Shape changing and accelerating solitons in integrable variable mass sine-Gordon model

Sine-Gordon (SG) models with variable mass or perturbed soliton appear in many physical situations, which however breaks the integrability of the model. A class of such inhomogeneous models with accelerating and shape changing solitons is constructed, which are integrable both at the classical and the quantum level with exact solutions. PACS: 05.45.Yv,03.65.Fd,11.10.Lm,11.55.Ds Among the exclusive families of nonlinear integrable systems the sine-Gordon (SG) model enjoys a special status and continues receiving attention till today, for its inherent richness and wide range of applications in different fields [1, 2, 3, 4]. Apart from the fascinating properties of integrable systems in general, e.g., Lax pair, infinite number of independent conserved charges, exact N-soliton solution obtainable by inverse scattering (IS) and Hirota’s bilinear method etc, [6], the SG model possesses special properties, like relativistic invariance, integer-valued topological charge represented by solutions like kink, antikink, breather etc. [7]. Ultralocality with r-matrix formulation and leads to its quantum integrability described by the quantum Yang-Baxter equation (QYBE) R( μ )Uj(λ)⊗ Uj(μ) = Uj(μ) ⊗ Uj(λ)R( μ), j = 1, 2, ..., N, which for the SG model results to the well known quantum suq(2) algebra [8, 9]. Since a physical oscillator going beyond the simple harmonic motion is described by nonlinear equation: ẍ + sinx = 0, in a chain of such coupled oscillators the SG equation appears naturally in the continuum. This is the generic reason why the SG model can describe many physical events like current through Josephson junction (JJ), spin-wave in ferromagnet, charge-density wave, DNA transcription etc. [1, 2], apart from playing an important role in nonperturbative QFT [10]. Solitons in the SG model, as in other integrable systems, move with constant velocity and shape. In the realistic situations however under the influence of external forces or inhomogeneities soliton velocity may change [2, 3], which can be used as a desirable effect for fast transport, fast communication, or even for the possibility of a soliton gun [4]. Inhomogeneities can appear due to impurity, dislocation, defect or incommensuration in the media, producing additional terms in the SG equation or modifying the existing ones, with diverse consequences [2, 3, 11]. In a Josephson junction dissipation of fluxons or local enhancing of the Josephson current may occur, or the variable mass SG (VMSG) equation can also appear in modeling the propagation of domain walls, dislocations, fluxons etc. in the presence of noise, defect of the order parameter etc. [2]. Inhomogeneous xxz spin chains can arise in the Cooper-pair pumping in a linear array of JJ [5] or when the interaction strength varies periodically along the spin-chain. If the periods are incommensurating such inhomogeneities can lead to specific form of VMSG model in the continuum [11]. Soliton velocity of the SG model can change under the influence of force or stepwise defect [3, 4]. However the inhomogeneities as well as variable soliton velocity tend to destroy the integrability of the SG model and hence its exact solutions, which are the most cherishable properties of this model and the related results can at best be perturbative [2, 3]. Therefore to meet the challenge of building SG model with accelerated soliton or variable mass and keeping its integrability preserved both at the classical and the quantum level, we observe that, the spoiling effect of variable velocity can be compensated for by a variable mass. The solitons of such integrable VMSG model can exhibit intriguing properties as shown in Fig. 1i-iii). The quantum integrability of the model can also be made valid, since the modifications do not affect its quantum R-matrix. Since our strategy is to respect integrability, we start from the linear spectral problem