A Riemann theta function formula with its application to double periodic wave solutions of nonlinear equations

A Riemann theta function form ula with its application to dou- ble p erio dic w a v e solutions of nonlinear equations Engui F an a 1 and Kw ok Wing Chow b a. Sc ho ol of Mathematica l Sciences and Key La bora tory of Mathematics for Nonlinear Science, F udan Univ ersity , Shanghai, 2 00433, P .R. China b. Department of Mec hanical Eng ineering, Univ ersity of Hong K ong, Pokfulam, Hong Kong Abstract. Based on a Riemann theta function and Hirota’s bilinear form, a lucid and straightforward wa y is presented to explicitly construct double p erio dic wav e solutions for b oth nonlinear differ e n tial and difference equations . Once such a equation is written in a bilinear form, its p erio dic wa ve solutions can b e directly obtained by using an unified theta function formula. The r elations b etw een the per iodic wa v e solutions a nd so liton solutions a re rig orously e s tablished. The efficiency of our prop osed metho d ca n be demons trated on a class v ariety of non- linear e quations such as those considered in this paper , shall water wa ve equation, (2+1)-dimensiona l Bo goy avlenskii-Schiff eq uation and differen tial-difference KdV equation. Keyw ords: Nonlinear equations; Hirota’s bilinea r metho d; Riemann theta function; do uble p erio dic wa ve solutions; soliton so lutions. P ACS num b ers: 11 . 30 . Pb; 05. 45. Yv; 02. 30. Gp; 45. 10 . -b. 1. In t ro duction The bilinear deriv ativ e metho d d ev elop ed b y Hirota is a p o werful and direct appr oac h to constru ct exact solution of nonlinear equations. Once a non lin ear equation is wr itte n in bilinear forms b y a dep endent v ariable transformation, then multi -soliton solutions are usually obtained [1]-[6]. It wa s based on Hirota forms that Nak amura prop osed a con v en ient w a y to constru ct a kind of quasi-p erio dic solutions of nonlinear equations [7, 8 ], where the p erio dic wa ve solutions of the KdV equation and the Boussinesq equatio n were obtained. Suc h a metho d indeed exhib its some ad v an tages. F or example, it do es not need an y Lax pairs and Riemann surface for the co nsidered equation, allo ws the explicit construction of m ulti-p erio dic w a v e solutions, only relies on the existence of the Hirota’s bilinear form, as well 1 E-mail address: faneg@fudan.edu.cn 1 2 as all parameters app earing in Riemann matrix are arbitrary . Recent ly , fu rther d ev elopment w as made to inv estigate the discrete T o da lattice, (2+1)-dimensional Kadomtsev- P etviash vili equation and Bogo ya vlenskii’s b r eaking soliton equation [9]-[14]. Ho w ev er, w here rep etitiv e recursion and computation m ust b e preformed for eac h equation [7]-[14]. The m otiv ati on of th is pap er is to considerably im p ro ves the key steps of the ab o v e existing metho ds. T o ac hiev e this aim, w e d evise a theta