A Class of Mixed Integrable Models

arXiv:0903.0579v2 [nlin.SI] 16 Apr 2012 A Class of Mixed Integrable Models J.F. Gomes1, G.R. de Melo1,2 and A.H. Zimerman1 1 Instituto de F´ısica Te´orica-UNESP Rua Pamplona 145 fax (55)11 31779080 01405-900 S˜ao Paulo, Brazil 2 Faculdade Metropolitana de Cama¸cari - FAMEC, Av. Eixo Urbano Central, Centro, 42800-000, Cama¸cari, BA, Brazil. Abstract The algebraic structure of the integrable mixed mKdV/sinh-Gordon model is dis- cussed and extended to the AKNS/Lund-Regge model and to its corresponding su- persymmetric versions. The integrability of the models is guaranteed from the zero curvature representation and some soliton solutions are discussed. 1 Introduction The mKdV and the sine-Gordon equations are non-linear differential equations belonging to the same integrable hierarchy representing different time evolutions [1]. The structure of its soliton solutions present the same functional form in terms of ρ = ekx+kntn, (1.1) which carries the space-time dependence. Solutions of different equations within the same hierarchy differ only by the factor kntn in ρ. For instance n = 3 correspond to the mKdV equation and n = −1 to the sinh-Gordon. For n > 0 a systematic construction of integrable hierarchies can be solved and classified according to a decomposition of an affine Lie algebra, ˆG and a choice of a semi-simple constant element E (see [2] for review). Such framework 1 was shown to be derived from the Riemann-Hilbert decomposition which later, was shown to incorporate negative grade isospectral flows n < 0 [3] as well. The mixed system φxt = α3 4  φxxxx −6φ2 xφxx  + 2η sinh(2φ) (1.2) is a non-linear differential equation which represents the well known mKdV equation for η = 0 ( v = −∂xφ) and the sinh-Gordon equation for α3 = 0. It was introduced in [4] where, employing the inverse scattering method, multi soliton solutions were constructed by modification of time dependence in ρ. Solutions (multi soliton) were also considered in [5] by Hirota’s method. Moreover, two-breather solution was discussed in [6] in connection with few-optical-cycle pulses in transparent media. The soliton solutions obtained in [4], [5] and [6] indicates integrability of the mixed model (1.2). In this paper we consider the mixed system mKdV/sinh-Gordon (1.2) within the zero curvature representation. We show that a systematic solution for the mixed model is obtained by the dressing method and a specific choice of vacuum solution. Such formalism is extended to the mixed AKNS/Lund-Regge and to its supersymmetric versions as well. In the last section we discuss the coupling of higher positive and negative flows general- izing the examples given previously. 2 The mixed mKdV/sinh-Gordon model Let us consider a non-linear system composed of a mixed sinh-Gordon and mKdV equation given by eqn. (1.2) and the following zero curvature representation, [∂x + E(1) + A0, ∂t + D(3) 3 + D(2) 3 + D(1) 3 + D(0) 3 + D(−1) 3 ] = 0 (2.3) 2 where E(2n+1) = λn(Eα + λE−α), A0 = vh and E±α and h are sl(2) generators satisfying [h, E±α] = ±2E±α, [Eα, E−α] = h. According to the grading operator Q = 2λ d dλ + 1 2h, D(j) 3 is a graded j Lie algebra valued and eqn. (2.3) decomposes into 6 independent equations (decomposing grade by grade): [E, D(3) 3 ] = 0, [E, D(2) 3 ] + [A0, D(3) 3 ] + ∂xD(3) 3 = 0, [E, D(1) 3 ] + [A0, D(2) 3 ] + ∂xD(2) 3 = 0, [E, D(0) 3 ] + [A0, D(1) 3 ] + ∂xD(1) 3 = 0, [E, D(−1) 3 ] + [A0, D(0) 3 ] + ∂xD(0) 3 −∂tA0 = 0, [A0, D(−1) 3 ] + ∂xD(−1) 3 = 0. (2.4) where E ≡E(1). In order to solve (2.4) let us propose D(3) 3 = α3  λEα + λ2E−α  + β3  λEα −λ2E−α  , D(2) 3 = σ2λh, D(1) 3 = α1 (Eα + λE−α) + β1 (Eα −λE−α) , D(0) 3 = σ0h. (2.5) Substituting (2.5) in (2.4) we obtain β3 = 0, α3 = const and β1 = α3 2 vx, α1 = −α3 2 v2, σ0 = α3 4 (vxx −2v3), σ2 = α3v. (2.6) In order to solve the last eqn. in (2.4) we parametrize A0 = −∂xBB−1 = −∂xφh, B = eφh (2.7) 3 and D(−1) 3 = ηBE(−1)B−1 = ηλ−1(e2φEα + λe−2φE−α) (2.8) The zero grade projection in (2.4) yields the time evolution equation (1.2). Notice that in order to solve the last eqn. (2.5) we have introduced the sinh-Gordon variable φ in (2.7) and in (2.8) such that v = −∂xφ. Let us now recall some basic aspects of the dressing method which provides systematic construction of soliton solutions. The zero curvature representation implies in a pure gauge configuration. In particular, the vacuum is obtained by setting φvac = 0 or vvac = 0 which, when in (2.3) implies, ∂xT0T −1 0 = E(1), ∂tT0T −1 0 = α3E(3) + ηE(−1) (2.9) and after integration T0 = exp  t(α3E(3) + ηE(−1))  exp  xE(1) , E(2n+1) = λn(Eα + λE−α). (2.10) If we identify v = −∂xφ eqn. (1.2) represents a coupling of mKdV and sinh-Gordon equations and becomes a pure mKdV when η = 0 and pure sinh-Gordon when α3 = 0. Tracing back those two limits from (2.6) and (2.8) it becomes clear that the sinh-Gordon limit (η = 0) in (1.2) is responsible for the vanishing of D(−1) 3 . On the other hand, α3 = 0 implies D(j) 3 = 0, j = 0, · · · 3. Inspired by the

…(Full text truncated)…

This content is AI-processed based on ArXiv data.