A direct method for solving the generalized sine-Gordon equation Yoshimasa Matsuno Division of Applied Mathematical Science, Graduate School of Science and Engineering 0 Yamaguchi University, Ube, Yamaguchi 755-8611, Japan 1 0 E-mail address: [email protected] 2 n a J Abstract 8 2 The generalized sine-Gordon (sG) equation was derived as an integrable generalization of ] the sG equation. In this paper, we develop a direct method for solving the generalized sG S P equation without recourse to the inverse scattering method. In particular, we construct . n i multisoliton solutions in the form of parametric representation. We obtain a variety l n of solutions which include kinks, loop solitons and breathers. The properties of these [ 1 solutions are investigated in detail. We find a novel type of solitons with a peculiar v 8 structure that the smaller soliton travels faster than the larger soliton. We also show that 2 1 the short pulse equation describing the propagation of ultra-short pulses is reduced from 5 . the generalized sG equation in an appropriate scaling limit. Subsequently, the reduction 1 0 to the sG equation is briefly discussed. 0 1 : v Xi MSC: 35Q51, 37K10, 37K40 r a 1 1. Introduction We consider the following generalized sine-Gordon (sG) equation u = (1+ν∂2)sin u, (1.1) tx x where u = u(x,t) is a scalar-valued function, ν is a real parameter, ∂2 = ∂2/∂x2 and x the subscripts t and x appended to u denote partial differentiation. The generalized sG equation has been derived for the first time in [1] using bi-Hamiltonian methods. Quite recently, equation (1.1) with ν < 0 was shown to be a completely integrable partial differential equation (PDE) [2]. Indeed, constructing a Lax pair associated with it, the initial value problem of the equation was solved for decaying initial data. In the process, the Riemann-Hilbert formalism was developed to obtain eigenfunctions of the Lax pair. Soliton solutions are obtainable in principle, but their derivation needs a very complicated procedure. Although some qualitative features of traveling-wave solutions arediscussed in a different context from the Riemann-Hilbert formalism, explicit expressions of solutions are not available as yet. The purpose of this paper is to obtain exact solutions of equation (1.1) with ν < 0 and discuss their properties. We consider real and nonperiodic solutions. In our analysis, we take ν = 1 without loss of generality. The exact method of solution used here is − the bilinear transformation method which is a very powerful tool in obtaining special solutions of soliton equations [3, 4]. The method has wide applications ranging from continuous to discrete soliton equations. The central problem in the bilinear formalism is the construction of tau-functions which are introduced through dependent variable transformations. This paper is organized as follows. In section 2, we develop an exact method of solution. Specifically, we use a hodograph transformation to transform equation under consideration into a more tractable form. The transformed equation is further put into a system of bilinear equations by introducing appropriate dependent variable transfor- mations. We then construct explicit solutions of the bilinear equations by means of a standard procedure in the bilinear formalism. The multisoliton solutions are obtained in the form of parametric representation. In section 3, we describe properties of solutions. 