On a generalization of Jacobi's elliptic functions and the Double Sine-Gordon kink chain
aa r X i v : . [ m a t h - ph ] S e p On a generalization of Jacobi’s elliptic functions and the Double Sine-Gordon kinkchain
Michael Pawellek
Institut f¨ur Theoretische Physik III,Universit¨at Erlangen-N¨urnberg,Staudtstr.7, D-91058 Erlangen, Germany ∗ Abstract
A generalization of Jacobi’s elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the specialproperties of these functions, present addition theorems and give a list of indefinite integrals. As a physical application weshow that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in termsof generalized Jacobi functions. ∗ Electronic address: [email protected] . INTRODUCTION The inversion of the Abelian integral u = Z xx d tR ( t ) p P ( t ) , (1)where R ( t ) is a rational function and P ( t ) is a polynom of degree p , is a problem, which has attracted many math-ematicians e.g. Euler, Jacobi and, of course, Abel. So for p = 2 and P ( x ) = (1 − x ) the inversion of (1) gives theperiodic trigonometric functions x = sin( u ) and x = cos( u ). For p = 4 and P ( x ) = (1 − x )(1 − k x ) (1) becomes anelliptic integral and its inversion leads to the doubly-periodic Jacobi elliptic functions x = sn( u ) , x = cn( u ), etc.For p > x ( t − t ) of a point particle ina potential V ( x ) given by a polynomial of degree greater than four needs the inversion of the integral [5]: t − t = Z xx d x ′ p E − V ( x ′ )) . (2)There are special cases for p > { s ( u ) , c ( u ) , d ( u ) , d ( u ) } , which are inversions of certain hyperellipticintegrals, where P ( x ) are polynomials of degree 6. As we will show, they can be understood as generalizations of theJacobi elliptic functions of the case p = 4. For example, the relationsn ′ ( u ) = cn( u )dn( u ) (3)will be extended to s ′ ( u ) = c ( u ) d ( u ) d ( u ) . (4)This generalization of Jacobi’s elliptic functions were recently identified as special solutions of a generalization ofLam´e’s differential equation [6] and in the following sections we will continue to discuss the mathematical propertiesof these functions.As a physical application of these functions we will show, that several periodic kink solutions of the double sine-Gordon model [7, 8, 9], which are expressible as nested combinations of text book functions are just reincarnations ofone unique generalized Jacobi function. II. DEFINITIONS
In this section we introduce the generalized Jacobi functions and clarify some of their properties, which were alreadyused in [6].
Definition II.1
Consider without loss of generality > k > k > as moduli parameter.a) The generalized Jacobi elliptic function x = s ( u, k , k ) and their companion functions c ( u, k , k ) , d ( u, k , k ) and d ( u, k , k ) are defined by the inversion of the hyperelliptic integrals u ( x, k , k ) = Z x = s ( u )0 d t p (1 − t )(1 − k t )(1 − k t ) , (5) u ( x, k , k ) = Z x = c ( u ) d t q (1 − t )( k ′ + k t )( k ′ + k t ) , (6) u ( x, k , k ) = k Z x = d ( u ) d t q (1 − t )( t − k ′ )( k − k + k t ) , (7) u ( x, k , k ) = k Z x = d ( u ) d t q (1 − t )( t − k ′ )( k − k + k t ) , (8)2 IG. 1: s ( x, k , k ) for k = 0 . respectively.b) The generalized amplitude function a ( u, k , k ) is the given by the inversion of u ( ϕ , k , k ) = Z ϕ = a ( u )0 d ψ q (1 − k sin ψ )(1 − k sin ψ ) (9) with s ( u, k , k ) = sin( a ( u, k , k )) . Without solving the integrals (5) explicitly, one can derive certain properties of these functions.
