Negative Even Grade mKdV Hierarchy and its Soliton Solutions

Abstract

In this paper we provide an algebraic construction for the negative even mKdV hierarchy which gives rise to time evolutions associated to even graded Lie algebraic structure. We propose a modification of the dressing method, in order to incorporate a non-trivial vacuum configuration and construct a deformed vertex operator for $\hat{sl}(2)$, that enable us to obtain explicit and systematic solutions for the whole negative even grade equations.

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Negative Even Grade mKdV Hierarchy and its Soliton
Solutions
J.F. Gomes, G.Starvaggi Fran¸ ca, G. R. de Melo and A.H. Zimerman
Instituto de F´ ısica Te´ orica - IFT/UNESP
Rua Dr. Bento Teobaldo Ferraz, 271, Bloco II
01140-070, S˜ ao Paulo - SP, Brazil
ABSTRACT
In this paper we provide an algebraic construction for the negative even mKdV hierarchy
which gives rise to time evolutions associated to even graded Lie algebraic structure. We
propose a modification of the dressing method, in order to incorporate a non-trivial vacuum
configuration and construct a deformed vertex operator forˆsl(2), that enable us to obtain
explicit and systematic solutions for the whole negative even grade equations.
0jfg@ift.unesp.br, guisf@ift.unesp.br, gmelo@ift.unesp.br, zimerman@ift.unesp.br
arXiv:0906.5579v1 [hep-th] 30 Jun 2009

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1 Introduction
The odd mKdV hierarchy consists of a series of non-linear equations of motion associated
to certain odd graded Lie algebraic structure such that, each equation correspond to a time
evolution according to time t = t2n+1[1]. Since these are directly associated to odd graded
operators, they are dubbed odd order mKdV hierarchy.
In this paper we employ the algebraic technique which, for positive order, the graded
structure of the zero curvature representation imposes severe restrictions so that only odd
times are allowed. For negative order however, the structure is less restrictive and gives also
rise to a subclass of equations of motion, described by time evolutions associated to negative
even grades. These are constructed by the Lax operator in terms of an affine graded Lie
algebra,ˆsl(2) which, from the zero curvature representation generates systematically a series
of nonlinear integrable equations.
By considering a special case of Zakharov-Shabat AKNS spectral problem and using
recursion techniques, a class of integrable equations were considered [2] in order to develop
negative order mKdV hierarchy as well as to obtain some parametric type of solutions.
Here, in our approach, the simplest case of the even mKdV where t = t−2, is studied in
detail and a crucial observation that a trivial zero constant solution is not admissible, lead
us to extend the dressing method to incorporate non-zero constant vacuum solutions. This
implies in deforming the usual vertex operators preserving the nilpotency property peculiar
in solving for soliton solutions. Employing the modified dressing formalism, we construct
multi-soliton solutions for the whole negative even grade mKdV hierarchy.
In Sect. 2 we discuss the algebraic formalism for positive and negative hierarchies [3, 4].
In particular, the construction of the equation of motion for t = t−2. Such equation agrees
with the one proposed in [2] using recursion operator techniques. In Sect. 3 we discuss
the dressing formalism [5, 6, 7, 8, 9] to construct soliton solutions for the odd hierarchy.
In this case the formalism is based upon a constant zero vacuum solution which, by gauge
transformation, generate multi-soliton solutions. In Sect. 4 we extend the dressing formalism
to the negative even hierarchy, by introducing a non-zero constant vacuum configuration.
We then construct the deformed vertex operators which generate explicitly the multi-soliton
solutions. Some details involving the explicit calculation of matrix elements of products of
vertex operators and the proof of their nilpotency are described in the appendix.