2 First, we consider 1-soliton solutions which include kink and loop soliton solutions as well as a new type of multivalued functions. Throughout this paper we use the term ”soli- ton” as a generic name of elementary solutions such as kink, loop soliton and breather solutions. A novel feature of regular kink solutions is found which has never been seen in the sG kinks. Next, the 2-soliton solutions are discussed. We address the kink-kink and kink-loop soliton solutions together with the 1-breather solution. Last, we explore the general multisoliton solutions. Our particular concern is the multikink solution for which the large-time asymptotic is derived and the associated formula for the phase shift is obtained. A recipe for constructing the multibreather solutions is briefly described. As examples of the multisoliton solutions, we present a solution describing the interaction between a soliton and a breather as well as a 2-breather solution which are reduced from the 3 and 4-soliton solutions, respectively. In section 4, we point out a close relationship between the generalized sG equation and the short pulse equation which models the prop- agationof ultra-short optical pulses. We show that the generalized sG equation is reduced to the short pulse equation by taking an appropriate scaling limit. The parametric mul- tiloop soliton solution of the short pulse equation presented in [5] is reproduced from the corresponding one for the generalized sG equation. The similar limiting procedure is also applied to the formula for the phase shift. Subsequently, the reduction of the generalized sG equation to the sG equation is discussed shortly. Section 5 is devoted to conclusion. In the appendix A, we show that the tau-functions for the multisoliton solutions obtained in section 3 satisfy a system of bilinear equations. The proof is carried out by means of an elementary method using various formulas for determinants. In the appendix B, we derive the 1-soliton solutions by an alternative method and demonstrate that they reproduce the corresponding 1-soliton solutions obtained in section 3. 2. Exact method of solution 2.1. Hodograph transformation We introduce the new dependent variable r in accordance with the relation r2 = 1+u2, (r > 0), (2.1) x 3 to transform equation (1.1) with ν = 1 into the form − r +(r cos u) = 0. (2.2) t x We then define the hodograph transformation (x,t) (y,τ) by → dy = rdx r cos udt, dτ = dt. (2.3) − The x and t derivatives are then rewritten in terms of the y and τ derivatives as ∂ ∂ ∂ ∂ ∂ = r , = r cos u . (2.4) ∂x ∂y ∂t ∂τ − ∂y With the new variables y and τ, (2.1) and (2.2) are recast into the form r2 = 1+r2u2, (2.5) y 1 (cos u) = 0, (2.6) y r − (cid:18) (cid:19)τ respectively. Further reduction is possible if one defines the variable φ by π π u = sin φ, φ = φ(y,τ), < φ < , mod 2π . (2.7) y −2 2 (cid:16) (cid:17) It follows from (2.5) and (2.7) that 1 = cos φ. (2.8) r Substituting (2.7) and (2.8) into equation (2.6), we find φ = sin u. (2.9) τ If we eliminate the variable φ from (2.7) and (2.9), we obtain a single PDE for u u τy = sin u. (2.10) 1 u2 − y Similarly, elimination of the variablepu gives a single PDE for φ φ τy = sin φ. (2.11) 1 φ2 − τ p 4 By inverting the relationship (2.4) and using (2.8), equation that determines the inverse mapping (y,τ) (x,t) is found to be governed by the system of linear PDEs for x = → x(y,τ) x = cos φ, (2.12a) y x = cos u. (2.12b) τ Note that the integrability of the system of equations (2.12) is assured by (2.7) and (2.9). A sequence of transformations described above are almost the same as those employed for solving the short pulse equation [5]. The underlying idea is to transform the orignal equation to the (possibly) integrable equation. In the case of the short pulse equation, the transformrd equation is the sG equation whereas in the present case, the corresponding equations are (2.10) and (2.11). The soliton solutions of the latter equations will be construcred here for the first time. Given u and φ, the most difficult problem is how to integrate the system of equations (2.12). This becomes the core part of the present paper and will be resolved by Theorem 2.1. 