Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations

arXiv:nlin/0406029v1 [nlin.SI] 15 Jun 2004 Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations Niurka R. Quintero a,b,1and El´ıas Zamora-Sillero a,c,2 aDepartamento de F´ısica Aplicada I, Escuela Universitaria Polit´ecnica, Universidad de Sevilla, Virgen de ´ Africa 7, 41011, Sevilla, Spain bInstituto Carlos I de F´ısica Te´orica y Computacional Universidad de Granada. E-18071 Granada, Spain cDepartamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Sevilla, Apartado de Correos 1160, Sevilla E-41080, Spain Abstract We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism together with the Rice Ansatz to derive the equations of motion of the collective coordinates (CCs) for the perturbed sine-Gordon (sG) and φ4systems. For the Ncollective coordinates, regardless of the Ansatz used, we show that, for the nonlinear Klein-Gordon equations, this approach is equivalent to the Generalized Traveling Wave Ansatz (GTWA). Key words: Collective coordinates, Solitons, Solitary waves, Perturbed Nonlinear Klein-Gordon equations. PACS: 03.40.Kf, 04.25.-g, 04.20.Fy 1 Introduction The solitons and solitary waves under the external perturbations have been extensively studied in the last three decades (see e.g. [1,2,3] and references there in). This has been more than justified since in the physical real systems, modeled by the equations related with these solutions, the dissipation 1Corresponding author. E-mail: [email protected] 2E-mail: el[email protected] Preprint submitted to Physica D 6 February 2008 and the external forces are unavoidably present [4,5]. In this work we use the Lagrangian formalism to study the effect of damping and external forces in the nonlinear Klein-Gordon systems, and in particular we use as examples sine-Gordon (sG) and φ4(this method can be extended to other nonlinear Klein-Gordon equations such as, the double sine-Gordon (DsG) [3] or the asymmetric double sine-Gordon (ADSG) [6]). To illustrate the advantage of using this method over previous ones used in this kind of problem, let us revise the situation of the perturbation theory and collective coordinates (CCs in short) on the aforesaid systems. First of all, the authors of [4] developed a technique to investigate the influence of external perturbations, such as damping, dc force and spatial inhomogeneity, on solitary waves. This method is based on the expansion of the solution of the perturbed problem in the complete set of eigenfunctions of the Sturm-Liouville problem associated with the linearized partial differential equation (PDE) around a kink solution. Later, the dynamic of the breather and kink-antikink solution in the perturbed sG equation was also considered using a new perturbative analysis [5,7,8], where the radiation field was also included [5,7]. All these methods involve cumbersome calculations, even more so if we extend them to the φ4equation where the internal mode is present. In particular, if it is assumed that the perturbations only alter the center of the kink, X(t), the calculations can be simplified. In this case, it was shown that the perturbative method is equivalent to the variation of the energy of the system [5]. However, the presence of the internal mode in φ4and some phonon’s modes in sG are also able to change the kink’s width, l(t). Then, the solitary waves under certain perturbations can exhibit resonance’s phenomena [9,10,11] and it is not appropriate to only consider one collective coordinate, for example, the center of the kink, in order to describe its dynamics. These kind of resonances due to the action of the ac force and damping were successfully explained by using a more general Antsatz, the so called Generalized Traveling Wave Ansatz (GTWA) [9,10,12] together with the Rice approximated solution [7,13]. This latter method is easier to apply than the former perturbation theory, however it is only supported in the projection technique. Recently, the equivalence of the GTWA and the variation of the momentum and the energy (two constants of motion of the unperturbed problem) of the nonlinear Klein-Gordon systems when two CCs are considered [14], has been shown. When it is necessary to use more that two CCs, this equivalence is unclear due to the absence of (or lack of knowledge of) the constants of motion for the general unperturbed nonlinear Klein-Gordon equation (except for the integrable sG equation). In order to solve the perturbed nonlinear Klein-Gordon equation, we need to assume how explicit the approximated solution of the problem is, i.e. the Ansatz. Usually the Ansatz involves a number of unknown variables (collective coordinates) and by using either the perturbation theory or CCs approach we obtain the equations of motion (system of ordinary differential equations) that the CCs satisfy. 