2 Positive and Negative Hierarchies
Consider the positive mKdV hierarchy given by the zero curvature representation
[∂x+ E(1)+ A0,∂tn+ D(n)+ D(n−1)+ ··· + D(0)] = 0,
where E(2n+1)= λn(Eα+ λE−α), A0 = vh contains the field variable v = v (x,tn), and
{E±α,h} are sl(2) generators satisfying [h,E±α] = ±2E±α, [Eα,E−α] = h. The grading
operator Q = 2λd
ˆG = ⊕iGi,
G2m= {h(m)= λmh},
(2.1)
dλ+1
2h decomposes the affine Lie algebraˆsl(2) into graded subspaces,
G2m+1= {λm(Eα+ λE−α), λm(Eα− λE−α)}
(2.2)
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m = 0,±1,±2,... and D(j)∈ Gj. A more subtle structure arises if one consider the decom-
positionˆG = K ⊕ M where K = {λn(Eα+ λE−α)} denotes the Kernel of E ≡ E(1), i.e.,
K = {k ∈ˆG | [E,k] = 0} and M is its complement. We assume that E is semi-simple in the
sense that this second decomposition is such that
[K,K] ⊂ K,[K,M] ⊂ M,[M,M] ⊂ K. (2.3)
Eqn. (2.1) can be decomposed grade by grade and solved for D(j). For instance, the highest
grade in (2.1) yields
[E,D(n)] = 0 ⇒ D(n)= D(n)
Since by (2.2) K has grade 2m + 1, this last equation implies that n = 2m + 1 and hence
tn= t2m+1, showing that only odd grades are admissible for positive mKdV hierarchy (2.1).
Moving down grade by grade and using the symmetric space structure (2.3), eqn. (2.1) allows
one to solve for all D(j)= D(j)
in M yields the equation of motion
∂tnA0− ∂xD(0)
where we have taken into account that A0∈ M. Eqn. (2.5) represents a series of nonlinear
evolution equations associated with time t2m+1. Choosing m = 1 for example, we will obtain
the well known mKdV equation [1, 3].
The same, however, does no happen for the negative mKdV hierarchy [3, 4], i.e., for n < 0.
Let us consider the zero curvature representation
K ∈ K. (2.4)
K+ D(j)
M, j = 0...n. In particular, the zero grade projection
M− [A0,D(0)
K] = 0 (2.5)
[∂x+ E(1)+ A0,∂t−n+ D(−n)+ D(−n+1)+ ··· + D(−1)] = 0.
Here, the lowest grade projection,
(2.6)
∂xD(−n)+ [A0,D(−n)] = 0(2.7)
yields a nonlocal equation for D(−n). The second lowest projection of grade −n + 1 leads to
∂xD(−n+1)+ [A0,D(−n+1)] + [E(1),D(−n)] = 0(2.8)
which determines D(−n+1). The same mechanism works recursively until we reach the zero
grade equation
∂t−nA0+ [E(1),D(−1)] = 0
which gives the time evolution for the field in A0according to time t−n. The simplest model
of this sub-hierarchy is obtained for n = 1, for which the following equations arrive from the
zero curvature (2.6),
(2.9)
∂xD(−1)+ [A0,D(−1)] = 0,
∂t−1A0− [E(1),D(−1)] = 0. (2.10)
These equations can be solved in general if we parametrize the fields as
D(−1)= B−1E(−1)B,A0= B−1∂xB,B = exp(G0) (2.11)
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in terms of the zero grade subalgebra G0. Space-time is associated to the light-cone coor-
dinates ¯ z,z as x = ¯ z, t−1 = z. The time evolution is then given by the Leznov-Saveliev
equation,
∂t−1
which forˆsl(2) with principal gradation Q = 2λd
?
B−1∂xB
?
= [E(1),B−1E(−1)B] (2.12)
dλ+1
2h, yields the sinh-Gordon equation [7]
∂t−1∂xφ = e2φ− e−2φ,B = eφh. (2.13)
Note that from the definition of A0and the parametrization (2.11), we find the following
relation between φ and v: A0= vh = B−1∂xB ⇒ v = ∂xφ.
We now propose the first nontrivial example for the negative even sub-hierarchy:
∂xD(−2)+ [A0,D(−2)] = 0, (2.14)
(2.15)
(2.16)
∂xD(−1)+ [A0,D(−1)] + [E(1),D(−2)] = 0,
∂t−2A0− [E(1),D(−1)] = 0.
From grading (2.2) and decomposition (2.3) we find
D(−2)
D(−1)
= c−2λ−1h,
= a−1
?