2.2. Bilinear formalism Here, we develop a method for solving a system of PDEs (2.7) and (2.9). We use the bilinear transformation method [3, 4]. Let σ and σ be solutions of the sG equation ′ σ = sin σ, σ = σ(y,τ), (2.13a) τy σ = sin σ , σ = σ (y,τ). (2.13b) τ′y ′ ′ ′ We then put 1 u = (σ +σ ), (2.14a) ′ 2 1 φ = (σ σ ). (2.14b) ′ 2 − In terms of σ and σ , equations (2.7) and (2.9) can be written as ′ 1 1 (σ +σ ) = sin (σ σ ), (2.15a) ′ y ′ 2 2 − 1 1 (σ σ ) = sin (σ +σ ). (2.15b) ′ τ ′ 2 − 2 5 It should be remarked that the system of PDEs (2.15) constitutes a B¨acklund transfor- mation of the sG equation with a B¨acklund parameter taken to be 1 [6]. The real-valued solutions of the sG equations (2.13) can be put into the form [7-10] f ∗ σ = 2i ln , (2.16a) f g ∗ σ = 2i ln , (2.16b) ′ g where f and g denote the complex conjugate of f and g, respectively. The tau-functions ∗ ∗ f andg playacentralroleinouranalysis. Theyarefundamentalquantitiesinconstructing solutions. For soliton solutions, they satisfy the following bilinear equations [7]: 1 D D f f = (f2 f 2), (2.17a) τ y ∗ · 2 − 1 D D g g = (g2 g 2), (2.17b) τ y ∗ · 2 − where the bilinear operators D and D are defined by τ y DτmDynf ·g = (∂τ −∂τ′)m(∂y −∂y′)nf(τ,y)g(τ′,y′)|τ′=τ,y′=y, (m,n = 0,1,2,...). (2.18) Now, we seek solutions of equations (2.7) and (2.9) of the form F ∗ u = i ln , (2.19a) F G ∗ φ = i ln , (2.19b) G where F and G are new tau-functions. It turns out from (2.14), (2.16) and (2.19) that f g F ∗ ∗ ∗ 2u = σ +σ = 2i ln = 2i ln , (2.20a) ′ fg F f g G ∗ ∗ 2φ = σ σ = 2i ln = 2i ln . (2.20b) ′ − fg G ∗ By taking into account (2.20), we may assume the following relations among the tau- functions f,g,F and G: F = fg, (2.21a) 6 G = fg . (2.21b) ∗ The above expressions lead to an important relation F F = G G. (2.22) ∗ ∗ Substituting (2.19) into equations (2.7) and (2.9) and using (2.22), we see that F and G satisfy a system of bilinear equations 1 D F F = (G2 G 2), (2.23a) y ∗ ∗ · −2 − 1 D G G = (F2 F 2). (2.23b) τ ∗ ∗ · −2 − Thus, the problem under consideration is reduced to obtain solutions of equations (2.23) subjected to the condition (2.22). After some trials, however, we found that this proce- dure for constructing solutions is difficult to perform. Hence, we employ an alternative approach. To begin with, we impose the following bilinear equations for f and g which turn out to be the starting point in our analysis 1 D f g = (fg f g), (2.24a) y ∗ ∗ ∗ · 2 − 1 D f g = (fg f g ). (2.24b) τ ∗ ∗ · 2 − With (2.24) at hand, the following proposition holds: Proposition 2.1. If f and g satisfy the bilinear equations (2.24), then the tau-functions F and G defined by (2.21) satisfy the bilinear equations (2.23). Proof. First, we prove (2.23a). We substitute (2.21a) into the left-hand side of (2.23a) and rewrite it in terms of bilinear operator to obtain D F F = (D f g )f g +(D f g)fg . (2.25) y ∗ y ∗ ∗ y ∗ ∗ · − · · By virtue of (2.24a), the right-hand side of (2.25) becomes (1/2) (fg )2 (f g)2 which ∗ ∗ − { − } is equal to the right-hand side of (2.23a) by (2.21b). The proof of (2.23b) can be done in (cid:3) the same way by using (2.24b). 2.3. Parametric representation 7 We demonstrate that the solution of equation (1.1) with ν = 1 admits a parametric − representation. The following relation is crucial to integrate (2.12): Proposition 2.2. cos φ is expressed in terms of f and g as g g ∗ cos φ = 1+ ln . (2.26) f f (cid:18) ∗ (cid:19)y Proof. Using (2.24a), one obtains g g D f g D f g ∗ y ∗ y ∗ ln = · · f f − fg − f g (cid:18) ∗ (cid:19)y ∗ ∗ 1(fg )2 +(f g)2 ∗ ∗ = 1. (2.27) 2 f fg g − ∗ ∗ On the other hand, it follows from (2.19b) and (2.21b) that 1 G G ∗ cos φ = + 2 G G (cid:18) ∗ (cid:19) 1(fg )2 +(f g)2 ∗ ∗ = . (2.28) 2 f fg g ∗ ∗ (cid:3) The relation (2.26) follows immediately by comparing (2.27) and (2.28). Integrating (2.12a) coupled with (2.26) by y, we obtain the expression of x g g ∗ x = y +ln +d(τ), (2.29) f f ∗ where d is an integration constant which depends generally on τ. The expression (2.29) now leads to our main result: Theorem 2.1. The real-valued solution of equation (1.1) with ν = 1 can be written by − the parametric representation f g ∗ ∗ u(y,τ) = i ln , (2.30a) fg g g ∗ x(y,τ) = y +τ +ln +y , (2.30b) 0 f f ∗ where the tau-functions f and