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Construction of Doubly Periodic Solutions Via the Poincare Mathematics and Computers in Simulation 57 (2001) 239–252 Construction of doubly periodic solutions via the Poincare–Lindstedt method in the case of massless ϕ4 theory Oleg Khrustalev a,∗, Sergey Vernov b a Department of Physics, Institute for Theoretical Problems of Microphysics of Moscow State University, Moscow State University, Vorobievy Gory, Moscow 119899, Russia b Institute of Nuclear Physics, Moscow State University, Vorobievy Gory, Moscow 119899, Russia Abstract Doubly periodic (periodic both in time and in space) solutions for the Lagrange–Euler equation of the (1 + 1)-dimensional scalar ϕ4 theory are studied. Provided that the nonlinear term is small, the Poincare–Lindstedt asymptotic method can be used to find asymptotic solutions in the standing wave form. The principal resonance problem, which arises for zero mass, is solved if the leading-order term is taken in the form of Jacobi elliptic function. To obtain this leading-order term the system REDUCE is used. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: φ4 Theory; Periodic solutions; Standing waves; Poincaré uniform expansion 1. Introduction 1.1. Periodic solutions of nonlinear equations At present time periodic solutions of the nonlinear wave equation: ∂2ϕ(x,t) ∂2ϕ(x,t) − − g(ϕ) = 0, (1) ∂x2 ∂t2 where g(ϕ) denotes a continuous function on R such that g(0) = 0, are studied intensively. Coron [1] has proved that T -periodic (in time) solutions of the following problem: ∂2ϕ(x,t) ∂2ϕ(x,t) − − g(ϕ) = 0,x,t∈ R,g(0) = 0, ∂x2 ∂t2 (2) lim|x|→∞ ϕ(x,t) = 0,x,t∈ R, ϕ(x,t + T)= ϕ(x,t), x,t,T ∈ R, ∗ Corresponding author. E-mail addresses: [email protected] (O. Khrustalev), [email protected] (S. Vernov). 0378-4754/01/$ – see front matter © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S0378-4754(01)00342-1 240 O. Khrustalev, S. Vernov / Mathematics and Computers in Simulation 57 (2001) 239–252 can exist only when π 2 g ( )> 2 . 0 T In particular, if g(ϕ) = sin(ϕ), there are no T -periodic solutions when T<2π (for any period T>2π an explicit solution is known, see e.g. [2]). When g(ϕ) =−sin(ϕ) or g(ϕ) =±ϕ3, then ϕ ≡ 0 is the only solution of (2) (for any period T ). To obtain periodic solutions of massless ϕ4 theory one has to change the boundary conditions. Generalizing the Rabinowitz theorem [3], Brezis et al. [4] have proved that for each T which is a rational multiple of π, the following problem: ∂2ϕ(x,t) ∂2ϕ(x,t) − − g(ϕ) = 0,x,t∈ R,g(0) = 0, ∂x2 ∂t2 ϕ( ,t)= ϕ(π,t) = ,t∈ R, 0 0 (2a) ϕ(x,t + T)= ϕ(x,t), x,t,T ∈ R, with continuous nondecreasing functions g(ϕ) such that g(ϕ) lim =∞, |ϕ|→∞ ϕ and ∃α, β > 0 such that 1 ϕ ∀ϕ ∈ R : ϕ · g(ϕ) − g(ψ)dψ ≥ β|g(ϕ)|−α 2 0 has a nontrivial weak T -periodic solution. By a weak solution we mean such function ϕ that T π (ϕ(vtt − vxx) + g(ϕ)v)dt dx = 0 0 0 for all v ∈ C2([0,π] × R) satisfying the boundary and periodicity conditions. The proof in [4] is simpler than the original proof of Rabinowitz. In the case g(ϕ) =|ϕ|n · ϕ, where n>0, this theorem has been proved also in [5]. 1.2. Two kinds of doubly periodic solutions Our investigation is dedicated to the construction of doubly periodic classical fields in the (1 + 1)-dimensional ϕ4 theory. We study the model of an isolated real scalar field ϕ(x,t), described by the Lagrangian density: ε 1 2 2 2 2 4 L(ϕ) = ϕ,t (x, t) − ϕ,x(x, t) − M ϕ (x, t) − ϕ (x, t) . 2 2 The associated Lagrange–Euler equation is: ∂2ϕ(x,t) ∂2ϕ(x,t) − − M2ϕ(x,t) − εϕ3(x, t) = 0. (3) ∂x2 ∂t2 O. Khrustalev, S. Vernov / Mathematics and Computers in Simulation 57 (2001) 239–252 241 There are two classes of doubly periodic solutions for this equation. If we seek fields in the traveling wave form: ϕ(x,t) = ϕ(x − vt), where v is the velocity of the wave motion, then Eq. (3) reduces to the Duffing’s equation [6]. As is known [7], periodic solutions for this equation are Jacobi elliptic functions. Such solutions are well known for both (1 + 1)-dimensional [8] and (3 + 1)-dimensional [9] variants of ϕ4 theory. They are used in the Yang–Mills theory [10]. The second class of doubly periodic solutions consists of functions in the standing wave form: ∞ ∞ ϕ(x,t) ≡ Cnj sin(n(x − x0)) sin(jω(t − t0)), (4) n=1 j=1 where x0 and t0 are constants determined by boundary and initial conditions. Eq. (3) is a translation-invariant, so we can restrict our consideration to the case of zero x0 and t0 without loss of generality. We suppose that the function ϕ(x,t) is 2π-periodic in space and seek its period in time. Exact standing wave solutions are not known. Approximate solutions for Eq. (3) with ε = 1 and M = 1 have been found under the assumption that all highest harmonics are zeros: ∀n, j > N : Cnj = 0, where N is a large number. Numerical calculations have been made [11] for several values of N. The obtained solutions are very close to each other, but convergence of the sequence of these solutions as N tends to infinity has yet to be proved. 