f unction b ilinear formula, which actually pr o vides us a lu cid and straightforw ard w a y for app lying in a class of nonlinear equations. O nce a nonlinear equation is written in bilinear forms, then the double p erio dic w a v e solutions of the nonlin ear equation can b e obtained directly b y using th e formula. Moreo ver, w e prop ose a simple and effectiv e metho d to analyze asymp totic prop erties of the p erio dic solutions. As illustrativ e examples, we shall construct d ou b le p erio dic w a v e solutions to the shall w ater wa ve equation, (2+1)-dimensional Bogo ya vlenskii-Sc hiff equation and differentia l-difference K d V equation. The organization of this pap er is as follo w s. In section 2, we b riefly introd uce a Hirota bilinear op erator and a Riemann theta fu nction. In particular, we pr ovide a ke y form ula for constru cting d ouble p erio dic w a v e solutions for b oth d ifferen tial and difference equa- tions. As applications of our metho d, in sections 3-5, we constru ct double p eriod ic wa v e solutions to the shall w ater wa ve equation, (2+1)-dimensional Bogo ya vlenskii-Sc hiff equation and d ifferential- difference KdV equation, resp ectiv ely . In addition, it is rigorously sho wn that the doub le p erio dic wa v e solutions tend to the soliton solutions un der sm all amp litude limits. 2. Hirota bilinear op erator and Riemann theta function T o fi x the n otat ions w e recall br iefly some notions th at will b e used in this pap er. Th e Hirota bilinear op erators D x , D t and D n are defined as follo ws: D m x D k t f ( x, t ) · g ( x, t ) = ( ∂ x − ∂ x ′ ) m ( ∂ t − ∂ t ′ ) k f ( x, t ) g ( x ′ , t ′ ) | x ′ = x,t ′ = t e δD n f ( n ) · g ( n ) = e δ ( ∂ n − ∂ ′ n ) f ( n ) g ( n ′ ) | n ′ = n = f ( n + δ ) g ( n − δ ) , cosh( δ D n ) f ( n ) · g ( n ) = 1 2 ( e δD n + e − δD n ) f ( n ) · g ( n ) , sinh( δ D n ) f ( n ) · g ( n ) = 1 2 ( e δD n − e − δD n ) f ( n ) · g ( n ) . 3 Prop osition 1. The Hirota bilinear op erators D x , D t and D n ha v e pr op erties [1]-[6] D m x D k t e ξ 1 · e ξ 2 = ( α 1 − α 2 ) m ( ω 1 − ω 2 ) k e ξ 1 + ξ 2 , e δD n e ξ 1 · e ξ 2 = e δ ( ν 1 − ν 2 ) e ξ 1 + ξ 2 , cosh( δ D n ) e ξ 1 · e ξ 2 = cosh[ δ ( ν 1 − ν 2 )] e ξ 1 + ξ 2 , sinh( δ D n ) e ξ 1 · e ξ 2 = sinh[ δ ( ν 1 − ν 2 )] e ξ 1 + ξ 2 , where ξ j = α j x + ω j t + ν j n + σ j , and α j , ω j , ν j , σ j , j = 1 , 2 are p arameters and n ∈ Z is a discrete v ariable. More generally , we ha ve F ( D x , D t , D n ) e ξ 1 · e ξ 2 = F ( α 1 − α 2 , ω 1 − ω 2 , exp[ δ ( ν 1 − ν 2 )]) e ξ 1 + ξ 2 , (2 . 1) where F ( D x , D t , D n ) is a p olynomial ab out op erators D x , D t and D n . This prop erties are useful in deriving Hirota’s bilinear form and constructing p erio dic wa v e solutions of nonlinear equations. In th e follo wing, we introduce a general Riemann theta function and discuss its p eriod ic- it y , wh ic h pla ys a central role in the construction of p erio dic solutions of n on lin ear equations. The Riemann theta fu nction r eads ϑ  ε s  ( ξ , τ ) = X m ∈ Z exp { 2 π i ( ξ + ε )( m + s ) − π τ ( m + s ) 2 } . (2 . 