Corollar II.1
Given the generalized Jacobi elliptic functions s ( u ) , c ( u ) , d ( u ) and d ( u ) as defined by (5). Then c ( u ) = 1 − s ( u ) , d ( u ) = 1 − k s ( u ) , d ( u ) = 1 − k s ( u ) , (10) d i ( u ) − k i c ( u ) = 1 − k i , i = 1 , k d ( u ) − k d ( u ) = k − k . (11) The first derivatives of these functions are given by s ′ ( u ) = c ( u ) d ( u ) d ( u ) , c ′ ( u ) = − s ( u ) d ( u ) d ( u ) ,d ′ ( u ) = − k s ( u ) c ( u ) d ( u ) , d ′ ( u ) = − k s ( u ) c ( u ) d ( u ) , (12) Proof
By the substitutions x = p − y , x = q − k y , x = q − k y (13)in (5) one obtains the three other integrals where the relations (10) and (11) can be read off. Relation (12) followsfrom the differential versions of (5). (cid:3) The functions s ( u, k , k ) , c ( u, k , k ) , d ( u, k , k ) and d ( u, k , k ) are generalizations of the classic Jacobi ellipticfunctions sn( u, k ) , cn( u, k ) and dn( u, k ) and they reduce to them for k → k = k . For fixed k i we will us theabbreviated notations s ( u ) ≡ s ( u, k , k ), etc.So far we have only stated some formal relations between the inverted hyperelliptic integrals (5), provided thesefunction exist, which we have to show next. For this we note that the differential of the hyperelliptic integral, whichdefines s ( u ) is an Abelian differential of the first kind d η = d xy (14)3 IG. 2: s ( x, k , k ) for k = 0 . c ( x, k , k ) for k = 0 . d ( x, k , k ) for k = 0 . y = (1 − x )(1 − k x )(1 − k x ) (15)It is holomorphic on the hyperelliptic curve C defined by C = { ( y, x ) ∈ C | y = (1 − x )(1 − k x )(1 − k x ) } , (16)which can be modelled as a Riemann surface of genus 2. The important observation is that the hyperelliptic curve C is also a double cover C π −→ E of the elliptic curve E defined by E = { ( w, z ) ∈ C | w = z (1 − z )(1 − k z )(1 − k z ) } , (17)with covering map π ( y, x ) given by ( w, z ) = π ( y, x ) = ( xy, x ) . (18)The differential (14) is therefore the pullback of the elliptic differential of the first kindd η = d zw , (19)and the inversion of its integral gives a double-valued function, which can now be expressed in terms of ellipticfunctions: Theorem II.1
The generalized Jacobi elliptic functions exist and are given by s ( u, k , k ) = sn( k ′ u, κ ) q k ′ + k sn ( k ′ u, κ ) , c ( u, k , k ) = k ′ cn( k ′ u, κ ) p − k cn ( k ′ u, κ ) ,d ( u, k , k ) = p k − k dn( k ′ u, κ ) q k − k dn ( k ′ u, κ ) , d ( u, k , k ) = p k − k q k − k dn ( k ′ u, κ ) , (20) and the generalized amplitude function is a ( u, k , k ) = arctan[ k ′ − sc( k ′ u, κ )] = arctan[ k ′ − tan(am( k ′ u, κ ))] (21) with κ = ( k − k ) / (1 − k ) , k ′ = p − k and ≤ k ≤ k ≤ . They have branch-cuts along ( u , u ) and ( u , u ) with u = i cn − ( k , κ ′ ) k ′ , u = − u + 2 i K ( κ ′ ) k ′ , u = u + 2 K ( κ ) k ′ , u = u + 2 K ( κ ) k ′ , (22) where K ( k ) is the complete elliptic integral of the first kind and κ ′ = √ − κ . Proof
From the discussion above follows that by substituting t = √ τ , the hyperelliptic integral (5) can be reducedto the following elliptic integral: u ( x, k , k ) = 12 Z x d τ p τ (1 − τ )(1 − k τ )(1 − k τ ) , (23)where the inverse function is given [11] by the first expression of (20). The sign of the root in the denominator ischosen in such a way that for k → s ( u, k , k ) → sn( u, k ). The other three expressions are obtained byapplying (10). The branch points are a result of the zeros of the denominators in (20). (cid:3) Figures 1 to 4 show example plots of these functions for selected values of the moduli k and k .As the Jacobi representation (20) shows, the introduction of generalized Jacobi functions is mathematically re-dundant. Nevertheless it would be not obvious in the Jacobi representation that among these four functions suchelementary relations as (12) are fulfilled. It is therefore advantageous to use (10) to (12) when working with thesefunctions and not representation (20). With this set-up algebraic manipulations become very simply and straight-forward. 5 FIG. 5: A fundamental cell in the complex plane with ’short cuts’ and the four periods
Further, the generalized Jacobi functions serve as prototype examples of meromorphic functions on a genus twoRiemann surface. This can be seen as follows. Consider the two points u = u and u = u + 2 K ( κ ) k ′ . There exist twodifferent paths for analytic continuation to obtain the value of s ( u ) from s ( u ). Path a avoids the branch cut andpath a goes through one cut, see Figure 5. After passing the cut ( u , u ) one has to use the other branch of thesquare root. Let ( u, +) and ( u, − ) denote points lying in the two different branches of the square root. Then one gets s (( u, − )) = sn( k ′ u ) − q k ′ + k sn ( k ′ u ) = sn( k ′ u + 2 K ) q k ′ + k sn ( k ′ u + 2 K ) = s (( u + 2 K ( κ ) /k ′ , +)) , (24)where we have used the anti-periodicity of the sn-function. Thus by identifying the points ( u, − ) ∼ ( u + 2 K ( κ ) /k ′ , +)of the two branches, the path a enters the cut ( u , u ) and appears at the other cut ( u , u ). The branch cuts areshort cuts and depending on the path of analytic continuation one gets: s ( u + 4 K ( κ ) /k ′ ) a = s ( u ) , s ( u + 2 K ( κ ) /k ′ ) a = s ( u ) . (25)Thus the generalized Jacobi functions are realizations of functions with two real periods 2 K ( κ ) /k ′ and 4 K ( κ ) /k ′ ,depending on the path of analytic continuation. The identification of the nontrivial cycles as in Figure 5 makes itclear that the generalized Jacobi functions are one-valued functions on the corresponding genus two Riemann surface.The cycles b and b in Figure 5 correspond to the imaginary period 2 i K ( κ ′ ) /k ′ . One can think of this surface as atorus with an additional handle attached connecting the branch cuts. III. PROPERTIES
In this section we present addition theorems, special values and indefinite integrals of the generalized Jacobi ellipticfunctions.