2. Asymptotic solutions in the standing wave form Let us consider periodic solutions for Eq. (3), provided that ε is small or, equivalently, that these solutions have small amplitudes. An asymptotic expansion, containing only bounded functions, is called a uniform expansion. The possibility to obtain a uniform expansion, using standard asymptotic methods [12], for example, Poincare– Lindstedt [13] or Krylov–Bogoliubov [14] methods, depends on values of the frequencies in the zero approximation. If ε = 0, then Eq. (3) is linear and has periodic solutions of the form (4) with frequencies in time Ωj = j 2 + M2, where j ∈ IN. There are two fundamentally different cases. If the ratio Ωi/Ωj is irrational for all i and j, then it is a non-resonant case. The periodic asymptotic solutions in non-resonant case are well known. To obtain these solutions to any degree of accuracy, standard asymptotic methods can be used. The resonance case, when there exist two frequencies Ωj , whose relation is rational, is more difficult. The Krylov–Bogoliubov method and the standard variant of the Poincare–Lindstedt method can be used to find periodic solutions only to a few leading orders in ε. 4 The important example of resonance case is the massless ϕ theory when Ωj = j and all relations of frequencies are rational, which is the principal resonant case. Using standard asymptotic methods, one cannot construct a periodic solution even to the first order in ε. It is possible to transform the differential equations to a system of nonlinear algebraic equations in Fourier coefficients and frequencies using the Poincare–Dulac’s normal form method. The algorithm of this procedure was constructed [15,16] and 242 O. Khrustalev, S. Vernov / Mathematics and Computers in Simulation 57 (2001) 239–252 realized in the software for symbolic and algebraic computation REDUCE [17–19] (the system NORT [20,21]). But an algorithm to solve the resulting algebraic system has yet to be created. 3. The standing wave solutions for the massless ϕ4 theory 3.1. Construction of asymptotic solutions via the Poincare–Lindstedt method The purpose of this article is the construction of standing wave solutions of Eq. (3) with M = 0, using asymptotic methods when ε 1. We use the Poincare–Lindstedt method: introduce the new time t˜ ≡ ωt and look for a doubly periodic solution of Eq. (3). The ϕ(x,t)˜ and a frequency (in time) ω in the form of power series in ε: ∞ n ϕ(x,t,ε)˜ ≡ ϕn(x, t)ε˜ , n=0 ∞ n ω(ε) ≡ 1 + ωnε . n=1 Expanding the Lagrange–Euler equation in a power series in ε gives a sequence of equations. The leading two equations are: • to zero order in ε the equation for ϕ0 is: ∂2ϕ (x, t)˜ ∂2ϕ (x, t)˜ 0 − 0 = 0; (5) ∂x2 ∂t˜2 • to first order in ε the equation in ϕ0, ϕ1 and ω1 is: ∂2ϕ (x, t)˜ ∂2ϕ (x, t)˜ ∂2ϕ (x, t)˜ 1 − 1 = 2ω 0 + ϕ3(x, t).˜ (6) ∂x2 ∂t˜2 1 ∂t˜2 0 ˜ ˜ Eq. (5) has many periodic solutions. If we select ϕ0(x, t) = sin(x) sin(t), then the second equation has no periodic solution, because the frequency of the external force sin(3x)sin(3t)˜ is equal to the frequency of its own oscillations and it is impossible to put this resonance harmonic to zero by selecting only ω1. ˜ In the massless case, the correct selection of not only the frequency ω(ε), but also the function ϕ0(x, t) allows us to find a uniform expansion. 3.2. The condition of existence of periodic solutions The general solution, in the standing wave form (4), for Eq. (5) is the function ∞ ˜ ˜ ϕ0(x, t) = an sin(nx) sin(nt) n=1 O. Khrustalev, S. Vernov / Mathematics and Computers in Simulation 57 (2001) 239–252 243 ˜ with arbitrary an. We have to find coefficients an so that the function ϕ1(x, t) is a periodic solution for ˜ Eq.
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