g satisfy both (2.17) and (2.24) simultaneously and y is 0 an arbitrary constant independent of y and τ. Proof. The expression (2.30a) for u is a consequence of (2.19a) and (2.21a). To prove (2.30b), we substitute (2.29) into (2.12b) and obtain the relation g g ∗ cos u = ln +d(τ). (2.31) ′ f f (cid:18) ∗ (cid:19)τ 8 The left-hand side of (2.31) can be expressed by f and g in view of (2.30a) whereas the right-hand side is modified by using (2.24b). After a few calculations, we find that most terms are cancelled, leaving the equation d(τ) = 1. Integrating this equation, one obtains ′ d(τ) = τ +y , which, substituted into (2.29), gives the expression (2.30b) for x. (cid:3) 0 The parametric solution (2.30) would produce in general a multi-valued function as happned in the case of the short pulse equation [5]. To derive a criterion for single-valued functions, we calculate u with use of (2.7) and (2.8) and obtain x u = ru = tan φ. (2.32) x y Thus, if the inequality π/2 < φ < π/2 (mod π) holds for all y and τ, then u becomes a − regular function of x and t. By virtue of the identity G ImG i ln ∗ = 2 tan 1 , (2.33) − G ReG (cid:18) (cid:19) as well as the relation (2.21b), the above condition for regularity can be written as Im(fg ) ∗ 1 < < 1. (2.34) − Re(fg ) ∗ In the case of 1-soliton solutions discussed in the next section, the above condition is found explicitly in terms of the parameters characterizing solutions. 2.4. Multisoliton solutions The last step in constructing solutions is to find the tau-functions f and g for the sG equationwhichsatisfysimultaneouslythebilinearequations(2.24). Thefollowingtheorem establishes this purpose: Theorem 2.2. The tau-functions f and g given below satisfy both the bilinear forms (2.17) of the gG equation and the bilinear equations (2.24) N π f = exp µ ξ + i + µ µ γ , (2.35a) j j j k jk 2 " # µX=0,1 Xj=1 (cid:16) (cid:17) 1≤jX<k≤N N π g = exp µ ξ 2d + i + µ µ γ , (2.35b) j j j j k jk − 2 " # µX=0,1 Xj=1 (cid:16) (cid:17) 1≤jX<k≤N 9 where 1 ξ = p y + τ +ξ , (j = 1,2,...,N), (2.36a) j j j0 p j 2 p p eγjk = j − k , (j,k = 1,2,...,N;j = k), (2.36b) p +p 6 (cid:18) j k(cid:19) 1 p e 2dj = − j, (j = 1,2,...,N). (2.36c) − 1+p j Here, p and ξ are arbitrary real parameters satisfying the conditions p = p for j = k j j0 j k 6 6 and N is an arbitrary positive integer. The notation implies the summation over µ=0,1 all possible combination of µ = 0,1,µ = 0,1,...,µ = 0,1. 1 2 N P Proof. It has been shown that f and g given by (2.35a) and (2.35b) satisfy the bilinear equations (2.17a) and (2.17b), respectively [7]. Thus, it is sufficient to prove that they satisfy the bilinear equations (2.24). The proof is carried out by a lengthy calculation (cid:3) using various formulas for determinants. It will be summarized in the appendix A. 2.5. Remark The tau-functions f and g given by (2.35) yield real-valued solutions since all the param- eters p and ξ (j = 1,2,...,N) are chosen to be real numbers. If one looks for breather j j0 solutions, for example, one needs to introduce complex parameters (see sections 3.2.3, 3.3.2 and 3.3.3). Even in this case, however, the analysis developed here can be applied as well without making essential changes. Actually, we may use the tau-functions f and ′ g instead of f and g , respectively where f and g are obtained simply from f and ′ ∗ ∗ ′ ′ g by replacing i by i, but all the parameters in the tau-functions are assumed to be − complex numbers. The solutions of the sG equations (2.13a) and (2.13b) can be written as σ = 2i ln(f /f) and σ = 2i ln(g /g), respectively, where the tau-functions f,f ,g and ′ ′ ′ ′ g satisfy the following systems of bilinear equations ′ 1 1 D D f f = (f2 f 2), D D f f = (f 2 f2), (2.37a) τ y ′ τ y ′ ′ ′ · 2 − · 2 − 1 1 D D g g = (g2 g 2), D D g g = (g 2 g2). (2.37b) τ y ′ τ y ′ ′ ′ · 2 − · 2 − The bilinear equations corresponding to (2.24) are then given by 1 1 D f g = (fg f g), D f g = (fg f g ), (2.38a)) y ′ ′ ′ τ ′ ′ · 2 − · 2 − 10