A. Relation to classic Jacobi elliptic functions
From (20) we can state the following
Corollar III.1
The 12 classic Jacobi elliptic functions are given by the nontrivial quotients of the generalized Jacobifunctions, e.g. one has s ( u, k , k ) d ( u, k , k ) = k ′− sn( k ′ u, κ ) , c ( u, k , k ) d ( u, k , k ) = cn( k ′ u, κ ) , d ( u, k , k ) d ( u, k , k ) = dn( k ′ u, κ ) , (26) where the modulus of the resulting Jacobi elliptic functions is κ . For the remaining nine quotients see Table I. ABLE I: The ratios of generalized Jacobi functions with moduli k and k give the twelve Jacobi elliptic functions withmodulus κ s ( u ) c ( u ) d ( u ) d ( u ) s ( u ) 1 k ′ cs( k ′ u ) k ′ ds( k ′ u ) k ′ ns( k ′ u ) c ( u ) k ′− sc( k ′ u ) 1 dc( k ′ u ) nc( k ′ u ) d ( u ) k ′− sd( k ′ u ) cd( k ′ u ) 1 nd( k ′ u ) d ( u ) k ′− sn( k ′ u ) cn( k ′ u ) dn( k ′ u ) 1 This looks very similar to the definition of the Jacobi functions by theta functions [10]:sn( u ) = ϑ ϑ ϑ ( u/ϑ ) ϑ ( u/ϑ ) , cn( u ) = ϑ ϑ ϑ ( u/ϑ ) ϑ ( u/ϑ ) , dn( u ) = ϑ ϑ ϑ ( u/ϑ ) ϑ ( u/ϑ ) , (27)where ϑ i = ϑ i (0). More similarity with theta functions can be found, when one notice that from (12) especially followsthe identity s ′ (0) = c (0) d (0) d (0) , (28)which is also very similar to the famous theta constant identity [10] ϑ ′ (0) = ϑ (0) ϑ (0) ϑ (0) . (29)Nevertheless a similar relation as (12) does not hold for theta functions: ϑ ′ ( u ) = ϑ ( u ) ϑ ( u ) ϑ ( u ) , (30)which is a crucial difference to the generalized Jacobi functions. B. Addition theorems
Theorem III.1 (Addition theorem)
The generalized Jacobi functions with moduli k , k fulfill the following addi-tion theorems: s ( u ± v ) = s ( u ) d ( u ) c ( v ) d ( v ) ± s ( v ) d ( v ) c ( u ) d ( u ) p [ d ( u ) d ( v ) − κ k ′ s ( u ) s ( v )] + k [ s ( u ) d ( u ) c ( v ) d ( v ) ± s ( v ) d ( v ) c ( u ) d ( u )] c ( u ± v ) = c ( u ) d ( u ) c ( v ) d ( v ) ∓ k ′ s ( u ) d ( u ) s ( v ) d ( v ) p [ d ( u ) d ( v ) − κ k ′ s ( u ) s ( v )] + k [ s ( u ) d ( u ) c ( v ) d ( v ) ± s ( v ) d ( v ) c ( u ) d ( u )] d ( u ± v ) = d ( u ) d ( u ) d ( v ) d ( v ) ∓ κ k ′ s ( u ) c ( u ) s ( v ) c ( v ) p [ d ( u ) d ( v ) − κ k ′ s ( u ) s ( v )] + k [ s ( u ) d ( u ) c ( v ) d ( v ) ± s ( v ) d ( v ) c ( u ) d ( u )] d ( u ± v ) = d ( u ) d ( v ) − κ k ′ s ( u ) s ( v ) p [ d ( u ) d ( v ) − κ k ′ s ( u ) s ( v )] + k [ s ( u ) d ( u ) c ( v ) d ( v ) ± s ( v ) d ( v ) c ( u ) d ( u )] Proof
Write the addition theorem for sn( u ) with the help of (26) assn( k ′ u ± k ′ v, κ ) = k ′ s ( u ) d ( u ) c ( v ) d ( v ) ± s ( v ) d ( v ) c ( u ) d ( u ) d ( u ) d ( v ) − κ k s ( u ) s ( v ) (31)The addition theorem for d ( u ) follows then immediately by using (31) in d ( u ± v ) = k ′ k ′ + k sn ( k ′ u ± k ′ v, κ ) . (32)7ow one can use the addition theorem for d ( u ) in order to get the corresponding theorem for s ( u ) from s ( u ± v ) = k ′− sn( k ′ u ± k ′ v, κ ) d ( u ± v ) , (33)and similar for c ( u ) and d ( u ). (cid:3) A special case of the addition theorems is the following