2 The equations of the CCs have been also derived by using the Lagrangian formalism. In spite of its simplicity, this method has only been used to study the kink-antikink collisions in the unperturbed problem [15] and the kinkimpurity interactions in the sG and φ4models [16,17]. Furthermore, it has been used to analyze kink-antikink collisions in the damped φ4and sG models driven by an external force, however the projection technique was used to treat the dissipation separately [18]. Probably, the main difficulty in using this method was the fact that there is not any systematic way to construct the Lagrangian density when a generic perturbation is added into the sG or φ4equations. The aim of this work is to extend the Lagrangian formalism to the generic perturbed nonlinear Klein-Gordon equation in order to obtain the evolution equations that obey the CCs. We explicitly consider a damping term as a perturbation. In this case, we are able to write the Lagrangian density by introducing a new time variable and calculate the Lagrangian as a function of the CCs for a given Ansatz (see the section 2). Some examples are also considered in section 2, where we analyze perturbed sG and φ4equations and we show the equivalence between the GTWA and the Lagrangian formalism for the nonlinear Klein-Gordon equations. Finally, in section 3 we discuss and summarize the main results of this work. 2 Lagrangian formalism In this section we study the approximated solution of the following perturbed nonlinear Klein-Gordon equation φtt −φxx =−dU dφ −βφt+z(x, t, φ),(1) where the subindex tand xindicate the partial derivatives with respect to time and space, respectively; U(φ) is the nonlinear Klein-Gordon potential, βis the damping coefficient and z(x, t, φ) represents a generic perturbation on the system. Notice that, by making the change of variable in time τ= exp(−βt) [19], Eq. (1) becomes in dissipationless equation, φτ τ −φxx β2τ2=−1 β2τ2 dU dφ +˜z(x, τ, φ) β2τ2.(2) Let us to remark that the idea to suppress the damping in the nonlinear systems have been already used in [20], where the authors showed that the 3 effect of damping in the Landau-Lifshitz equation is only a rescaling of time by a complex constant. Now by using the Euler-Lagrange equation [21] it is not difficult to show that Eq. (2) can be obtained from the Lagrangian L=Z+∞ −∞ dxL=Z+∞ −∞ dx (1 2φ2 τ−1 2 φ2 x β2τ2−U(φ) β2τ2+W(x, τ)φx β2τ2),(3) where Lis the Lagrangian density, and W(x, τ) = −Zx x0 ˜z(y, τ, φ)dy, (4) where φin the last expression represents the approximated solution of Eq. (1), i.e. the Ansatz. Notice that, in general the expression (3) is an approximated Lagrangian density corresponding to the Eq. (2). In particular, if z(x, t, φ) is not a function of the field φ, the Eq. (3) represents an exact Lagrangian density corresponding to Eq. (1). To proceed we need to assume an approximated solution for Eq. (2). The Ansatz we use involves NCCs (~ Y=Y1, Y2, ..., YN) and so, by inserting it in the Eq. (3) we obtain the Lagrangian as a function of the NCCs and their derivatives with respect to the new time variable τ, L(~ Y , ~ Y′). From this moment on with the prime and the dot we denote the time derivative in relation to τand t, respectively. The next step is to derive the equations of motion for a given CC Yi(i= 1,2, ..., N) by using the Lagrange equation d dτ ∂L ∂Y ′ i!−∂L ∂Yi = 0.(5) In particular, we will analyze two cases, U(φ) = 1 −cos(φ) corresponding to sG equation, and U(φ) = (1/4)(1 −φ2)2to φ4one. In both examples, we use the Rice’s Ansatz [7,13]. This approximated solution reads φ(x, τ) = 4 atan exp "x−X(τ) l(τ)#!,(6) for the sG and φ(x, τ) = tanh "x−X(τ) l(τ)#,(7) 4 for φ4equation, where X(τ) and l(τ) represent the center and the width of the kink, respectively. Substituting Eqs. (6) and (7) in (3) and integrating we obtain L(X, X′, l, l′) = M0l0 2l(X′)2+αM0l0 2l(l′)2−M0 2β2τ2 l0 l+l l0!+ (8) 1 β2τ2Z+∞ −∞ W(X+θl, τ)φθdθ, where M0is the mass of the kink, l0represents the width of the static unperturbed kink and αis a coefficient. In particular, M0= 8, l0= 1 and α=π2/12 for the sG and M0= 4/(3√2), l0=√2 and α= (π2−6)/12 for φ4. From the Eq. (5) and the Lagrangian (8), the equations of motion for the CCs X(τ) and l(τ) are given by dP dτ =−1 β2τ2Z+∞ −∞ ˜z(X+θl, τ, φ)φθdθ, P (τ) = M0l0 l(τ)X′,(9) αM0l0 l l′′ −(l′)2 2l!+P2 2M0l0 +M0 2β2τ2l l l0−l0 l!= 1 β2τ2Z+∞ −∞ ∂W ∂l (X+θl, τ)φθdθ, or equivalent, in a more compact form, the equations for X(t) and l(t) read dP dt =−βP(t)−Z+∞ −∞ z(X+θl, τ, φ)φθdθ, P(t) = M0l0˙ X l(t),(10) α[˙ l2−2βl˙ l−2l¨ l] = l2 l2 0 P2 M2 0 + 1!