We also touch upon the
implications of these results for the KdV equation.
Submitted to: J. Phys. A: Math. Gen.
On Some Classes of mKdV Periodic Solutions 2
1. Introduction
Among the soliton bearing nonlinear integrable equations, sine-Gordon, nonlinear
Schro¨dinger (NLS), Korteweg-de Vries (KdV) and the modified Korteweg-de Vries
(mKdV) are of special interest [1, 2, 3, 4, 5]. These equations possess exact breather
solutions. Therefore, they may also have exact solutions in the form of a spatially
periodic array of single breathers, i.e. breather lattices. Similarly to the sine-Gordon,
NLS and KdV, the mKdV equation is also important in many physical contexts.
For example, it appears in the context of ion acoustic solitons [6], van Alfve´n waves
in collisionless plasma [7], Schottky barrier transmission lines [8], models of traffic
congestion [9] as well as phonons in anharmonic lattices [10]. Furthermore, the
modelling of a subclass of hyperbolic surfaces [11], slag-metallic bath interfaces [12] as
well as meandering ocean jets [13] is also related to the mKdV equation. The dynamics
of thin elastic rods has also been demonstrated to be reducible to the mKdV equation
[14]. Finally, if one studies the examples of surface dynamics that are purely local,
yet maintain global constraints like conservation of perimeter and enclosed area, one
finds that these dynamics are closely related to the KdV and mKdV hierarchies [15].
The nonlinear term in the mKdV equation (6u2ux) may assume either a positive
or a negative sign. We will classify the equation as “positive mKdV” and “negative
mKdV” if the prefactor is +1 or −1, respectively. In an earlier study, the stability of
the sine-Gordon breather lattice was examined [16]. Recently, a particular form of an
exact breather lattice solution was obtained in [17] for the positive mKdV equation
and its stability was discussed.
The aim of the present Letter is to present a new class of periodic solutions both
for the positive and for the negative mKdV equations. We also intend to examine
the dynamical stability of these novel classes of solutions and particularly to illustrate
that many of them can be dynamically stable. This is notably different from the
behavior observed previously for breather lattice solutions in the models of [16, 17].
Furthermore, it is worth noting that the solutions of the negative mKdV are related
to those of the KdV equation via the Miura transform [18]. Thus, our solutions can
be translated into exact periodic solutions of the KdV equation as well. The latter is
also a ubiquitous equation in a variety of fields ranging from conformal field theory to
plasma physics.
Our presentation is structured as follows. In Sec. 2 we examine the breather
lattice and two-soliton lattice solutions of the mKdV equation with positive sign of
nonlinearity and discuss their stability. In Sec. 3 we examine periodic solutions of
the mKdV equation with the negative sign of the nonlinearity. In Sec. 4 we briefly
comment on the corresponding KdV solutions and follow that with our conclusions in
Sec. 5.
2. Breather and Soliton Lattices of the Positive mKdV Equation
For a field u(x, t), the positive mKdV equation is given by
ut + 6u
2ux + uxxx = 0. (1)
Using the ansatz
u = −2 d
dx
[tan−1 φ], (2)
On Some Classes of mKdV Periodic Solutions 3
we are able to obtain several exact spatially periodic solutions. The first one is a
breather lattice solution [17]
φ = αsn(ax + bt + a0, k)dn(cx + dt + c0,m), (3)
where a0, c0 are arbitrary constants. In fact, all the solutions discussed in this paper
admit such constants even though we will not always display them hereafter. In this
solution:
α = − c
a
,
c4
a4
=
k
(1−m) ,
b
a
= [a2(1 + k)− 3c2(2−m)],
d
c
= [3a2(1 + k)− c2(2−m)] , (4)
while sn(x, k), (cn(x, k) below), and dn(x,m) are Jacobi elliptic functions with modulus
k and m, respectively. This solution was studied in detail previously in [17]. For
completeness, we discuss briefly the relevant results. In order to ensure periodicity of
the solution (in space), a commensurability condition was postulated in that work. In
its strict form (strong commensurability), this condition demands that the two elliptic
