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

Lo calized Solutions of the Non-Linear Klein-Gordon Equati on in Man y Dimensions M.V. P erel, I.V. Fialk o vsky ∗ July 21, 2021 Abstract W e present a new complex non- stationary particle-like solution of the non-linear Klein-Gordon equation with several spatial v aria bles. The con- struction is based on reduction to an ordinary d ifferential equ ation. The problem of finding or proving the existence o f lo calized solutions of the non-linear Klein-Gordo n equation in ma ny spatial dimensio ns was disc ussed in many pap ers from mathematical, physical a nd numerical p o ints of view [1]-[5]. The bo o k [3] is dev oted to complex asymptotic solutions of non-linear equations. W e use the approach to construction of lo calized solutions of linear equations [6, 7]. Here we give a metho d o f ca lculating complex lo calized solutions of the non-linear Klein- Go rdon equation. F or moderate time this s olution has simple explicit exp onentially decreasing asymptotic behavior outside so me area moving with the gro up sp eed. The first term of this as ymptotics is the exact solution of the linear Klein-Gordo n equation presented earlier in [7] whic h decrea s e exp o- nent ially aw ay from the p oint moving a long the straight line. Inside the moving area this s olution can b e found numerically from an ordinary differential e q ua- tion o f some complex v ariable dep ending on the time and spatial co ordinates. P article-lik e s olution on the l inear Klein-Gordo n equation in tw o dimensi ons. W e co nsider the linear Klein-Gor don equation with constant co- efficients c − 2 v tt − △ v + m 2 v = 0 , △ v = v xx + v z z . (1) The eq uation (1) has the solution dep ending on a single v ar iable s (see [7]) v = exp ( ims ) s (2) ∗ Ph ysics F aculty , St. Pet ersburg Unive rsity , Ulyan ov sk a y a 1-1, Pet ro dv orets, St. Pe ters- burg, 198904, R uss ia; E-mail: ifialk@gmail.com, p erel@mph.phys.spbu.ru 1 with s dep ending on the spatia l co ordina tes and time as follows s = i r ( z − ik b ) 2 + x 2 − ( ct − i ω c b ) 2 = i p m 2 b 2 + x 2 + z 2 − c 2 t 2 + 2 ib ( ω t − kz ) . (3) Here k , b are free par ameters and ω = c √ k 2 + m 2 . It is shown in [7] that the solution (2) has finite energ y when b and k are real and Im s > 0. If the time is small enough | t | ≪ b m 2 /ω than the solution decreases exponentially for | x | → ∞ and | z | → ∞ . If | x | ≪ bm and | z | ≪ min ( bm 2 /k , bm ) then the expansion of the form ims ∼ − bm 2 − ( z − v gr t ) 2 ∆ 2 k − x 2 ∆ 2 ⊥ − i ( ω t − k z ) (4) holds. W e use the following notations ∆ k = √ 2 bmc/ω , ∆ ⊥ = √ 2 b . F rom (4) and (2) it follows that the so lution represents a wav e packet with the Ga us sian env elope filled with oscillations. It mov es with the gr oup sp eed v gr = dω / dk in the p os itive direction o f the z axis. This is demonstrated b y the numerical calculations of the solution (2) in succes sive times, see Fig.1 where the r esults are presented for the par ameters m = 5 , c = 1, k = 2, b = 15 in the conv entional units. Non-line ar Klein-Gordon equation in tw o spacial di mensions . W e search now the solution o n the non-linear Klein-Go r don equation in tw o dimen- sional s pace c − 2 u tt − △ u + f ( u ) = 0 (5) depe nding on the s patial co ordinates and tim e o nly through the complex v ari- able s defined b y (3). Then the pa rtial differential equation (5) reduces to the ordinary differ e n tial equa tion u ss + 2 s u s + f ( u ) = 0 . (6) Cho osing for the sake o f definiteness the function f ( u ) as follows f ( u ) = m 2 u + γ u 3 , γ = const, (7) we prove that there exists the exact solution o n non-linea r equatio n (5) having an estimate u ( s ) = C exp ( ims ) s (1 + O ( q exp ( − 2 a )) , C = const, (8) if q exp ( − 2 a ) is small enough, wher e Re( ims ) ≤ ( − a ) < 0 , q = γ C 2 / ( m | S | ) . (9) 2 Figure 1: Particle-like solution on the Klein-Gor do n equation in the s uccessive times in conven tio nal units 3 W e use here the technique of integral equations. In conditions of the v a lidit y o f (4) the inequality (9) can b e written as fo llows ( z − v gr t ) 2 ∆ 2 k + x 2 ∆ 2 ⊥ ≥ a − b m 2 . (10) The a s ymptotics (8 ) is v alid for the solution of (6) outside the ellipse (10). Inside the ellipse (10 ) the equation (6) should b e so lved numerically . Non-line ar Kl ein-Gordon equation i n man y di mensional space. The Klein-Gordo n e quation in many dimensional s pace c − 2 u tt − △ u + f ( u ) = 0 , △ u = u x 1 x 1 + u x 2 x 2 + . . . + u x n x n , (11) can be trea ted analogous ly to the case of t wo dimensiona l spa ce. Seeking the solution o f (11 ) u a s the function o f the single complex v ariable s s = i r ( x 1 − ik b ) 2 + x 2 2 + . . . + x 2 n − ( ct − i ω c b ) 2 . (12) we obtain the ordinar y differential e q uation u ss + n s u s + f ( u ) = 0 . (13) W e supp ose that f is defined by (7). F or mo dera te v a lues o f t a solution on the equation (13 ) exists with the asymptotics wr itten in terms of the Hankel function u ( s ) = s − ( n − 1) / 2 H (1) ( n − 1) / 2 ( ms )(1 + O (exp( − 2 a ))) , a → ∞ (14) which is v alid outside the moving area ( x 1 − v gr t ) 2 / ∆ 2 k + ( x 2 2 + . . . + x 2 n ) / ∆ 2 ⊥ ≥ ( a − bm 2 ) where ∆ k , ∆ ⊥ , a are defined a bove. Lo ca lization o f the so lution for mo derate times follows from the a symptotics of the Ha nkel function. 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