Corollar III.2 (Half argument) s ( u/
2) = d ( u ) − c ( u ) d ( u ) − k c ( u ) + k ′ d ( u ) , (34) c ( u/
2) = k ′ c ( u ) + d ( u ) d ( u ) − k c ( u ) + k ′ d ( u ) , (35) d ( u/
2) = ( k − k ) c ( u ) + d ( u ) k d ( u ) − k d ( u ) + ( k − k ) c ( u ) , (36) d ( u/
2) = ( k − k ) c ( u ) + d ( u ) k d ( u ) − k d ( u ) + ( k − k ) c ( u ) . (37) C. Special Values
Definition III.1
The generalization of the complete elliptic integral of the first kind is K := K ( k , k ) = 1 k ′ K ( κ ) = Z d t p (1 − t )(1 − k t )(1 − k t ) . (38) Define also K ′ = K ( κ ′ ) and κ ′ = 1 − κ = − k − k . From the definition of the generalized Jacobi functions follows s ( K ) = 1 , c ( K ) = 0 , d ( K ) = k ′ , d ( K ) = k ′ . (39)In Table II we summarize analytic expressions for the generalized Jacobi functions evaluated at specific points. As anexample we will demonstrate that s ( K /
2) = (1 + k ′ k ′ ) − . For this we choose u = v = K / c ( u + v ). One gets c ( K / d ( K / − k ′ s ( K / d ( K /
2) = 0 . (40)This can be written as ( k + k k ′ ) s ( K / − s ( K /
2) + 1 = 0 , (41)with solution s ( K /
2) = ± s ± k ′ k ′ k + k ′ k ′ . (42)Considering the limit k → K /
2) = (1 + k ′ ) − , which fixes the signs such as s ( K /
2) = + s − k ′ k ′ k + k k ′ . (43)By writing the denominator as k + k k ′ = 1 − k ′ + k k ′ = 1 − k ′ k ′ = (1 + k ′ k ′ )(1 − k ′ k ′ ) , (44)the promised result s ( K /
2) = (1 + k ′ k ′ ) − is obtained.The other values in Table II can be shown in similar ways using the addition theorems appropriately.8 ABLE II: Special values for generalized Jacobi functions s ( u ) c ( u ) d ( u ) d ( u ) u = K / k ′ k ′ ) − ( k ′ k ′ ) (1 + k ′ k ′ ) − k ′ ( k ′ + k ′ ) (1 + k ′ k ′ ) − k ′ ( k ′ + k ′ ) (1 + k ′ k ′ ) − u = K k ′ k ′ u = 3 / K (1 + k ′ k ′ ) − − ( k ′ k ′ ) (1 + k ′ k ′ ) − k ′ ( k ′ + k ′ ) (1 + k ′ k ′ ) − k ′ ( k ′ + k ′ ) (1 + k ′ k ′ ) − u = i K ′ / i ( κk ′ − k ) − r κk ′ − k + k κk ′ − k √ κ (1 − k κk ′ ) − (1 − k κk ′ ) − u = i K ′ k − ik − k ′ ik − ( k − k ) u = K / i K ′ / r k + ik ′ k ′ κk − k + k k r − k k ′ − ik ′ k ′ κk − k + k k r k ′ ( k − k ) − ik k ′ k ′ κk − k + k k r k − k − ik k ′ k ′ κk − k + k k u = K / i K ′ (1 − k ′ k ′ ) − − i ( k ′ k ′ ) (1 − k ′ k ′ ) − − ik ′ ( k ′ − k ′ ) (1 − k ′ k ′ ) − − ik ′ ( k ′ − k ′ ) (1 − k ′ k ′ ) − u = K + i K ′ k − ik − k ′ k − ( k − k ) Together with the addition theorems one finds further s ( u + K ) = c ( u ) q d ( u ) − k k ′ s ( u ) , c ( u + K ) = − k ′ k ′ s ( u ) q d ( u ) − k k ′ s ( u ) ,d ( u + K ) = k ′ d ( u ) q d ( u ) − k k ′ s ( u ) , d ( u + K ) = k ′ d ( u ) q d ( u ) − k k ′ s ( u ) . (45) D. The integrals of generalized Jacobi functions