function terms of Eq. (3) have the same period, i.e. 4K(k)/a = 2K(m)/c (where
K(k) denotes the complete elliptic integral of the first kind). In its weaker form (weak
commensurability), the condition demands that the periodicities are rational multiples
of each other i.e.,
4p
K(k)
a
= 2q
K(m)
c
(5)
with p, q ∈ Z⋆+, note that p = q = 1 yields the strong commensurability as a special
case. Both strong and weak forms, however, resulted in unstable dynamical evolutions
of the breather lattice [17] (induced by means of numerical perturbations to the exact
solution). We further showed that this solution could be stabilized through ac driving
and damping. In the limit k → 0,m→ 1, but with k/(1−m) = v4, this solution goes
over to the well known single breather (bion) solution
φ = −v sin(ax+ bt+ a0)sech(cx + dt + c0). (6)
Remarkably, it turns out that there is a different breather lattice solution to the
positive mKdV equation given by
φ = αcn(ax + bt + a1, k)cn(cx + dt + c1,m), (7)
where
α2 =
mk
(1 −m)(1− k) ,
c4
a4
=
k(1 − k)
m(1 −m) ,
b
a
= [a2(1− 2k) + 3c2(1− 2m)], d
c
= [3a2(1− 2k) + c2(1 − 2m)]. (8)
Notice that in the limit k → 0,m→ 1, a1 = pi/2+a0 c1 = c0 but with k/(1−m) = v4,
this solution also goes over to the same breather (bion) solution (6). However, for
other values of k,m, the two breather lattice solutions (3) and (7) are quite different
(see also the comparison of the right panel of Fig. 1). We should note here that the
characterization of the solutions that we present as breather or two-soliton lattices is
given on the basis of what their limiting (see e.g., Eq. (6)) profile looks like, as the
elliptic functions asymptote to trigonometric/hyperbolic ones.
The dynamical stability of the breather lattice solution (7) was examined for
both strong and weak commensurability by means of direct numerical simulations.
On Some Classes of mKdV Periodic Solutions 4
−30 −20 −10 0 10 20 30
−0.5
0
0.5
x
u
(x,
0)
−30 −20 −10 0 10 20 30
−1
−0.5
0
0.5
1
x
u
(x,
55
)
−30 −20 −10 0 10 20 30
−3
−2
−1
0
1
2
3
x
u
(x)
Figure 1. The left panel shows the x-t (i.e., space-time) evolution of the contour
of the unstable breather lattice solution of the form of Eqs. (7)-(8); a = 1, k = 0.5,
c = 1.707 and m = 0.03 (such that K(k)/a = 2K(m)/c) were used in the initial
conditions. The middle panel shows the spatial profile at t = 0 (initial condition)
and at t = 55 (after the instability has set in). Finally, the right panel shows
with a solid line the initial condition and with a dashed line the (other breather
lattice) solution of Eqs. (3)-(4) for the same parameters.
The numerical scheme used here, motivated by the KdV discretization of Ref. [19]
was analyzed previously in [17]. However, the results were also verified with different
discretizations of the integrable model such as the ones proposed in [20]. In Fig. 1,
we demonstrate a typical example of the dynamical evolution of the solution of Eqs.
(7)-(8). We observe that similarly to the previously obtained breather lattice solution
of [17], the breather lattice identified above is dynamically unstable in the mKdV
equation and results in a few more strongly and many weakly localized peaks. Hence,
the instability of mKdV breather lattices appears to be generic.
Using the ansatz in terms of Eq. (2), we can obtain yet another novel family of
solutions, namely a two-soliton lattice
φ = αsc(ax + bt, k)dn(cx + dt,m), (9)
where α = −c/a, sc(x,m) ≡ sn(x,m)/cn(x,m) and
c4
a4
=
1− k
1−m,
b
a
= −[a2(2− k) + 3c2(2−m)],
d
c
= −[c2(2−m) + 3a2(2− k)]. (10)
This solution can also be obtained from the breather lattice solution (3) by taking
a→ ia, b→ ib, α→ −iα and using the well known relations
sn(ix,m) = i
sn(x, 1−m)
cn(x, 1−m) , dn(ix,m) =
dn(x, 1−m)
cn(x, 1−m) ,
cn(ix,m) =
1
cn(x, 1−m) . (11)
In the limit k → 1, m→ 1 but with (1− k)/(1−m) = v4 this solution goes over to
φ = −v sinh[a(x− [1 + 3v2]t)]sech[av(x− [3 + v2]t)] , (12)
which, except for v = 1, is the well-known 2-soliton solution (hence the classification
of this solution as a 2-soliton lattice). For v = 1, however, it corresponds to the
one-soliton solution. The latter implies k = m and c = a from Eq. (10).
