The construction of a nonautonomous mixed mKdV/sine-Gordon model is proposed by employing an infinite dimensional affine Lie algebraic structure within the zero curvature representation. A systematic construction of soliton solutions is provided by an adaptation of the dressing method which takes into account arbitrary time dependent functions. A particular choice of those arbitrary functions provides an interesting solution describing the transition of a pure mKdV system into a pure sine-Gordon soliton.
Sometime ago, the study of nonlinear effects in lattice dynamics under the influence of a weak dislocation potential has lead to a mixed mKdV/sine-Gordon equation [1]. The system was shown to admit multisoliton solutions and an infinite set of conservation laws [1]. More recently the two-breather solution was discussed in [2] in connection with the propagation of few cycle pulses (FCP) in non linear optical media. According to ref. [2] the general mKdV/sine-Gordon equation, in fact, describes the propagation of a ultrashort optical pulses in a Kerr media.
Moreover, it was shown in [3] that, when the ressonance frequency of atoms in the physical system are well above or well below the characteristic duration of the pulse, the propagation is described by the mKdV or sine-Gordon equations respectively. The main object of this paper is to provide a systematic construction of soliton solutions that describe the transition between the two regimes, i.e. governed by the mKdV and sine-Gordon equations. This is accomplished by considering the mixed integrable model proposed in [1] with two arbitrary time-dependent coefficients. In this paper we show the integrability of the mixed model with time dependent coefficients and that, by suitable choice of these coefficients as a smooth step-type functions (as shown in figs. 1 and 3) we obtain exact solutions for the mKdV-SG transition and hence a more realistic description of such phenomena.
In ref. [4] the algebraic structure of the mixed mKdV/sine-Gordon equation was formulated within the zero curvature representation and a graded infinite dimensional Lie algebraic structure as we shall now briefly review. Consider the associated G = sl(2) Lie algebra with gen-
m = 0, ±1, ±2, . . .. In [4], a simple proof that a mixed mKdV/sine-Gordon hierarchy is indeed an integrable model follows from the zero curvature representation of the integrable hierarchy generated by
where D (j) ∈ G j and A 0 = vh contains the field variable v = v (x, t). According to the subspace decomposition (2) for N = 3 and t = t 3 which corresponds to the mixed mKdV-SG equation.
Let us parametrize,
The grade by grade decomposing of eqn. (2) leads to
together with the equation of motion
In solving eqns. (4) we find
where a 3 (t) is an arbitrary function of t. Introducing (8) in the first eqn. ( 5), we obtain
which implies that
where f 1 (t) is another arbitrary function of t. It therefore follows that
Substituting (9) in the second eqn. (5), we get
Adding and subtracting eqns. (6), we obtain
where we have denoted
Without loss of generality we may solve (11) by changing the variable
which leads us to a ± = f -1 (t)e ∓2φ , where f -1 (t) is another arbitrary function of t. Writing
we find
Substituting ( 10), ( 12) and ( 13) in ( 7), we finally obtain
Considering f 1 (t) = 0, a 3 (t) = constant and f -1 (t) = constant we find the usual mixed mKdV/sine-Gordon equation. For f 1 (t) = 0, a 3 (t) a given numerical constant = 0 we recover eqn. (10) of ref. [5]. Moreover for f -1 (t) = 0, we recover equation considered in [6] with a choice of coefficients that makes the model integrable.
We should point out that by change of coordinates (see for instance [7]) (x, t) → (x, t) = (x+V (t), t) where V t = f 1 (t) followed by a subsequently change t → T = a 3 ( t)d t and re-scaling
Although eqn. (15) corresponds to the equation discussed in [5] the object of this paper is to consider a class of solutions that interpolates between the mKdV and sine-Gordon equations.
This is more conveniently accomplished by employing eqn. ( 14) where the two arbitrary functions a 3 (t) and f -1 (t) (with f 1 (t) = 0) can be chosen as step-like limiting functions (Figs. 1 and3) as we shall see.
In order to construct, in a systematic manner, the soliton solutions of the mixed model let us now recall some basic aspects of the dressing method (see for instance [8]). The zero curvature representation (2) implies in a pure gauge configuration, i.e.,
In particular, the vacuum is obtained by setting φ vac = 0 1 which implies,
which after integration yields
Following the dressing method explained in [8] and employied in [4] we define the tau-functions
where λ n , n = 0, 1 are fundamental weights of the full affine Kac-Moody algebra ŝl(2), g is a constant group element which classifies the soliton solutions and B is a zero grade group 1 For a general member of the hierarchy evolving according t = t 2n+1 , the vacuun configuration implies
element containing the physical fields. In order to ensure heighest weight representations we now introduce central extensions within the affine Lie algebra, characterized by ĉ, i.e.,
-α ] = h (n+m) + nδ n+m,0 ĉ, [h (n) , h (m) ] = 2nδ n+m,0 ĉ, and define highest weight representations, i.e.,
n = 0, 1. Under this affine picture the group element B acquires a central term contribution,
In order to obtain explicit space-time dependence from the r.h.s. of (19) we consider the verte
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