It is easy to see that the integral of d ( u ) is closely related to the incomplete elliptic integral of the third kind: Z d ud ( u ) = Z d u k k ′ sn ( k ′ u, κ ) = 1 k ′ Π (cid:18) k ′ u, − k k ′ , κ (cid:19) . (46)Using (10) we get the corresponding integrals of s ( u ) , c ( u ) and d ( u ), see Table III. IV. GENERALIZED JACOBI FUNCTIONS AS DOUBLE SINE-GORDON KINKS
We are now able to discuss the (quasi-) periodic kink solutions of the double sine-Gordon model (DSG) L = 12 ∂ µ φ∂ µ φ − V ( φ ) (47)where the potential is given by V ( φ ) = µβ cos( βφ ) − λβ cos (cid:18) β φ (cid:19) + C. (48)9 ABLE III: A integral table of generalized Jacobi elliptic functionsf(u) F(u) f(u) F(u) s ( u ) − k ′ k sn − “ k c ( u ) d ( u ) , κk ” s ( u ) d ( u ) k ln ( d ( u ) − k c ( u )) c ( u ) k sn − “ k s ( u ) , k k ” c ( u ) d ( u ) k arctan “ k s ( u ) d ( u ) ” d ( u ) sn − ( s ( u ) , k ) c ( u ) d ( u ) k arctan “ k s ( u ) d ( u ) ” d ( u ) sn − ( s ( u ) , k ) d ( u ) d ( u ) a ( u ) s ( u ) c ( u ) k k ln ( k d ( u ) − k d ( u )) d ( u ) k ′ Π “ k ′ u, − k k ′ , κ ” s ( u ) d ( u ) k ln ( d ( u ) − k c ( u )) d ( u ) “ − k k ” u + k ′ k k Π “ k ′ u, − k k ′ , κ ” s ( u ) k u − k ′ k Π “ k ′ u, − k k ′ , κ ” d ( u ) d ( u ) k ′ h E ( k ′ u, κ ) − k k k ′ u + “ k ′ + k k ” Π “ k ′ u, − k k ′ , κ ” ++ k k ′ s ( u ) c ( u ) d ( u ) d ( u ) i c ( u ) k k ′ Π “ k ′ u, − k k ′ , κ ” − k ′ k u We will choose the constant C in order to set the minima of the potential to zero, which gives C = λ − µβ , ( µ, λ > λ µ >
1) or µ < , λ > , (49) C = 1 β ( λ µ + µ ) , µ > , | λ | µ < , (50) C = − λ + µβ , µ, λ < , (51)The signs of the different terms of (48) are chosen, so that for µ → λ > V ( φ ) µ → −→ λβ (cid:18) − cos (cid:18) β φ (cid:19)(cid:19) . (52)The kinks are solutions of the first order equation of motion12 (cid:18) d φ d x (cid:19) − V ( φ ( x )) = A, (53)where A is some integration constant. This model possesses a rich phase structure depending on the parameters λ and µ [12, 13], e.g. for λ/ µ > φ = πnβ with in particular φ = 0 as minimumand φ = πβ as maximum.We will show in this section that the kink solutions and corresponding energy densities get a unique canonicalexpression in terms of generalized Jacobi functions.By shifting ¯ φ ( x ) = βφ ( x ) − π the first order equation of motion for static kink configurations (53) can uniformlybe brought to the form d ¯ φ q (1 − k sin φ )(1 − k sin φ ) = 2 √ µ d x, (54)with solution φ ( x ) = 2 πβ + 4 β a (cid:18) √ µk k ( x − x ) , k , k (cid:19) , (55)10 E cl H R L FIG. 6: Classical energy for λ = 4 and µ = 0 .