2) Here the in teger v alue m ∈ Z , complex parameter s, ε ∈ C , and complex phase v ariables ξ ∈ C ; The τ > 0 whic h is called the p erio d matrix of the Riemann th eta f u nction. In the definition of th e theta function (2.2), for the case s = ε = 0, hereafter we us e ϑ ( ξ , τ ) = ϑ  0 0  ( ξ , τ ) for simplicit y . Moreo ver, we ha v e ϑ  ε 0  ( ξ , τ ) = ϑ ( ξ + ε, τ ). Definition 1. A fu n ction g ( t ) on C is said to b e quasi-p erio dic in t with fu ndamen tal p erio ds T 1 , · · · , T k ∈ C , if T 1 , · · · , T k are linearly dep endent o v er Z and there exists a fun ctio n G ( y 1 , · · · , y k ), such that G ( y 1 , · · · , y j + T j , · · · , y k ) = G ( y 1 , · · · , y j , · · · , y k ) , for all y j ∈ C , j = 1 , · · · , k . G ( t, · · · , t, · · · , t ) = g ( t ) . In p articular, g ( t ) is called double p erio dic as k = 2, and it b ecome s p erio dic w ith T if and only if T j = m j T , j = 1 , · · · , k .  Let’s first see the p erio dicit y of the theta fu n ction ϑ ( ξ , τ ). Prop osition 2. [15] The theta function ϑ ( ξ , τ ) h as th e p erio dic prop erties ϑ ( ξ + 1 + iτ , τ ) = exp( − 2 π iξ + π τ ) ϑ ( ξ , τ ) . (2 . 3) 4 W e regard the vect ors 1 and iτ j as p erio ds of the theta function ϑ ( ξ , τ ) w ith m ultipliers 1 and exp( − 2 π iξ + π τ ), resp ectiv ely . Here, iτ is not a p erio d of theta fu nction ϑ ( ξ , τ ), but it is the p erio d of the fun ctions ∂ 2 ξ ln ϑ ( ξ , τ ), ∂ ξ ln[ ϑ ( ξ + e, τ ) /ϑ ( ξ + h, τ )] and ϑ ( ξ + e, τ ) ϑ ( ξ − e, τ ) /ϑ ( ξ + h, τ ) 2 . Prop osition 3. The meromorph ic fu nctions f ( ξ ) on C are as follo w ( i ) f ( ξ ) = ∂ 2 ξ ln ϑ ( ξ , τ ) , ξ ∈ C , ( ii ) f ( ξ ) = ∂ ξ ln ϑ ( ξ + e, τ ) ϑ ( ξ + h, τ ) , ξ , e, h ∈ C . ( ii ) f ( ξ ) = ϑ ( ξ + e, τ ) ϑ ( ξ − e, τ ) ϑ ( ξ , τ ) 2 , ξ , e, h ∈ C . then in all three cases (i)–(iii), it h olds that f ( ξ + 1 + iτ ) = f ( ξ ) , ξ ∈ C , (2 . 4) that is, f ( ξ ) is a double p erio dic function with 1 and iτ . Pr o of. By using (2.3), it is easy to see th at ∂ ξ ϑ ( ξ + 1 + iτ , τ ) ϑ ( ξ + 1 + iτ , τ ) = − 2 π i + ∂ ξ ϑ ( ξ , τ ) ϑ ( ξ , τ ) , or equiv alen tly ∂ ξ ln ϑ ( ξ + 1 + iτ , τ ) = − 2 π i + ∂ ξ ln ϑ ( ξ , τ ) . (2 . 5) Differen tiating (2.5) with r esp ectiv e to ξ again imm ed iate ly pro v es th e formula (2.4) for the case (i). The formula (2.4) can b e prov ed for the cases (ii) and (iii) in a similar manner.  Theorem 1. Supp ose th at ϑ  ε ′ 0  ( ξ , τ ) and ϑ  ε 0  ( ξ , τ ) are t w o R iemann theta f u nctions, in which ξ = αx + ω t + ν n + σ . Then Hirota bilinear op erators D x , D t and D n exhibit the follo win g p erfect prop erties when they act on a pair of th eta f unctions D x ϑ  ε ′ 0  ( ξ , τ ) · ϑ  ε 0  ( ξ , τ ) =   X µ =0 , 1 ∂ x ϑ  ε ′ − ε − µ/ 2  (2 ξ , 2 τ ) | ξ =0   ϑ  ε ′ + ε µ/ 2  (2 ξ , 2 τ ) , (2 . 