An example of the dynamical evolution for the two-soliton lattice solution is shown
in Fig. 2. As it can be seen, this solution persists unchanged for long dynamical
On Some Classes of mKdV Periodic Solutions 5
−30 −20 −10 0 10 20 30
−1
0
1
2
3
4
x
u
(x,
0)
−30 −20 −10 0 10 20 30
−1
0
1
2
3
4
x
u
(x,
15
0)
Figure 2. The left panel shows the spatio-temporal evolution of the contour plot
of a two-soliton lattice in the case of the positive mKdV equation for a = 0.25,
k = 0.1, c = 1.596 and m = 0.999, such that K(k)/a = 2K(m)/c. The right panel
shows the t = 0 (top panel) and the t = 150 (bottom panel) spatial profiles.
evolutions (i.e., for times of the order of 150 in the arbitrary time units of the time
evolution of Fig. 2), hence our numerical simulations indicate that it is dynamically
stable.
3. Lattice Solutions of the Negative mKdV Equation
The negative mKdV equation is given as
ut − 6u2ux + uxxx = 0 . (13)
In this case, to identify the corresponding solutions, we start with the ansatz
u = −2 d
dx
[tanh−1 φ] . (14)
One can then show that the field φ satisfies the equation
(1− φ2)[φt + φxxx] + 6φx[φφxx − φ2x] = 0. (15)
Using an ansatz in terms of Jacobi elliptic functions we obtain the following new
periodic solution:
φ = αdn(ax + bt, k)dn(cx + dt,m), (16)
where
c4
a4
=
1− k
1−m,
b
a
= −[a2(2− k) + 3c2(2−m)],
α2 =
1√
(1 − k)(1−m)
,
d
c
= −[c2(2−m) + 3a2(2− k)]. (17)
Unfortunately, this solution may be singular for a finite x (for a given t), depending on
the parameters. In particular, the derivative of Eq. (14) induces a term ∼ 1/(1− u2).
However, from the properties of the elliptic functions, one can obtain that:√
(1− k)(1 −m) < u2 < 1√
(1− k)(1 −m) . (18)
On Some Classes of mKdV Periodic Solutions 6
−30 −20 −10 0 10 20 30
−1.5
−1
−0.5
0
0.5
1
1.5
x
u
(x,
0)
−30 −20 −10 0 10 20 30
−1.5
−1
−0.5
0
0.5
1
1.5
x
u
(x,
15
0)
Figure 3. Same as Fig. 2 but for the periodic solution of the negative
mKdV equation with a = 1, k = 0.5, c = 1.719 and m = 0.057 (such that
K(k)/a = 2K(m)/c). The dynamical evolution is stable.
Equation (18), in turn, implies (given the continuity of u) that u will, typically, assume
the value 1 for a certain x, hence that the corresponding solution will, generically, be
singular. We thus do not consider it further here.
Another, more interesting solution of the negative mKdV equation is given by
φ = αsn(ax + bt, k)sn(cx + dt,m), (19)
where
c4
a4
=
k
m
,
b
a
= [a2(1 + k) + 3c2(1 +m)],
α2 =
√
km,
d
c
= [c2(1 +m) + 3a2(1 + k)]. (20)
For k = m, this solution degenerates to the well-known soliton-lattice solution which
in the limit k = m = 1 goes over to the famous one-soliton solution of mKdV with
negative sign.
We have examined the solution (19) for strong, as well as weak commensurability.
We have performed numerical simulations for various (p, q) pairs including the
degenerate case with c = a amd k = m. The principal feature of the temporal
evolution of this solution lies in its dynamical stability, in the sense that for the
duration of the numerical simulation (of the order of t = 150 time units) the structure
robustly maintains its character, see Fig. 3.