01 (solid), µ = 0 . µ = 0 .
99 (dashed) which depends implicitly on the radius R = 2 k k √ µ K ( k , k ) . (56)The corresponding energy density can be analytically expressed as E ( x, k , k , A ) = 16 µβ k k d (cid:18) √ µk k x, k , k (cid:19) d (cid:18) √ µk k x, k , k (cid:19) − A, (57)where d ( x ) and d ( x ) are the previous introduced generalized Jacobi functions. (55) and (57) are the unique solutionof the first order differential equation (54). The only thing one has to do, is to work out the explicit dependence of themoduli k , k on the parameters µ, λ, β and the integration constant A of the potential (48) in the different sectors.The solution has the following (quasi-)periodic properties, depending on the integration constant A : φ ( x + R ) = φ ( x ) + 4 πβ , A > , (58) φ ( x + 2 R ) = φ ( x ) , A < . (59)Depending on the physical situation these solutions can be used to describe kink chains on an infinite line or a kinksolution on the compact circle with circumference R . Although (55) is in principle valid for all values of k , k we willgive in addition for all cases an expression in text book functions where the elliptic modulus κ lies in the fundamentalinterval between 0 and 1. This will establish the connection with previous obtained expressions for periodic solutionsof the DSG model [7, 8, 9]. A. Case: λ, µ > and λ > µ In this region of the parameter space the potential (48) has only one type of minima. The moduli are given by k , = 1 β A + 2 λ h µ + λ ± p ( λ − µ ) − µβ A i , (60)with following properties k k = 8 µβ A + 2 λ , k + k = 8 µ + 2 λβ A + 2 λ , k + k − k k = 2 λβ A + 2 λ , k ′ k ′ = β Aβ A + 2 λ . (61)11 -
10 10 20 Φ H Φ L - -
10 10 20 x51015202530 E H R L FIG. 7: The DSG potential V ( φ ) and the kink-chain energy density E ( x ) for λ > µ < A < ( λ − µ ) / (8 µ ) In this case is 0 < k < k < φ ( x ) = 2 πβ + 4 β arctan (cid:20) k ′ − sc (cid:18) k ′ √ µk k ( x − x ) , κ (cid:19)(cid:21) , (62)depending on the radius R = 2 k k √ µ k ′ K ( κ ) . (63)This solution can be interpreted as an infinite kink chain on the line with distance R . The energy of this fieldconfiguration on S is E ( k , k ) = 16 √ µk ′ β k k " E ( κ ) − (cid:18) k k + 1 − k (cid:19) K ( κ ) + k ′ + k k ! Π − k k ′ , κ ! . (64)With (60) the radius (56) and energy (64) become functions of A, β, µ and λR = R ( A ; β, λ, µ ) , E = E ( A ; β, λ, µ ) , (65)which can for given β, λ, µ be plotted with parameter A (see Figure 5). A = 0 This is the decompactification limit since from k = 1 and k = 4 µ/λ and (56) follows R → ∞ . Then the kinksolution reduces to a single DSG kink on the infinite line φ ( x ) → πβ + 4 β arctan "s λλ − µ sinh r λ − µ ( x − x ) ! , (66)which is the solution found in [13]. The corresponding topological charge Q = φ (+ ∞ ) − φ ( −∞ ) is Q = 4 πβ (67)12 . A = ( λ − µ ) / (8 µ ) This is the trigonometric point, since the moduli are given by k = k = k = 8 µλ + 4 µ . (68)and in the kink solution all elliptic functions degenerate to trigonometric functions: φ ( x ) → πβ + 4 β arctan (cid:20) k ′ tan (cid:18) k ′ k √ µ ( x − x ) (cid:19)(cid:21) , R → π √ µ k k ′ . (69)The energy is E → π √ µβ (cid:20) − k k + 2 2 − k k ′ + 1 − k (cid:21) . (70) • µ = 0This is the sine-Gordon limitFrom (60) one can see k → φ ( x ) → πβ + 4 β am p λ/ k ( x − x ) , k ! (71)and R → k p λ/ K ( k ) (72)with mass parameter m = p λ/
