A novel approach for solving the three-dimensional sine-Gordon equation

A no v el approac h for solving the three-dimensional sine-Gordon equation Sergey G Artyshev Department of Applied Mathemat i cs, Natio nal Research Nuclea r Universit y MEPhI 31 Kashirskoe Shosse, 1154 0 9, Mosco w, Russian F ederati on E-mai l : SGArtyshev@mephi.ru Abstract A new wa y for finding analytical solutions of the three-dimensional sine-Gordon equation is p resen ted. The metho d is based on the es- tablished relation b et w een the solutions of the three-dimensional w a v e equation and solutions of the three-dimensional sine-Gordon equation. Some examples of the solutions th us obtained are sho w. The sine-Gordon equation has b een used to de scrib e with a go o d ap- pro ximation a n um b er of ph ysic al phenomena [1, 2, 3, 4]. The applications can additionally b e extended under the condition: dimension of the equation greater t ha n 1. Ho w ev er the metho ds already kno wn for s olving of the one- dimensional sine-Gordon equation cannot b e used for finding a solutions of the n-dimensional sine-Go r do n equation ( n > 1). First of all, w e s how how a solution of the w av e equation ∂ 2 F ∂ t 2 − ∂ 2 F ∂ x 2 − ∂ 2 F ∂ y 2 − ∂ 2 F ∂ z 2 = 0 F = F ( x, y , z , t ) (1) can b e used for finding a solution of t he three-dimensional sine-Gordon equa- tion ∂ 2 u ∂ t 2 − ∂ 2 u ∂ x 2 − ∂ 2 u ∂ y 2 − ∂ 2 u ∂ z 2 + s in u = 0 u = u ( x, y , z , t ) . (2) 1 The substitution [5] u = 4 tan − 1 σ σ = σ ( x, y , z , t ) (3) leads to the follow ing n onlinear partial differential equation for σ : (1 + σ 2 )[ ∂ 2 σ ∂ t 2 − ∂ 2 σ ∂ x 2 − ∂ 2 σ ∂ y 2 − ∂ 2 σ ∂ z 2 ] − 2 σ [( ∂ σ ∂ t ) 2 − ( ∂ σ ∂ x ) 2 − ( ∂ σ ∂ y ) 2 − ( ∂ σ ∂ z ) 2 ] = σ 3 − σ. (4) One p ossibilit y [5 ] for splitting equation (4) in to tw o is the following: ∂ 2 σ ∂ t 2 − ∂ 2 σ ∂ x 2 − ∂ 2 σ ∂ y 2 − ∂ 2 σ ∂ z 2 = − σ (5) ( ∂ σ ∂ t ) 2 − ( ∂ σ ∂ x ) 2 − ( ∂ σ ∂ y ) 2 − ( ∂ σ ∂ z ) 2 = − σ 2 . (6) Prop osition L et F satisfies the wa v e e quation (1) and an e quation ( ∂ F ∂ t ) 2 − ( ∂ F ∂ x ) 2 − ( ∂ F ∂ y ) 2 − ( ∂ F ∂ z ) 2 = − 1; (7) then σ = e F (8) is a solution of the system (5) , (6) . Pro of. Com bining ( ∂ F ∂ t ) 2 − ( ∂ F ∂ x ) 2 − ( ∂ F ∂ y ) 2 − ( ∂ F ∂ z ) 2 = [( ∂ σ ∂ t ) 2 − ( ∂ σ ∂ x ) 2 − ( ∂ σ ∂ y ) 2 − ( ∂ σ ∂ z ) 2 ] /σ 2 and ∂ 2 F ∂ t 2 − ∂ 2 F ∂ x 2 − ∂ 2 F ∂ y 2 − ∂ 2 F ∂ z 2 = [ ∂ 2 σ ∂ t 2 − ∂ 2 σ ∂ x 2 − ∂ 2 σ ∂ y 2 − ∂ 2 σ ∂ z 2 ] /σ − ( ∂ F ∂ t ) 2 +( ∂ F ∂ x ) 2 +( ∂ F ∂ y ) 2 +( ∂ F ∂ z ) 2 w e get the prop osition. Consequen tly , if F ( x, y , z , t ) is a solution of system (1),(7); then the func- tion u ( x, y , z , t ) is a solution of (2). This are illustrated by following exam- ples. Example 