4. Periodic Solution of the KdV Equation
Through the Miura transform v = u2 ± ux [18] we can obtain the periodic solutions
of the KdV equation (v) from the corresponding solutions of the negative mKdV
equation (u). In particular, to the solution (19) of negative mKdV, corresponds the
KdV solution of the form:
v = u2 + ux = 2
[2φ′2(x) − (1 + φ)φ′′(x)]
(1− φ2)(1 + φ) , (21)
where prime denotes derivative with respect to the argument. The other solution
u2 − u′(x) is obtained from here simply by changing α to -α. Using the solution (19)
On Some Classes of mKdV Periodic Solutions 7
of the negative mKdV equation, we find that the solution (21) can be expressed in the
form
v =
2α
[1 + αsn(ax + bt, k)sn(cx + dt,m)]2
[
(a2(1 + k) + c2(1 +m))sn(ax + bt, k)
sn(cx + dt,m) + 2a2αsn2(cx + dt,m) + 2c2αsn2(ax + bt, k)
− 2accn(ax + bt, k)dn(ax + bt, k)cn(cx + dt,m)dn(cx + dt,m)
]
. (22)
Perhaps more interestingly, the Miura transform is an exact transformation
between the solution of the negative mKdV equation (∀(x, t)) and the one of the
KdV equation. This implies that the numerically observed stability of the above
mentioned lattice solution of the negative mKdV equation carries over to the existence
and stability of such a solution in the setting of the KdV equation.
5. Conclusion
In this Letter, we have obtained new classes of periodic solutions for the positive
and negative mKdV equations. The positive mKdV solutions (7) and (9) could
be identified as the breather lattice and two-soliton lattice solutions, since in the
appropriate limit they reduce to single breather and two-soliton solutions, respectively.
In the case of the negative mKdV equation, lattice solutions have been identified in
the form of Eqs. (16) and (19). However, the latter have not been designated as
breather or soliton lattices (as the process of obtaining limiting expressions is less
straightforward for the negative mKdV case).
We have also examined the stability of the various periodic solutions of mKdV
equation with both positive and negative signs. We have found that the two-soliton
lattice solution of the positive mKdV and the lattice solution (19) of the negative
mKdV are quite robust with respect to perturbations in contrast with the breather
lattice solution of the positive mKdV equation. These results, and more specifically
the stability of the negative mKdV lattice solution, have direct implications for the
corresponding lattice solutions of the KdV equation.
This work was supported in part by the U.S. Department of Energy. PGK is
grateful to the Eppley Foundation for Research, the NSF-DMS-0204585 and the NSF-
CAREER program for financial support.
References
[1] P.G. Drazin and R.S. Johnson, Solitons: an introduction (Cambridge University Press,
Cambridge, U.K., 1989).
[2] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM,
Philadelphia, 1981).
[3] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University Press,
Cambridge, 1990).
[4] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Waves
(Academic Press, London, 1982).
[5] A. Scott, Nonlinear Science (Oxford University Press, New York, 1999).
[6] K. E. Lonngren, Optical and Quantum Electronics, 30, 615 (1998).
[7] A. H. Khater, O. H. El-Kalaawy, and D. K. Callebaut, Phys. Script. 58, 545 (1998).
[8] V. Ziegler, J. Dinkel, C. Setzer, and K. E. Lonngren, Chaos, Solitons and Fractals 12, 1719
(2001).
On Some Classes of mKdV Periodic Solutions 8
[9] T. S. Komatsu and S. I. Sasa, Phys. Rev. E 52, 5574 (1995); T. Nagatani, Physica A 265, 297
(1999).
[10] H. Ono, J. Phys. Soc. Jpn. 61, 4336 (1992).
[11] W. K. Schief, Nonlinearity 8, 1 (1995).
[12] M. Agop and V. Cojocaru, Mater. Trans. JIM 39, 668 (1998).
[13] E. A. Ralph and L. Pratt, J. Nonlin. Sci. 4, 355 (1994).
[14] S. Matsutani and H. Tsuru, J. Phys. Soc. Jpn. 60 (1991) 3640.
[15] R.E. Goldstein and D.M. Petrich, Phys. Rev. Lett. 67, 3203 (1991); J. Langer and R. Perline,
Phys. Lett. A 239, 36 (1998); K. S. Chou and C. Z. Qu, Physica D 162, 9 (2002).