Φx(x, λ) = U(λ, x)Φ(x, λ), Φt(x, λ) = V (λ, x)Φ(x, λ), with the Lax pair of the SG model: U = i 4 ( −utσ +mk1 cos u2σ −mk0 sin u2σ ) , V = i 4 ( −uxσ −mk0 cos u2σ +mk1 sin u2σ ) , where k0 = 2λ + 1 2λ , k1 = 2λ − 1 2λ , with λ as the spectral parameter. Compatibility Φxt = Φtx leads to the flatness condition Ut − Vx + [U, V ] = 0 , yielding the SG equation for constant mass m and λ. Recall that in the IS method applied to integrable systems the solitons are obtained as a reflection-less potential with discrete spectrum λn, n = 1, 2, . . . N , representing poles of the transmission coefficient 1 a(λ) , a(λ = λn) = 0 and the velocity of SG soliton (kink) is linked to λ1 = i 2e θ as vs = tanh θ. Therefore for accommodating variable soliton velocity one should have a variable λ , which however violates in general the flatness condition. Making m also variable, we get on the other hand the constraint: (k0m)t + (mk1)x = 0, (k1m)t + (mk0)x = 0, with a solution m(x, t) = mf+f−, k0(x, t) = cosh(θ − ρ(x, t)), k1(x, t) = sinh(θ − ρ(x, t)), ρ(x, t) = ln f+ f− , (1) where m, θ are constants and f± are arbitrary smooth functions of x± t, respectively. This constructs an integrable VMSG equation utt − uxx +m(x, t)sinu = 0, m(x, t) = mf+f−, (2) the relativistic invariance of which is lost in general. Nevertheless under a Lorenz transformation (x, t) → (x, t) the form of the equation (2) remains the same with the replacement f+f− → f ′ +f ′ − and therefore choosing the functions f± we can control the variable mass, as suitable to physical situations. Fig. 1 i-iv) show the dynamics of soliton with different mass functions. For the exact solution of the VMSG model, let us apply both Hirota’s bilinearization and the IS formalism, the former being a direct method, while the later is an indirect one for more general solutions. Hirota’s solution for the SG equation may be expressed as u = −2i ln g+ g− , where g± are conjugate functions with expansion in plane-wave solutions for its soliton solution. For the VMSG model (2) the same ansatz seems to work, only the plane waves should be replaced by their generalized form: g(n) = cn λn e i 2 (X(λn,x,t)−T n, where X(λn, x, t) = ∫ x dxm(x, t)k1n(x , t), T (λn, x, t) = ∫ t dtm(x, t′)k0n(x, t ). (3) This gives the soliton solutions through the expansion: g± = 1± g, for the kink solution . g± = 1± (g + g) + s( θ1 − θ2 2 )gg, for 2-soliton solutions (4) etc. with the scattering matrix s(θ) = tanh2θ and λn = i 2e θn , n = 1, 2, for the kink-kink and λ2 = −λ1 = ηe, for the kink-antikink bound state (breather solution). Similarly we can apply the IS formalism to the inhomogeneous SG model, for which the crucial step is to analyze and use the analytic properties of the solutions for Jost function Φ, identified by their asymptotic at space infinities. The required analyticity based on the behavior at λ → ∞, should hold equally for the inhomogeneous extension, replacing again the asymptotic plane waves by their generalized form. Therefore, going parallel to the standard SG model [6], we get for N-soliton (r(λ) = 0) with discrete spectrum λn , the solution for the Jost function component ψ (1) n as ψ = 1 2 [(1 + V ) + (1− V )]e i 4X , (5) denoting column vectors as ψ = {ψ1(λ1), . . . , ψN (λN )}, e = {eX(λ1 ,x,t), . . . , eX(λN ,x,t)} and V = {Vnm = ( cm(t) λn+λm )e i 4 nm}, with cm(t) = cm(0)e− i 2 T m, where the generalized X,T are as defined in (3). On the other hand comparing the leading term of ψ(1)(λ) at λ → 0 we can connect it with the field: sin u2 = i ∑ n cn(t) λn ψ (1) n e i 4 n, which derives using (5) the exact N-soliton solution for the SG field linked to the inhomogeneous model (2). We can get N = 1-soliton (kink) solution with λ1 = iη explicitly, either from the Hirota’s or from the IS method presented above, as sin u 2 = 1 coshζ , u = 4 tan(e), ζ = i 2 (X(iη, x, t) − T (iη, x, t)). (6) with variable soliton velocity vs(x, t) = − dt = k1(η,x.t k0(η,x.t) . To see the effect of different inhomogeneities on the