4. By using the limitlim k → k k " Π − k k ′ , κ ! − K ( κ ) = E ( k ) − K ( k ) , (73)the energy becomes E ( k , k ) → p λ/ β k (cid:2) ( k − K ( k ) + 2 E ( k ) (cid:3) (74) A > ( λ − µ ) / (8 µ ) Now the moduli k and k are complex conjugated with | k | = | k | = 8 µA + 2 µ (75)The explicit kink solution can be written as φ ( x ) = 2 πβ + 4 β arctan " ( k ′ k ′ ) − / sc p k ′ k ′ √ µk k x, i ( k ′ − k ′ )2 p k ′ k ′ ! dn p k ′ k ′ √ µk k x, i ( k ′ − k ′ )2 p k ′ k ′ ! , (76)where the radius is given by R = 2 k k √ µ p k ′ k ′ K i ( k ′ − k ′ )2 p k ′ k ′ ! (77)This is again a kink chain as in Case A.1, only the mathematical representation has changed.13 . − λ < A < In this case 0 < k < < k < ∞ and the solution can be written as φ ( x ) = 2 πβ + 4 β arctan " ( k − k ) − / sd p k − k √ µk k x, κ − ! , (78)where the radius is given by R = 2 k k √ µ p k − k K ( κ − ) (79)This solution can be interpreted as an infinite chain of kinks and anti-kinks on the line. A = − λ This is the endpoint for real valued solutions in the DSG model, where the moduli become k → ∞ , k = (1 − λ µ ) − , (80)and the kink solution reduces the constant field configuration φ ( x ) = 2 πβ , (81)with constant energy density E ( x ) = 2 λβ . (82)This happens at the critical value R = 2 π √ µ + λ . (83)Thus for R < R no non-trivial real valued periodic static field configuration exist in the DSG model. B. Case: | λ | < µ and µ > The potential (48) has now two different maxima and additional minima.The kink solution is again (55) with the moduli given by k , = 1 β A + µ ( λ + 4 µ ) h µ + λ ± p − µβ A i . (84)Since the DSG potential (48) has the symmetry V ( φ, λ ) = V (cid:18) φ + 2 πβ , − λ (cid:19) (85)the second solutions are given by φ II ( x, λ, A ) = φ I ( x, − λ, A ) − πβ . (86)14 . A > The moduli are complex conjugated with | k | = | k | = 8 µβ A + µ ( λ + 4 µ ) , (87)and the kink solution is φ I ( x ) = 2 πβ + 4 β arctan " ( k ′ k ′ ) − / sc p k ′ k ′ √ µk k x, i ( k ′ − k ′ )2 p k ′ k ′ ! dn p k ′ k ′ √ µk k x, i ( k ′ − k ′ )2 p k ′ k ′ ! , (88)where the radius is given by R I = 2 k k √ µ p k ′ k ′ K i ( k ′ − k ′ )2 p k ′ k ′ ! . (89)On S this solution represents a quasi-periodic kink. On the infinite line this solution represents a chain composed oftwo different types of kinks, a large and a small one, where the large kink lies around x = 0. This can be seen on theenergy density chart. The second solution φ II ( x ) is equivalent to φ I ( x ), but now the small kink lies around x = 0. A = 0 The moduli are k , = µ µ + λ and R → ∞ . The solution I reduces for λ > φ I ( x ) = 2 πβ + 4 β arctan s µ + λ µ − λ tanh s − (cid:18) λ µ (cid:19) √ µ x , (90)and for λ < II gives for λ > λ < φ II ( x ) = 4 β arctan s µ − λ µ + λ tanh s − (cid:18) λ µ (cid:19) √ µ x (91)The corresponding topological charge is given by Q I,II = 8 β arctan "s µ ± λ µ ∓ λ (92)The obvious relation Q I > Q II for λ > − ( λ − µ ) / (8 µ ) < A < The moduli are real with 1 < k < k < ∞ . Now there are two inequivalent solutions. The first one can now bewritten as φ I ( x ) = 2 πβ + 4 β arctan " ( k − − / sn p k − √ µk k x, κ ′− ! , R = 2 k k √ µ p k − K (cid:0) κ ′− (cid:1) . (93)The second one is φ II ( x, λ, A ) = φ I ( x, − λ, A ) − πβ . (94)Solution φ I represents for λ > R and for λ < R .15 -
10 10 20 Φ H Φ L - -
20 20 40 x2468 E H x L FIG. 8: The DSG potential V ( φ ) and energy density E ( x ) of a chain of large-kinks/small-kinks for λ < µ λ < and A = − ( λ − µ ) / (8 µ ) This is the endpoint of the kink/anti-kink chain of the small type. For the moduli we have k → ∞ , k = 4 µλ + 4 µ , (95)and the kink reduces to the constant field configuration φ ( x ) = 2 πβ , (96)with energy density E ( x ) = 2 µβ (cid:18) λ µ (cid:19) . (97)This happens at the critical value R = 2 π p −| λ | + 4 µ (98) λ > and − µ (1 + λ/ (4 µ )) < A < − ( λ − µ ) / (8 µ ) < In this case 0 < k < < k < ∞ and the solution can be written as φ ( x ) = 2 πβ + 4 β arctan " ( k − k ) − / sd p k − k √ µk k x, κ − ! , R = 2 k k √ µ p k − k K ( κ − ) . (99)This is the kink/anti-kink chain of the large type. λ > and A = − µ (1 + λ/ (4 µ )) This is the endpoint of the kink/anti-kink chain of the large type. For the moduli we have k → ∞ , k = 4 µλ + 4 µ , (100)and the kink reduces to the constant field configuration φ ( x ) = 2 πβ (101)at the critical radius R given by (83). 16 -