[16] P.G. Kevrekidis, A. Saxena, and A.R. Bishop, Phys. Rev. E 64, 026613 (2001).
[17] P.G. Kevrekidis, A. Khare, and A. Saxena, Phys. Rev. E 68, 047701 (2003).
[18] R. Miura, J. Math. Phys. 9, 1202 (1968).
[19] Y. Ohta and R. Hirota, J. Phys. Soc. Jpn. 60, 2095 (1991).
[20] M.J. Ablowitz and J.F. Ladik, J. Math. Phys., 16, 598 (1975); M.J. Ablowitz and J.F. Ladik,
J. Math. Phys., 17, 1011 (1976).


Indian Academy of Sciences

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On Some Classes of mKdV Periodic Solutions
P.G. Kevrekidis1, Avinash Khare2, A. Saxena3 and G.
Herring1
1 Department of Mathematics and Statistics, University of Massachusetts,
Amherst, MA 01003-4515, USA.
2 Institute of Physics, Bhubaneswar, Orissa 751005, India.
3 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA.
Abstract. We obtain exact periodic solutions of the positive and negative
modified Kortweg-de Vries (mKdV) equations. We examine the dynamical
stability of these solitary wave lattices through direct numerical simulations.
While the positive mKdV breather lattice solutions are found to be unstable, the
two-soliton lattice solution of the same equation is found to be stable. Similarly,
a negative mKdV lattice solution is found to be stable. We also touch upon the
implications of these results for the KdV equation.
Submitted to: J. Phys. A: Math. Gen.
On Some Classes of mKdV Periodic Solutions 2
1. Introduction
Among the soliton bearing nonlinear integrable equations, sine-Gordon, nonlinear
Schrödinger (NLS), Korteweg-de Vries (KdV) and the modified Korteweg-de Vries
(mKdV) are of special interest [1, 2, 3, 4, 5]. These equations possess exact breather
solutions. Therefore, they may also have exact solutions in the form of a spatially
periodic array of single breathers, i.e. breather lattices. Similarly to the sine-Gordon,
NLS and KdV, the mKdV equation is also important in many physical contexts.
For example, it appears in the context of ion acoustic solitons [6], van Alfvén waves
in collisionless plasma [7], Schottky barrier transmission lines [8], models of traffic
congestion [9] as well as phonons in anharmonic lattices [10]. Furthermore, the
modelling of a subclass of hyperbolic surfaces [11], slag-metallic bath interfaces [12] as
well as meandering ocean jets [13] is also related to the mKdV equation. The dynamics
of thin elastic rods has also been demonstrated to be reducible to the mKdV equation
[14]. Finally, if one studies the examples of surface dynamics that are purely local,
yet maintain global constraints like conservation of perimeter and enclosed area, one
finds that these dynamics are closely related to the KdV and mKdV hierarchies [15].
The nonlinear term in the mKdV equation (6u2ux) may assume either a positive
or a negative sign. We will classify the equation as “positive mKdV” and “negative
mKdV” if the prefactor is +1 or −1, respectively. In an earlier study, the stability of
the sine-Gordon breather lattice was examined [16]. Recently, a particular form of an
exact breather lattice solution was obtained in [17] for the positive mKdV equation
and its stability was discussed.
The aim of the present Letter is to present a new class of periodic solutions both
for the positive and for the negative mKdV equations. We also intend to examine
the dynamical stability of these novel classes of solutions and particularly to illustrate
that many of them can be dynamically stable. This is notably different from the
behavior observed previously for breather lattice solutions in the models of [16, 17].
Furthermore, it is worth noting that the solutions of the negative mKdV are related
to those of the KdV equation via the Miura transform [18]. Thus, our solutions can
be translated into exact periodic solutions of the KdV equation as well. The latter is
also a ubiquitous equation in a variety of fields ranging from conformal field theory to
plasma physics.
Our presentation is structured as follows. In Sec. 2 we examine the breather
lattice and two-soliton lattice solutions of the mKdV equation with positive sign of
nonlinearity and discuss their stability. In Sec. 3 we examine periodic solutions of
the mKdV equation with the negative sign of the nonlinearity. In Sec. 4 we briefly
comment on the corresponding KdV solutions and follow that with our conclusions in
Sec. 5.