λ−1Eα+ E−α
?
+ b−1
?
λ−1Eα− E−α
?
. (2.17)
From eqn. (2.14) and (2.15) we find, c−2= const. and
∂x(a−1+ b−1) + 2v (a−1+ b−1) − 2c−2 = 0,
∂x(a−1− b−1) − 2v (a−1− b−1) + 2c−2 = 0, (2.18)
which are ordinary differential equations with solution
a−1+ b−1 = 2c−2exp(−2d−1v)d−1?
a−1− b−1 = −2c−2exp(2d−1v)d−1?
In (2.19) we have denoted d−1f =
equation associated to time t−2is then given by eqn. (2.16):
∂t−2v + 2c−2e−2d−1vd−1?
Differentiating twice with respect to x, and setting c−2= 1 for convenience, we find the local
equation
vxxt−2− 4v2vt−2−vxvxt−2
Eqn. (2.21) was already obtained in [2] using recursion operator techniques.
exp(2d−1v)
?
,
?
exp(−2d−1v).(2.19)
?xf(x?)dx?. Having determined D(−1), the evolution
e2d−1v?
+ 2c−2e2d−1vd−1?
e−2d−1v?
= 0. (2.20)
v
− 4vx
v
= 0. (2.21)
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3 Odd Hierarchy Solutions
In order to employ the dressing method to construct soliton solutions, we now introduce the
fullˆsl(2) affine Kac-Moody algebra with central extensions:
[h(m),h(n)] = 2mδm+n,0ˆ c,
[h(m),E(n)
[E(m)
±α] = ±2E(m+n)
α ,E(n)
±α
,
−α] = h(m+n)+ mδm+n,0ˆ c(3.22)
together with the derivation operatorˆd such that,
[ˆd,T(n)
a ] = nT(n)
a ,T(n)
a
= {h(n), E(n)
±α}. (3.23)
The grading operator now reads Q = 2ˆd+1/2h(0). A well established method for determining
soliton solutions is to choose a vacuum solution and then to map it into a non trivial solution
by gauge transformation (dressing) [6],[7]. The zero curvature condition (2.1) or (2.6) implies
pure gauge connections, Ax= T−1∂xT = E + A0and Atn= T−1∂tnT = D(n)+ ··· + D(0)
or At−n= T−1∂t−nT = D(−n)+ ··· + D(−1), respectively. Suppose there exists a vacuum
solution satisfying
Ax,vac= E(1)− tkδk+1,0ˆ c,
where now [E(k),E(l)] =1
The solution for Ax,vac= T−1
Atk,vac= E(k), (3.24)
2(k − l)δk+l,0ˆ c, for (k,l) odd integers.
0 ∂xT0and Atk,vac= T−1
0 ∂tkT0is therefore given by
T0= exp(xE(1))exp(tkE(k)). (3.25)
The dressing method is based on the assumption of the existence of two gauge transforma-
tions, generated by Θ±, mapping the vacuum into non trivial configuration, i.e.
Ax = (Θ±)−1Ax,vacΘ±+ (Θ±)−1∂xΘ±,
Atk
= (Θ±)−1Atk,vacΘ±+ (Θ±)−1∂tkΘ±.
(3.26)
(3.27)
As a consequence we relate
Θ−Θ−1
+= T−1
0gT0
(3.28)
where g is an arbitrary constant group element. We suppose that Θ±are group elements of
the form
Θ−1
−= ep(−1)ep(−2)... ,Θ−1
+= eq(0)eq(1)eq(2)... (3.29)
where p(−i)and q(i)are linear combinations of grade (−i) and (i) generators, respectively
(i = 0,1,...). In considering Θ+, the zero grade component of (3.26) admits solution
eq(0)= B−1e−νˆ c
(3.30)
where we have used Ax= E(1)+ B−1∂xB + ∂xνˆ c − tkδk+1,0ˆ c. From eqn. (3.28) we find
...e−p(−2)e−p(−1)B−1e−νˆ ceq(1)eq(2)... = T−1
0gT0
(3.31)
4