10 10 20 Φ H Φ L - -
20 20 40 x - E H x L FIG. 9: The DSG potential V ( φ ) and energy density E ( x ) of a chain of kink-kink molecules for λ, µ < C. Case: λ, µ < and A > The potential (48) has now two different minima. The kink is again (55) with the following moduli: k , = 1 β A h µ + λ ± p ( λ − µ ) + 8 µ (2 λ − β A ) i , (102)with k < − < < k < . (103)Therefore an explicit representation of the kink in terms of text book functions is φ ( x ) = 2 πβ + 4 β arctan " k ′ − sc k ′ √− µk p − k ( x − x ) , κ ! , R = 2 k p − k k ′ √− µ K ( κ ) . (104)On the infinite line one can interpret this as a chain of two small kinks bounded in a kind of molecule, see Figure 9. D. Case: µ < and λ > The moduli are the same as for case A: k , = 1 β A + 2 λ h µ + λ ± p ( λ − µ ) − µβ A i , (105) A > The moduli are k < < k <
1. The kink solution is φ ( x ) = 2 πβ + 4 β arctan " k ′ − sc k ′ √− µk p − k ( x − x ) , κ ! , R = 2 k p − k √− µ k ′ K ( κ ) . (106) λ > | µ | and − λ < A < The moduli are k < < < k . The kink solution is φ ( x ) = 2 πβ + 4 β arctan " ( k − k ) − sd p k − k k p − k √− µ ( x − x ) , κ − ! , R = 2 k p − k √− µ p k − k K ( κ − ) . (107)This is a periodic bounce solution. 17 . CONCLUSION We introduced a generalization of Jacobi elliptic functions defined by the inversion of certain hyperelliptic integralswhich are reducible to elliptic integrals.As an example for their effectiveness in physics we have chosen the double sine-Gordon model. Its (quasi-)periodickink solution and corresponding energy densities can be described uniformly by a single generalized Jacobi function.The qualitative characteristics of the kink chains depend only on the moduli parameter k and k . Several solutionsof the DSG model obtained in the past [7, 8, 9] are just special cases of a unique generalized Jacobi function. Weobserved also a critial value R for kink/anti-kink chains, where for R < R no non-trivial static solution exists. [1] C.G.J. Jacobi, De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendetium Abelianaruminnititur, J. f. Math. (1834) 55-78[2] B. von Ludwig, Von der Umkehrung eines einzelnen Abelschen Integrals, J. f. Math (1926) 26-36[3] W. Gr¨obner, ¨Uber das Umkehrproblem der Abelschen Integrale, Math. Z. (1961) 101-105[4] H. F. Baker, Abelian Functions (Cambridge: University Press 1897)[5] Y. N. Fedorov and D. Gomez-Ullate, Dynamical systems on infinitely sheeted Riemann surfaces, Physica D (2007)120-134[6] M. Pawellek, Quasi-doubly periodic solutions to a generalized Lam´e equation, J. Phys. A (2007) 7673-7686[7] S. Iwabuchi, Commensurate-Incommensurate Phase Transition in Double Sine-Gordon System, Prog. Theo. Phys. (1983) 941-953[8] O. Hudak, Double sine-Gordon equation: A stable 2 pi -kink and commesurate-incommensurate phase transitions, Phys.Lett. A (1981) 95-96[9] M. Wang and X. Li, Exact solutions to the double sine-Gordon equation, Chaos, Solitons and Fractals (2006) 477-486[10] Whittaker and Watson, Modern Analysis (Cambridge: Cambridge University Press 1905)[11] P. F. and M. D. Friedman 1954
Handbook of elliptic integrals for engineers and physicists (Berlin: Springer)[12] C. A. Condat, R. A. Guyer and M. D. Miller, Double sine-Gordon chain, Phys. Rev.
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