6) exp( δ D n ) ϑ  ε ′ 0  ( ξ , τ ) · ϑ  ε 0  ( ξ , τ ) =   X µ =0 , 1 exp( δ D n ) ϑ  ε ′ − ε − µ/ 2  (2 ξ , 2 τ ) | ξ =0   ϑ  ε ′ + ε µ/ 2  (2 ξ , 2 τ ) , (2 . 7) where th e n ota tion P µ =0 , 1 represent s t w o d ifferen t transf orm atio ns corr esp onding to µ = 0 , 1. The bilinear formula for t is the same as (2.6) by rep lac ing ∂ x with ∂ t . 5 In general, for a p olynomial op erator F ( D x , D t , D n ) with resp ect to D x , D t and D n , we ha v e the follo wing useful form ula F ( D x , D t , D n ) ϑ  ε ′ 0  ( ξ , τ ) · ϑ  ε 0  ( ξ , τ ) = " X µ C ( ε ′ , ε, µ ) # ϑ  ε ′ + ε µ/ 2  (2 ξ , 2 τ ) , (2 . 8) in whic h, explicitly C ( ε, ε ′ , µ ) = X m ∈ Z N F ( M ) exp  − 2 π τ ( m − µ/ 2) 2 − 2 π i ( m − µ/ 2)( ε ′ − ε )  . (2 . 9) where we denote vec tor M = (4 π i ( m − µ/ 2) α, 4 π i ( m − µ/ 2) ω , exp[4 π i ( m − µ/ 2) δ ν ]) . Pr o of. Making use of Prop osition 1, w e obtain the relation D x ϑ  ε ′ 0  ( ξ , τ ) · ϑ  ε 0  ( ξ , τ ) = X m ′ ,m ∈ Z D x exp { 2 π im ′ ( ξ + ε ′ ) − π m ′ 2 τ } · exp { 2 π im ( ξ + ε ) − π m 2 τ } , = X m ′ ,m ∈ Z 2 π iα ( m ′ − m ) exp  2 π i ( m ′ + m ) ξ − 2 π i ( m ′ ε ′ + mε ) − π τ [ m ′ 2 + m 2 ]  By shifting su m ind ex as m = l ′ − m ′ , then ∆ = X l ′ ,m ′ ∈ Z 2 π iα (2 m ′ − l ′ ) exp  2 π il ′ ξ − 2 π i [ m ′ ε ′ + ( l ′ − m ′ ) ε ] − π τ [ m ′ 2 + ( l ′ − m ′ ) 2 ]  l ′ =2 l + µ = X µ =0 , 1 X l,m ′ ∈ Z 2 π iα (2 m ′ − 2 l − µ ) exp { 4 π iξ ( l + µ/ 2) − 2 π i [ m ′ ε ′ − ( m − 2 l − µ ) ε ] − π [ m ′ 2 + ( m ′ − 2 l − µ ) 2 ] τ } Finally letting m ′ = k + l , we conclude that ∆ = X µ =0 , 1 " X k ∈ Z 4 π iα [ k − µ/ 2] exp {− 2 π i ( k − µ/ 2)( ε ′ − ε ) − 2 πτ ( k − µ/ 2) 2 } # × " X l ∈ Z exp { 2 π i ( l + µ/ 2)(2 ξ + ε ′ + ε ) − 2 π τ ( l + µ/ 2) 2 # = " X µ =0 , 1 ∂ x ϑ  ε ′ − ε − µ/ 2  (2 ξ , 2 τ ) | ξ =0 # ϑ  ε ′ + ε µ/ 2  (2 ξ , 2 τ ) , b y using the follo win g relations k + l = ( k − µ/ 2) + ( l + µ/ 2) , k − l − µ = ( k − µ/ 2) − ( l + µ/ 2) . In a similar wa y , we can pro v e the formula (2.7). The formula (2.8) follo ws from (2.6) and (2.7).  Remark 1. The form ulae (2.8) and (2.9) show that if the follo wing equations are satisfied C ( ε, ε ′ , µ ) = 0 , (2 . 10) 6 for µ = 0 , 1, then ϑ  ε ′ 0  ( ξ , τ ) and ϑ  ε 0  ( ξ , τ ) are p erio dic wa v e solutions of the b ilinear equation F ( D x , D t , D n ) ϑ  ε ′ 0  ( ξ , τ ) · ϑ  ε 0  ( ξ , τ ) = 0 . The formula (2.10 ) con tains tw o equations wh ic h are called constrain t equations. Th is form ula actually p ro vid es us an unified app roac h to construct double p erio dic wa ve solutions for b oth differen tial and d ifference equ ations. Once a equation is written bilinear forms, then its p erio dic wa ve solutions can b e d ir ectl y obtained b y solving system (2.10). Theorem 2. Let C ( ε, ε ′ , µ ) and F ( D x , D t , D n ) b e giv en in Theorem 1, and mak e a c hoice such that ε ′ − ε = ± 1 / 2. Th en (i) If F ( D x , D t , D n ) is an ev en f unction in the form F ( − D x , − D t , − D n ) = F ( D x , D t , D n ) , then C ( ε, ε ′ , µ ) v anish es automatically for the case µ = 1, namely C ( ε, ε ′ , µ ) = 0 , for µ = 1 . (2 . 