2. Breather and Soliton Lattices of the Positive mKdV Equation
For a field u(x, t), the positive mKdV equation is given by
ut + 6u
2ux + uxxx = 0. (1)
Using the ansatz
u = −2 d
dx
[tan−1 φ], (2)
On Some Classes of mKdV Periodic Solutions 3
we are able to obtain several exact spatially periodic solutions. The first one is a
breather lattice solution [17]
φ = αsn(ax + bt + a0, k)dn(cx + dt + c0, m), (3)
where a0, c0 are arbitrary constants. In fact, all the solutions discussed in this paper
admit such constants even though we will not always display them hereafter. In this
solution:
α = − c
a
,
c4
a4
=
k
(1 − m) ,
b
a
= [a2(1 + k) − 3c2(2 − m)],
d
c
= [3a2(1 + k) − c2(2 − m)] , (4)
while sn(x, k), (cn(x, k) below), and dn(x, m) are Jacobi elliptic functions with modulus
k and m, respectively. This solution was studied in detail previously in [17]. For
completeness, we discuss briefly the relevant results. In order to ensure periodicity of
the solution (in space), a commensurability condition was postulated in that work. In
its strict form (strong commensurability), this condition demands that the two elliptic
function terms of Eq. (3) have the same period, i.e. 4K(k)/a = 2K(m)/c (where
K(k) denotes the complete elliptic integral of the first kind). In its weaker form (weak
commensurability), the condition demands that the periodicities are rational multiples
of each other i.e.,
4p
K(k)
a
= 2q
K(m)
c
(5)
with p, q ∈ Z⋆+, note that p = q = 1 yields the strong commensurability as a special
case. Both strong and weak forms, however, resulted in unstable dynamical evolutions
of the breather lattice [17] (induced by means of numerical perturbations to the exact
solution). We further showed that this solution could be stabilized through ac driving
and damping. In the limit k → 0, m → 1, but with k/(1−m) = v4, this solution goes
over to the well known single breather (bion) solution
φ = −v sin(ax + bt + a0)sech(cx + dt + c0). (6)
Remarkably, it turns out that there is a different breather lattice solution to the
positive mKdV equation given by
φ = αcn(ax + bt + a1, k)cn(cx + dt + c1, m), (7)
where
α2 =
mk
(1 − m)(1 − k) ,
c4
a4
=
k(1 − k)
m(1 − m) ,
b
a
= [a2(1 − 2k) + 3c2(1 − 2m)], d
c
= [3a2(1 − 2k) + c2(1 − 2m)]. (8)
Notice that in the limit k → 0, m → 1, a1 = π/2+a0 c1 = c0 but with k/(1−m) = v4,
this solution also goes over to the same breather (bion) solution (6). However, for
other values of k, m, the two breather lattice solutions (3) and (7) are quite different
(see also the comparison of the right panel of Fig. 1). We should note here that the
characterization of the solutions that we present as breather or two-soliton lattices is
given on the basis of what their limiting (see e.g., Eq. (6)) profile looks like, as the
elliptic functions asymptote to trigonometric/hyperbolic ones.
The dynamical stability of the breather lattice solution (7) was examined for
both strong and weak commensurability by means of direct numerical simulations.
On Some Classes of mKdV Periodic Solutions 4
−30 −20 −10 0 10 20 30
−0.5
0
0.5
x
u(
x,
0)
−30 −20 −10 0 10 20 30
−1
−0.5
0
0.5
1
x
u(
x,
55
)
−30 −20 −10 0 10 20 30
−3
−2
−1
0
1
2
3
x
u(
x)
Figure 1. The left panel shows the x-t (i.e., space-time) evolution of the contour
of the unstable breather lattice solution of the form of Eqs. (7)-(8); a = 1, k = 0.5,
c = 1.707 and m = 0.03 (such that K(k)/a = 2K(m)/c) were used in the initial
conditions. The middle panel shows the spatial profile at t = 0 (initial condition)
and at t = 55 (after the instability has set in). Finally, the right panel shows
with a solid line the initial condition and with a dashed line the (other breather
lattice) solution of Eqs. (3)-(4) for the same parameters.