1. Let F ( x, y , z , t ) = t sinh ψ + ( x cos α + y cos β + z cos γ ) cosh ψ + C , 2 where (cos α, cos β , cos γ ) is unit v ector, a nd ψ and C are constan ts. T aking (8) in to a ccount, solution (3) b ecomes u ( x, y , z , t ) = 4 t a n − 1 [ e t sinh ψ +( x cos α + y cos β + z cos γ ) cosh ψ + C ] . W e note tha t this f o rm ula denote a s three-dimensional v ersion w ell-kno w top ological soliton o f o ne- dimensional sine-Gordon equation[1, 2, 3, 4]. Example 2. Let F ( x, y , z , t ) = h ( x ± t ) + y cos α + z sin α, where h is an arbitrary smoo th function and α is constan t; then (3) is equiv- alen t to u ( x, y , z , t ) = 4 tan − 1 [ e h ( x ± t )+ y cos α + z sin α ] . W e assume that h = ln( f ) and α = 0 or α = π , then u ( x, y , t ) = 4 ta n − 1 [ f ( x ± t ) e ± y ] is a solution of the tw o - dimensional sine-Go rdon equation (see[5]). Example 3. The previous example may b e generalized. Supp ose a func- tion F ( x, y , z , t ) has the lo ok: F ( x, y , z , t ) = h ( x cos α 1 + y cos β 1 + z cos γ 1 ± t ) + x cos α 2 + y cos β 2 + z cos γ 2 , where (cos α 1 , cos β 1 , cos γ 1 ) and (cos α 2 , cos β 2 , cos γ 2 ) are m utually or t hgonal unit v ectors, and h is an arbitrary smo oth function. Using (3 ) w e get new solution u ( x, y , z , t ) = 4 t a n − 1 [ e h ( x cos α 1 + y cos β 1 + z cos γ 1 ± t )+ x cos α 2 + y cos β 2 + z cos γ 2 ] . Similarly , if h = ln( f ), o ne obtains the solution u ( x, y , z , t ) = 4 t a n − 1 [ f ( x cos α 1 + y cos β 1 + z cos γ 1 ± t ) e x cos α 2 + y cos β 2 + z cos γ 2 ] . Example 4. No w supp ose the function F ( x, y , z , t ) has the decomp osition in to the sum: F ( x, y , z , t ) = k X i =1 F i ( x, y , z , t ) . F or example, when k = 2 , F ( x, y , z , t ) = h 1 ( x cos α 1 + y cos β 1 + z cos γ 1 ± t )+( x cos α 2 + y cos β 2 + z cos γ 2 ) / √ 2 3 + h 2 ( x cos α 1 + y cos β 1 + z cos γ 1 ± t ) + ( x cos α 3 + y cos β 3 + z cos γ 3 ) / √ 2 , where h 1 , h 2 are arbitrar y smoo t h functions; (cos α 1 , cos β 1 , cos γ 1 ), (cos α 2 , cos β 2 , cos γ 2 ) and (cos α 3 , cos β 3 , cos γ 3 ) are m utually orthgonal unit v ectors. Substituting t his expression in to (8) and the result in to (3), we get new solutions of the three-dimensional sine-Gordon equation. The relations (1) and (7) can easily b e c hec k ed b y a direct calculation. References [1] Skyrme T A 1958 Pr o c.R oyal So c. A247 2 6 0–278 [2] Josephson BD 1974 R ev.Mo d.Phys. B46 2 51-254 [3] Kudry asho v NA 2008 Metho ds of Nonline ar Mathematic al Physics (Mosco w:MEPhI) [4] Shohet JL, Barmish BR, Ebraheem HK and Scott A C 20 0 4 Physics of Plasmas 11 3877-87 [5] Ouroushev D, Martinov N and Grigo r ov A 1 991 J.Phys.A:Math.Gen. 24 L527 4