11) (ii) If F ( D x , D t , D n ) is an o dd fu nction in the form F ( − D x , − D t , − D n ) = − F ( D x , D t , D n ) , then C ( ε, ε ′ , µ ) v anish es automatically for the case µ = 0, namely C ( ε, ε ′ , µ ) = 0 , for µ = 0 . (2 . 12) Pr o of. W e are going to consider the case where F ( D x , D t , D n ) is an even fun ctio n and pro v e the formula (2.11 ). T he form u la (2. 12) is analogous. Making transf ormatio n m = − ¯ m + µ , and noting F ( D x , D t , D n ) is ev en, we then deduce that C ( ε, ε ′ , µ ) = X ¯ m ∈ Z F ( − M ) exp  − 2 π τ ( ¯ m − µ/ 2) 2 + 2 π i ( ¯ m − µ/ 2)( ε ′ − ε )  = C ( ε, ε ′ , µ ) exp  4 π i ( ¯ m − µ/ 2)( ε ′ − ε )  = C ( ε, ε ′ , µ ) exp ( ± 2 π i ¯ m ) exp ( ± π iµ ) = − C ( ε, ε ′ , µ ) , whic h prov es th e formula (2.11).  Corollary 1. Let ε ′ j − ε j = ± 1 / 2 , j = 1 , · · · , N . Assume F ( D x , D t , D n ) is a linear com b ination of ev en and o dd functions F ( D x , D t , D n ) = F 1 ( D x , D t , D n ) + F 2 ( D x , D t , D n ) , 7 where F 1 ( D x , D t , D n ) is even and F 2 ( D x , D t , D n ) is o dd. In addition, C ( ε, ε ′ , µ ) corresp ond- ing (2.9) is give n b y C ( ε, ε ′ , µ ) = C 1 ( ε, ε ′ , µ ) + C 2 ( ε, ε ′ , µ ) , where C 1 ( ε, ε ′ , µ ) = X m ∈ Z N F 1 ( M ) exp  − 2 π τ ( m − µ/ 2) 2 − 2 π i ( m − µ/ 2)( ε ′ − ε )  , C 2 ( ε, ε ′ , µ ) = X m ∈ Z N F 2 ( M ) exp  − 2 π τ ( m − µ/ 2) 2 − 2 π i ( m − µ/ 2)( ε ′ − ε )  . Then C ( ε, ε ′ , µ ) = C 2 ( ε, ε ′ , µ ) for µ = 1 , (2 . 13) C ( ε, ε ′ , µ ) = C 1 ( ε, ε ′ , µ ) , for µ = 0 . (2 . 14) Pr o of. In a similar to th e pro of of Theorem 2, shifting sum index as m = − ¯ m + µ , and using F 1 ( D x , D t , D n ) ev en and F 2 ( D x , D t , D n ) o dd, w e hav e C ( ε, ε ′ , µ ) = C 1 ( ε, ε ′ , µ ) + C 2 ( ε, ε ′ , µ ) =  C 1 ( ε, ε ′ , µ ) − C 2 ( ε, ε ′ , µ )  exp ( ± π iµ ) . (2 . 15) Then for µ = 1, the equation (2.15) giv es C 1 ( ε, ε ′ , µ ) = 0 , whic h implies the f orm u la (2.13 ). The form ula (2.14) is is analogous.  The theorem 2 and corollary 1 are very useful to deal with coupled Hirota’s bilinear equations, which will b e seen in the follo wing section 4. 3. The shall w ater wa ve equation The shall wa ter w a ve equation tak es the form [16] u t − u xxt − 3 uu t + 3 u x Z ∞ x u t dx + u x = 0 , (3 . 1) whic h is lik e to the KdV equation in the family of shall wate r w a v e equations. Hirota and Satsuma ob tained soliton solutions of the equation b y means of bilinear metho d [17]. Here w e constru ct its a double p eriod ic w a v e solution and sho w that the on e-soliton solution can b e obtained as limiting case of the doub le p erio dic solution. T o ap p ly the Hirota b ilinear m ethod for constructing d ouble p erio dic wa ve solutions of the equation (3.1), we consider a v ariable transformation u = 2 ∂ 2 x ln f ( x, t ) . (3 . 