The numerical scheme used here, motivated by the KdV discretization of Ref. [19]
was analyzed previously in [17]. However, the results were also verified with different
discretizations of the integrable model such as the ones proposed in [20]. In Fig. 1,
we demonstrate a typical example of the dynamical evolution of the solution of Eqs.
(7)-(8). We observe that similarly to the previously obtained breather lattice solution
of [17], the breather lattice identified above is dynamically unstable in the mKdV
equation and results in a few more strongly and many weakly localized peaks. Hence,
the instability of mKdV breather lattices appears to be generic.
Using the ansatz in terms of Eq. (2), we can obtain yet another novel family of
solutions, namely a two-soliton lattice
φ = αsc(ax + bt, k)dn(cx + dt, m), (9)
where α = −c/a, sc(x, m) ≡ sn(x, m)/cn(x, m) and
c4
a4
=
1 − k
1 − m,
b
a
= −[a2(2 − k) + 3c2(2 − m)],
d
c
= −[c2(2 − m) + 3a2(2 − k)]. (10)
This solution can also be obtained from the breather lattice solution (3) by taking
a → ia, b → ib, α → −iα and using the well known relations
sn(ix, m) = i
sn(x, 1 − m)
cn(x, 1 − m) , dn(ix, m) =
dn(x, 1 − m)
cn(x, 1 − m) ,
cn(ix, m) =
1
cn(x, 1 − m) . (11)
In the limit k → 1, m → 1 but with (1 − k)/(1 − m) = v4 this solution goes over to
φ = −v sinh[a(x − [1 + 3v2]t)]sech[av(x− [3 + v2]t)] , (12)
which, except for v = 1, is the well-known 2-soliton solution (hence the classification
of this solution as a 2-soliton lattice). For v = 1, however, it corresponds to the
one-soliton solution. The latter implies k = m and c = a from Eq. (10).
An example of the dynamical evolution for the two-soliton lattice solution is shown
in Fig. 2. As it can be seen, this solution persists unchanged for long dynamical
On Some Classes of mKdV Periodic Solutions 5
−30 −20 −10 0 10 20 30
−1
0
1
2
3
4
x
u(
x,
0)
−30 −20 −10 0 10 20 30
−1
0
1
2
3
4
x
u(
x,
15
0)
Figure 2. The left panel shows the spatio-temporal evolution of the contour plot
of a two-soliton lattice in the case of the positive mKdV equation for a = 0.25,
k = 0.1, c = 1.596 and m = 0.999, such that K(k)/a = 2K(m)/c. The right panel
shows the t = 0 (top panel) and the t = 150 (bottom panel) spatial profiles.
evolutions (i.e., for times of the order of 150 in the arbitrary time units of the time
evolution of Fig. 2), hence our numerical simulations indicate that it is dynamically
stable.
3. Lattice Solutions of the Negative mKdV Equation
The negative mKdV equation is given as
ut − 6u2ux + uxxx = 0 . (13)
In this case, to identify the corresponding solutions, we start with the ansatz
u = −2 d
dx
[tanh−1 φ] . (14)
One can then show that the field φ satisfies the equation
(1 − φ2)[φt + φxxx] + 6φx[φφxx − φ2x] = 0. (15)
Using an ansatz in terms of Jacobi elliptic functions we obtain the following new
periodic solution:
φ = αdn(ax + bt, k)dn(cx + dt, m), (16)
where
c4
a4
=
1 − k
1 − m,
b
a
= −[a2(2 − k) + 3c2(2 − m)],
α2 =
1
√
(1 − k)(1 − m)
,
d
c
= −[c2(2 − m) + 3a2(2 − k)]. (17)
Unfortunately, this solution may be singular for a finite x (for a given t), depending on
the parameters. In particular, the derivative of Eq. (14) induces a term ∼ 1/(1− u2).
