Localized Solutions of the Non-Linear Klein-Gordon Equation in Many Dimensions

arXiv:0712.3284v1 [hep-th] 19 Dec 2007 Localized Solutions of the Non-Linear Klein-Gordon Equation in Many Dimensions M.V. Perel, I.V. Fialkovsky ∗ July 21, 2021 Abstract We present a new complex non-stationary particle-like solution of the non-linear Klein-Gordon equation with several spatial variables. The con- struction is based on reduction to an ordinary differential equation. The problem of finding or proving the existence of localized solutions of the non-linear Klein-Gordon equation in many spatial dimensions was discussed in many papers from mathematical, physical and numerical points of view [1]-[5]. The book [3] is devoted to complex asymptotic solutions of non-linear equations. We use the approach to construction of localized solutions of linear equations [6, 7]. Here we give a method of calculating complex localized solutions of the non-linear Klein-Gordon equation. For moderate time this solution has simple explicit exponentially decreasing asymptotic behavior outside some area moving with the group speed. The first term of this asymptotics is the exact solution of the linear Klein-Gordon equation presented earlier in [7] which decrease expo- nentially away from the point moving along the straight line. Inside the moving area this solution can be found numerically from an ordinary differential equa- tion of some complex variable depending on the time and spatial coordinates. Particle-like solution on the linear Klein-Gordon equation in two dimensions. We consider the linear Klein-Gordon equation with constant co- efficients c−2vtt −△v + m2v = 0, △v = vxx + vzz. (1) The equation (1) has the solution depending on a single variable s (see [7]) v = exp (ims) s (2) ∗Physics Faculty, St. Petersburg University, Ulyanovskaya 1-1, Petrodvorets, St. Peters- burg, 198904, Russia; E-mail: ifialk@gmail.com, perel@mph.phys.spbu.ru 1 with s depending on the spatial coordinates and time as follows s = i r (z −ikb)2 + x2 −(ct −iω c b)2 = i p m2b2 + x2 + z2 −c2t2 + 2ib(ωt −kz). (3) Here k, b are free parameters and ω = c √ k2 + m2. It is shown in [7] that the solution (2) has finite energy when b and k are real and Ims > 0. If the time is small enough |t| ≪bm2/ω than the solution decreases exponentially for |x| →∞and |z| →∞. If |x| ≪bm and |z| ≪ min (bm2/k, bm) then the expansion of the form ims ∼−bm2 −(z −vgrt)2 ∆2 ∥ −x2 ∆2 ⊥ −i(ωt −kz) (4) holds. We use the following notations ∆∥= √ 2bmc/ω, ∆⊥= √ 2b. From (4) and (2) it follows that the solution represents a wave packet with the Gaussian envelope filled with oscillations. It moves with the group speed vgr = dω/dk in the positive direction of the z axis. This is demonstrated by the numerical calculations of the solution (2) in successive times, see Fig.1 where the results are presented for the parameters m = 5, c = 1, k = 2, b = 15 in the conventional units. Non-linear Klein-Gordon equation in two spacial dimensions. We search now the solution on the non-linear Klein-Gordon equation in two dimen- sional space c−2utt −△u + f(u) = 0 (5) depending on the spatial coordinates and time only through the complex vari- able s defined by (3). Then the partial differential equation (5) reduces to the ordinary differential equation uss + 2 sus + f(u) = 0. (6) Choosing for the sake of definiteness the function f(u) as follows f(u) = m2u + γu3, γ = const, (7) we prove that there exists the exact solution on non-linear equation (5) having an estimate u(s) = C exp (ims) s (1 + O(q exp (−2a)), C = const, (8) if q exp (−2a) is small enough, where Re(ims) ≤(−a) < 0, q = γC2/(m|S|). (9) 2 Figure 1: Particle-like solution on the Klein-Gordon equation in the successive times in conventional units 3 We use here the technique of integral equations. In conditions of the validity of (4) the inequality (9) can be written as follows (z −vgrt)2 ∆2 ∥ + x2 ∆2 ⊥ ≥a −bm2. (10) The asymptotics (8) is valid for the solution of (6) outside the ellipse (10). Inside the ellipse (10) the equation (6) should be solved numerically. Non-linear Klein-Gordon equation in many dimensional space. The Klein-Gordon equation in many dimensional space c−2utt −△u + f(u) = 0, △u = ux1x1 + ux2x2 + . . . + uxnxn, (11) can be treated analogously to the case of two dimensional space. Seeking the solution of (11) u as the function of the single complex variable s s = i r (x1 −ikb)2 + x2 2 + . . . + x2n −(ct −iω c b)2. (12) we obtain the ordinary differential equation uss + n s us + f(u) = 0. (13) We suppose that f is defined by (7). For moderate values of t a solution on the equation (13) exists with the asymptotics written in terms of the Hankel function u(s) = s−(n−1)/2H(1) (n−1)/2(ms)(1 + O(exp(−2a))), a →∞ (14) which is valid outside the moving area (x1 −vgrt)2/∆2 ∥+ (x2 2 + . . . + x2 n)/∆2 ⊥≥ (a −bm2) where ∆∥, ∆⊥, a are defined above. Localization of the solution for moderate times follows from the asymptotics of the Hankel function. The research is supported by the grant RFBR 0001-00485. References [1] Strauss W.A. ”Decay

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