2) 8 Substituting (3.2) int o (3.1) and in tegrating with r esp ect to x , we then get the follo w ing Hirota’s bilinear form F ( D x , D t ) f · f = ( D x D t + D 2 x − D t D 3 x + c ) f · f = 0 , (3 . 3) where c is an int egration constan t. In the sp ecia l case of c = 0, starting from the bilinear equation (3.3), it is easy to fin d its one-soliton solution u 1 = 2 ∂ 2 x ln(1 + e η ) , (3 . 4) with phase v ariable η = px + p p 2 − 1 t + γ for every p and γ . Next, we turn to s ee the p erio dicit y of the solution (3.2), the fun ctio n f is c hosen to b e a Riemann theta fu nction, n amely , f ( x, t ) = ϑ ( ξ , τ ) , (3 . 5) where phase v ariable ξ = αx + ω t + σ. With Prop osition 3, we refer to u = 2 ∂ 2 x ln ϑ ( ξ , τ ) = 2 α 2 ∂ 2 ξ ln ϑ ( ξ , τ ) , (3 . 6) whic h shows that the solution u is a d ouble p erio dic f u nction with t w o fun damen tal p erio ds 1 and iτ . W e introd uce the notations by λ = e − π τ / 2 , ϑ 1 ( ξ , λ ) = ϑ (2 ξ , 2 τ ) = X m ∈ Z λ 4 m 2 exp(4 iπ mξ ) , ϑ 2 ( ξ , λ ) = ϑ  0 − 1 / 2  (2 ξ , 2 τ ) = X m ∈ Z λ (2 m − 1) 2 exp[2 iπ (2 m − 1) ξ ] , (3 . 7) where the phase v ariable ξ = αx + ω t + σ . Substituting (3.5) into (3.3), using formula (2.10) and (3.7) leads to a linear system ( corresp onding to µ = 0 and µ = 1, r esp ecti v ely) [ ϑ ′′ 1 (0 , λ ) α + ϑ (4) 1 (0 , λ ) α 4 ] ω + ϑ 1 (0 , λ ) c + ϑ ′′ 1 (0 , λ ) α 2 = 0 , [ ϑ ′′ 2 (0 , λ ) α + ϑ (4) 2 (0 , λ ) α 4 ] ω + ϑ 2 (0 , λ ) c + ϑ ′′ 2 (0 , λ ) α 2 = 0 , (3 . 8) where we ha v e denoted the deriv ativ e of ϑ j ( ξ , λ ) at ξ = 0 b y n otations ϑ ( k ) j (0 , λ ) = d k ϑ j ( ξ , λ ) dξ k | ξ =0 , j = 1 , 2; k = 1 , 2 , 3 , 4 . This system admits an explicit solution ( ω , c ). In this wa y , w e obtain an explicit p eriod ic w a v e solution (3.6) with parameters ω , c b y (3.8), w h ile other parameters α, σ, τ , σ are fr ee. In summ ary , double p erio dic w a v e (3.6) p ossesses the follo w ing features: (i) It is is one- dimensional, i.e. there is a single phase v ariable ξ . Moreov er, it has t w o fu ndamen tal p erio ds 9 1 and iτ in ph ase v ariable ξ , but it n eed not to b e p eriod ic in x and t . (ii) It can b e viewed as a parallel su p erp osition of o v erlapping one-soliton wa v es, p lace d one p erio d apart. In th e follo wing, w e further consider asymptotic prop erties of the p eriod ic w a v e solu- tion. In terestingly , the relation b etw een the p erio dic w a v e solution (3.6) and the one-soliton solution (3.4) can b e established as follo ws. Theorem 3. Supp ose th at the vec tor ( ω , c ) is a s olution of the system (3.8), and f or th e p erio dic wa ve solution (3.6), we let α = p 2 π i , σ = γ + π τ 2 π i , (3 . 9) where the p and γ are giv en in (3.4). Then we hav e the follo wing asymptotic prop erties c − → 0 , 2 π iξ − π τ − → η = px + p p 2 − 1 t + γ , ϑ ( ξ , τ ) − → 1 + e η , as λ → 0 . In other w ords, the doub le p erio dic solution (3.10) tends to the s oliton solution (3.4) under a small amplitude limit, that is, u − → u 1 , as λ → 0 . (3 . 