However, from the properties of the elliptic functions, one can obtain that:
√
(1 − k)(1 − m) < u2 < 1√
(1 − k)(1 − m)
. (18)
On Some Classes of mKdV Periodic Solutions 6
−30 −20 −10 0 10 20 30
−1.5
−1
−0.5
0
0.5
1
1.5
x
u(
x,
0)
−30 −20 −10 0 10 20 30
−1.5
−1
−0.5
0
0.5
1
1.5
x
u(
x,
15
0)
Figure 3. Same as Fig. 2 but for the periodic solution of the negative
mKdV equation with a = 1, k = 0.5, c = 1.719 and m = 0.057 (such that
K(k)/a = 2K(m)/c). The dynamical evolution is stable.
Equation (18), in turn, implies (given the continuity of u) that u will, typically, assume
the value 1 for a certain x, hence that the corresponding solution will, generically, be
singular. We thus do not consider it further here.
Another, more interesting solution of the negative mKdV equation is given by
φ = αsn(ax + bt, k)sn(cx + dt, m), (19)
where
c4
a4
=
k
m
,
b
a
= [a2(1 + k) + 3c2(1 + m)],
α2 =
√
km,
d
c
= [c2(1 + m) + 3a2(1 + k)]. (20)
For k = m, this solution degenerates to the well-known soliton-lattice solution which
in the limit k = m = 1 goes over to the famous one-soliton solution of mKdV with
negative sign.
We have examined the solution (19) for strong, as well as weak commensurability.
We have performed numerical simulations for various (p, q) pairs including the
degenerate case with c = a amd k = m. The principal feature of the temporal
evolution of this solution lies in its dynamical stability, in the sense that for the
duration of the numerical simulation (of the order of t = 150 time units) the structure
robustly maintains its character, see Fig. 3.
4. Periodic Solution of the KdV Equation
Through the Miura transform v = u2 ± ux [18] we can obtain the periodic solutions
of the KdV equation (v) from the corresponding solutions of the negative mKdV
equation (u). In particular, to the solution (19) of negative mKdV, corresponds the
KdV solution of the form:
v = u2 + ux = 2
[2φ′2(x) − (1 + φ)φ′′(x)]
(1 − φ2)(1 + φ) , (21)
where prime denotes derivative with respect to the argument. The other solution
u2 − u′(x) is obtained from here simply by changing α to -α. Using the solution (19)
On Some Classes of mKdV Periodic Solutions 7
of the negative mKdV equation, we find that the solution (21) can be expressed in the
form
v =
2α
[1 + αsn(ax + bt, k)sn(cx + dt, m)]2
[
(a2(1 + k) + c2(1 + m))sn(ax + bt, k)
sn(cx + dt, m) + 2a2αsn2(cx + dt, m) + 2c2αsn2(ax + bt, k)
− 2accn(ax + bt, k)dn(ax + bt, k)cn(cx + dt, m)dn(cx + dt, m)
]
. (22)
Perhaps more interestingly, the Miura transform is an exact transformation
between the solution of the negative mKdV equation (∀(x, t)) and the one of the
KdV equation. This implies that the numerically observed stability of the above
mentioned lattice solution of the negative mKdV equation carries over to the existence
and stability of such a solution in the setting of the KdV equation.
5. Conclusion
In this Letter, we have obtained new classes of periodic solutions for the positive
and negative mKdV equations. The positive mKdV solutions (7) and (9) could
be identified as the breather lattice and two-soliton lattice solutions, since in the
appropriate limit they reduce to single breather and two-soliton solutions, respectively.
In the case of the negative mKdV equation, lattice solutions have been identified in
the form of Eqs. (16) and (19). However, the latter have not been designated as
breather or soliton lattices (as the process of obtaining limiting expressions is less
straightforward for the negative mKdV case).
We have also examined the stability of the various periodic solutions of mKdV
equation with both positive and negative signs. We have found that the two-soliton
lattice solution of the positive mKdV and the lattice solution (19) of the negative
mKdV are quite robust with respect to perturbations in contrast with the breather
lattice solution of the positive mKdV equation. These results, and more specifically
the stability of the negative mKdV lattice solution, have direct implications for the
corresponding lattice solutions of the KdV equation.
This work was supported in part by the U.S. Department of Energy. PGK is
grateful to the Eppley Foundation for Research, the NSF-DMS-0204585 and the NSF-
CAREER program for financial support.
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On Some Classes of mKdV Periodic Solutions

On Some Classes of mKdV Periodic Solutions

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