−1 + 2l2 M0l0Z+∞ −∞ z(X+θl, τ, φ)θφθdθ. This system of equations has been also obtained by using the GTWA or the variation of the energy, E(t), and the momentum, P(t), together with the Rice’s Ansatz [22]. As we will later show the Lagrangian formalism and the GTWA are equivalent in a more general sense, i.e. the equivalence between both methods is regardless of the Ansatz that we use as approximated solution of Eq. (1) and also of how many CCs we use. Notice that, for the ac forces z(x, t, φ) = ǫ1sin(δt+δ0)+ǫ2sin(mδt+δ0) [9,10,23], for the parametric periodic force z(x, t, φ) = ǫsin(δt +δ0)φ[24], and for the driven z(x, t, φ) = −ǫ[25] we recover the equations related with the resonance phenomena and the ratchet effect studied in these references by using either the GTWA or the variation of E(t) and P(t). In order to establish the relation between GTWA and Lagrangian formalism, presented here for the nonlinear Klein-Gordon equations, let us assume 5 that the solution of Eq. (2) depends on the NCCs, so φ(x, ~ Y(τ)) with ~ Y= Y1, Y2, ..., YN. Then, the Lagrangian density in Eq. (3) is a function of φ(x, ~ Y(τ)), so that L=Z+∞ −∞ dx L(φ(x, ~ Y(τ))).(11) By inserting Eq. (11) in (5), after some straightforward calculations (see e.g. [18]a) we obtain Z+∞ −∞ dx (∂ ∂τ ∂L ∂φτ!+∂ ∂x ∂L ∂φx!−∂L ∂φ )∂φ ∂Yi .(12) This equation is just the projection of Eq. (2) into the functions ∂φ/∂Yi. By substituting the Ansatz φ(x, ~ Y) in Eq. (12) we obtain the following set of N equations for the NCCs by using the Lagrangian density defined in (3): Z+∞ −∞ dx    ∂φ ∂Yk N X i=1 ∂˜ ψ ∂Y ′ i Y′′ i+ N X i=1(i6=k)"∂φ ∂Yk ∂˜ ψ ∂Yi−∂˜ ψ ∂Yk ∂φ ∂Yi#Y′ i+ 1 β2τ2"∂φ ∂Yk δH δφ −βτ ∂˜ ψ ∂Yk δH δψ #−˜z β2τ2 ∂φ ∂Yk)= 0,(13) where k= 1,2, ..., N,˜ ψ≡φτ=˜ ψ(x, ~ Y , ~ Y′) and His the Hamiltonian corresponding to Eq. (1) with β= 0 and z= 0. In the variable tthis system of N equations reads Z+∞ −∞ dx    ∂φ ∂Yk N X i=1 ∂ψ ∂˙ Yi ¨ Yi+ N X i=1(i6=k)"∂φ ∂Yk ∂ψ ∂Yi−∂ψ ∂Yk ∂φ ∂Yi#˙ Yi+ ∂H ∂Yk−(z−βψ)∂φ ∂Yk)= 0,(14) where His the Hamiltonian density of the unperturbed Eq. (1) [put in (1) βand zequal to zero]. This set of equations for the NCCs can be obtained also by using the GTWA. This set of equations is reduced when two CCs, Y1≡X(t) and Y2≡l(t), are considered. In this case we recover the Eqs. (7) and (10) of [14], obtained by using either the GTWA or the variation of the energy and the momentum. 6 3 Conclusions In this work, using the Lagrangian formalism, we have developed an alternative method in order to obtain the system of ODEs that satisfy the CCs in the generic perturbed nonlinear Klein-Gordon models. The main advantage of this method over previous ones is that it involves less calculations, so that the equations of motion for the CCs can be derived straightforwardly. Indeed, given the generic perturbed nonlinear Klein-Gordon equation (1), we have shown that with a nonlinear change of the time variable in (1) it is possible to write the dissipationless equation (2) and its Lagrangian density (3). (In general Lis an approximated Lagrangian density of (1). In particular, when the perturbation z(x, t, φ) does not depend on φ, we are able to obtain an exact Lagrangian density corresponding to Eq. (1)). Furthermore, given an Ansatz for the solution of (1) we can, first, calculate the Lagrangian as a function of the NCCs by using the Lagrangian density defined by Eq. (3). Then, from the Lagrange equation (5) we can obtain the corresponding system of ODEs for the collective variables. By using the Rice Ansatz [7,13] we have obtained the system of ODEs that satisfy the collective variables for the perturbed sine-Gordon and φ4systems. These equations coincide with those obtained by the GTWA [12] or the variation of energy and the momentum [14]. In particular, for ac and dc forces we have recovered the results obtained in [9,10,23,24,25]. We would like to remark that this approach, presented here for the perturbed sG and φ4models with the Rice Ansatz can be extended to others perturbed nonlinear KleinGordon systems, for example, DsG and ADSG equations [6]. Furthermore, the Lagrangian formalism is not restricted to one dimensional system. Indeed, it recently have been applied to solve approximately a problem concerning to the vortex theory [26]. Finally for NCCs, we have shown the equivalence between this method and the GTWA, regardless of the Ansatz that we use for the approximated solution of Eq. (1). 4 Acknowledgments We would like to thank Renato ´ Alvarez-Nodarse and Franz Mertens for the useful discussion on this work. This work has been supported by the Ministerio de Ciencia y Tecnolog´ıa of Spain through grants BFM2001-3878-C02 and by the Junta de Andaluc´ıa under the project FQM-0207. 7 References [1] Yuri S. 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