properties of soliton, we consider some concrete cases. Notice that the choice of inhomogeneous functions as f+ = f− = f with f(x) = x n leads to the variable mass m(x2 − t2)n, preserved under relativistic motion. For the simplest case n = 1 we get the soliton and the kink solution (6) where ζ = m3 (2η(x − t)3 + 1 2η (x+ t)3). The evolution of this soliton is depicted in Fig 1i), showing clearly the intriguing change in its velocity and shape during the propagation. Since we have here ζ(x→ ±∞) → ±∞, the kink (antikink) solution corresponds to the usual topological charge Q = 1 2π (u(∞)− u(−∞)) = ±1. With the choice of inhomogeneity f− = 1, f+ = √ 2 cos(x + t), or similarly f+ = 1, f− = √ 2 cos q(x − t), we can get an integrable SG with variable mass √ 2m cos q(x ± t) and therefore may conclude that the nonintegrable physical model with mass m(x) = cos qx found in [11] can be tuned to an integrable one by making coupling strength to oscillate also periodically in time. For mass √ 2m cos q(x+ t) we can get soliton solution (6) with ζ = m2 (k0x−k1t+ 1 2q (k0−k1) sin 2q(x+ t)), ka = ka(η), a = 1, 2 with velocity vs and width d of the soliton changing periodically as vs = md(k1 + 1 ηq cos 2q(x+ t)), d = 1 m(k0+ 1 ηq cos 2q(x+t)) . The behavior of the solution is shown in Fig. 1ii). The choice of inhohomogeneity through exponential functions f+ = f− = f = exp( ρ 2x), leads to the only possible x-dependent mass as m(x) = meρ(x−x0) for an integrable SG equation . This demonstrates also why the VMSG with a different m(x) derived in [11] turned out to be nonintegrable. The soliton solution for this integrable case is obtained from (6) with ζ = 1 ρ k0(t)m(x), m(x) = exp(ρ(x− x0)), k0(t) = cosh(θ− ρ(t− t0)). Fig. 1iii) shows the soliton evolution with changing width d = 1 m(x)k0(t) , shape and velocity vs = tanh(θ − ρ(t− t0)), with acceleration and a bumeron [16] like property due to change in the direction of the velocity; ihe soliton speed however always remains less than the velocity of light: |vs| ≤ 1. For ρ > 0 we have ζ(x = ∞) = ∞, but ζ(x = −∞) = 0 and that makes the soliton to loose its usual localized form and the finite-energy solution needs proper normalization. The corresponding kink solution will have a topological charge Q = ± 2 and it requires 2-soliton to regain Q = ±1 and the localized form. Similar unusual properties can be observed for inhomogeneous mass m(x2 − t2)n with even n. At ρ→ 0: ζ = 1 ρ k0(t)m(x) → ζ = m(k0(x−x0)−k1(t− t0) and we recover the standard SG soliton and kink with m = const, vs = const. This standard case is shown in Fig. 1iv) for comparison. Figure 1: Exact soliton solutions for sin u2 of the SG equation with variable mass i) m(x 2 − t2)2, ii) 2m cos q(x+ t) iii) m exp(ρx) and iv) m = const. Now we switch over to explore the quantum integrability of our inhomogeneous SG model and follow the algebraic Bethe ansatz method developed for the standard model applicable to its exact lattice version [8]. Quantum lattice SG Lax operator Uj(λ,Sj,m), j = 1, 2, . . . , L involves operators S3 j (uj), S ± j (uj , pj ,m) with canonical momentum pj = u̇j and mass parameter m, which should be considered now as site dependent: mj [12]. Recall that QYBE with the quantum R-matrix ensures the quantum integrability, which for the SG model becomes equivalent to the quantum algebra. Find that, the trigonometric R-matrix associated with the SG model remains unchanged under our inhomogeneous extension, since this R( μ )-matrix depends on the ratio of two spectral parameters, in which x, t-dependence enters as multiplicative functions (1) and therefore cancels out. Moreover, QYBE being a local algebra (at each lattice site j) is not affected by inhomogeneity and yields the same quantum algebra suq(2); only with the replacement of m by a site-dependent function mj in its structure constant: [S j , S − k ] = δjkmj sinα2S j sinα . The aim of the algebraic Bethe ansatz is to solve exactly the eigenvalue problem of trT (λ), T (λ) =