10) Pr o of. Here we will d ir ectl y use the s y s tem (3.8) to analyze asymptotic prop erties of the p erio dic s olution (3.6). S ince the coefficien ts of s y s tem (3.8) are p o wer series ab out λ , its solution ( ω , c ) also sh ould b e a series ab out λ . W e explicitly expand the co efficien ts of system (3.8) as follo ws ϑ 1 (0 , λ ) = 1 + 2 λ 4 + · · · , ϑ ′′ 1 (0 , λ ) = − 32 π 2 λ 4 + · · · , ϑ (4) 1 (0 , λ ) = 512 π 4 λ 4 + · · · , ϑ 2 (0 , λ ) = 2 λ + 2 λ 9 + · · · ϑ ′′ 2 (0 , λ ) = − 8 π 2 λ + · · · , ϑ (4) 2 (0 , λ ) = 32 π 4 λ + · · · . (3 . 11) Let the solution of the system (3.8) b e in the form ω = ω 0 + ω 1 λ + ω 2 λ 2 + · · · = ω 0 + o ( λ ) , c = c 0 + c 1 λ + c 2 λ 2 + · · · = c 0 + o ( λ ) . (3 . 12) Substituting the expansions (3.11) and (3.12) into the s y s tem (3.8) (the second equ atio n is divided by λ ) and letting λ − → 0, we immediately obtain the follo win g relations c 0 = 0 , ( − 8 π 2 α + 32 π 4 α 3 ) ω 0 − 8 π 2 α 2 = 0 , whic h implies c 0 = 0 , w 0 = α 4 π 2 α 2 − 1 . (3 . 13) 10 Com bining (3.12) and (3.13) then yields c − → 0 , 2 π iω − → 2 π iα (2 π iα ) 2 − 1 = p p 2 − 1 , as λ → 0 . Hence w e conclude ˆ ξ = 2 π iξ − π τ = px + 2 π iω t + γ − → px + p p 2 − 1 t + γ = η , as λ → 0 . (3 . 14) It r emains to consider asymptotic prop erties of the p er io dic w a ve solution (3.6) u nder the limit λ → 0. By expanding the Riemann theta fun ction ϑ ( ξ , τ ) and u sing (3.14) , it follo ws that ϑ ( ξ , τ ) = 1 + λ 2 ( e 2 π iξ + e − 2 π iξ ) + λ 8 ( e 4 π iξ + e − 4 π iξ ) + · · · = 1 + e ˆ ξ + λ 4 ( e − ˆ ξ + e 2 ˆ ξ ) + λ 12 ( e − 2 ˆ ξ + e 3 ˆ ξ ) + · · · − → 1 + e ˆ ξ − → 1 + e η , as λ → 0 , whic h together with (3.6) leads to (3.10). Th er efore we conclude that the doub le p erio dic solution (3.6) ju s t go es to the one-soliton solution (3.4) as the amplitude λ → 0.  4. The mo dified B ogo ya vlenskii-Sc hiff equation W e consid er (2+1)-dimensional mo difi ed Bogo ya vlenskii-Schiff equation [18] u t − 4 u 2 u z − 2 u x ∂ − 1 x ( u 2 ) z + u xxz = 0 , (4 . 1) whic h was d educed from the Miura transformation [19]. Equ ation (4.1) is reduced to the mo dified KdV equation in the case of x = z . W e shall constru ct a d ouble p erio dic w a v e solution to the equation (4.1) b y u sing Th eorem 1 and 2. The equation (4.1) can b e describ ed b y a coupled system u = ψ x , ρ xx + ψ 2 x + c = 0 , ψ t + 2 ψ x ρ xz + ψ z ( ρ xx + ψ 2 x + c ) + ψ xxz = 0 . (4 . 2) W e p er f orm the dep endent v ariable transformations u = ψ x = ∂ x ln  f g  , ρ = ln( f g ) , (4 . 3) then equation (4.2) is reduced to the follo wing bilinear form F ( D x ) f · g = ( D 2 x + c ) f · g = 0 , G ( D t , D x , D z ) f · g = ( D t + D 2 x D z + cD z ) f · g = 0 , (4 . 4) where c is a constant. Th e equation (4.4) is a t yp e of coupled b ilinear equations, which is more d iffi cu lt to b e dealt with than the single bilinear equation (3.3) d ue to app earance of 11 t w o functions and t w o equations. W e will tak e fu ll adv antag es of Theorem 2 to r ed uce the n umber of constrain t equations. No w we tak e in to accoun t the p erio dicit y of the solution (4.3 ), in which we take f and g as f = ϑ ( ξ + e, τ ) , g = ϑ ( ξ + h, τ ) , e, h ∈ C , (4 . 5) where phase v ariable ξ = αx + β z + ω t + σ. By means of Prop osition 3, w e find that the solution u = α∂ ξ ln ϑ ( ξ + e, τ ) ϑ ( ξ + h, τ ) is a double p erio dic fun ctio n with tw o fundamenta l p erio ds 1 and iτ . In the sp ecial case of c = 0, the equation (4.2) admits one-soliton solution u 1 = ∂ x ln 1 + e η 1 − e η , (4 . 6) where η = px + q y − p 2 q t + γ for ev er y p, q and γ . W e take e = 0 , h = 1 / 2 in (4.5), and therefore f = ϑ ( ξ , τ ) = X m ∈ Z exp(2 π inξ − π m 2 τ ) , g = ϑ  1 / 2 0  ( ξ , τ ) = X m ∈ Z exp(2 π im ( ξ + 1 / 2) − π m 2 τ ) = X m ∈ Z ( − 1) m exp(2 π imξ − π m 2 τ ) . (4 . 7) Due to the fact th at F ( D x ) is an ev en function, its constraint equations in the f ormula (2.10) v anish automatically for µ = 1. Similarly the constrain t equations associated with G ( D t , D x , D z ) also v anish automatically f or µ = 0. T herefore, the Riemann theta fun ction (4.6) is a solution of the bilinear equ ation (4.4), pro vided the follo win g equations ϑ ′′ 1 (0 , λ ) α 2 + ϑ 1 (0 , λ ) c = 0 , ϑ ′ 2 (0 , λ ) ω + ϑ ′ 2 (0 , λ ) β c + ϑ ′′′ 2 (0 , λ ) α 2 β = 0 , (4 . 8) where we int ro duce the n ota tions b y λ = e − π τ / 2 , ϑ 1 ( ξ , λ ) = ϑ (2 ξ , 2 τ ) = X m ∈ Z λ 4 m 2 exp(4 iπ mξ ) , ϑ 2 ( ξ , λ ) = ϑ  1 / 2 − 1 / 2  (2 ξ , 2 τ ) = X m ∈ Z ( − 1) m λ (2 m − 1) 2 exp[2 iπ (2 m − 1) ξ ] . It is ob vious that equation (4.8) ad m its an explicit solution ω and c . In this w a y , a p erio dic w a v e solution r eads u = ∂ x ln ϑ ( ξ , τ ) ϑ ( ξ + 1 / 2 , τ ) , (4 . 9 ) 12 where p aramete rs ω and c are given b y (4.11), while other parameters α, β , τ , σ are fr ee. I n summary , doub le p erio dic w a v e (4.9) has the follo wing features: (i) It is one-dimensional an d has t wo fundamenta l p erio ds 1 and iτ in p hase v ariable ξ . (ii) It can b e view ed as a parallel sup erp osition of o verlapping one-soliton wa ves, placed one p erio d apart. In the follo wing, we fu rther consid er asymptotic pr op erties of th e d ouble p erio dic wa v e solution. The relation b et w een the p erio dic w a v e solution (4.9) and the one-solito n solution (4.6) can b e established as follo ws . Theorem 4. Supp ose th at th e ve ctor ( ω , c ) T is a solution of the system (4.8). In the p erio dic wa ve solution (4.9), we c ho ose parameters as α = p 2 π i , β = q 2 π i , σ = γ + π τ 2 π i , (4 . 10) where the p, q and γ are the same as those in (4.6). Then w e h a ve the follo win g asymptotic prop erties c − → 0 , ξ − → η + π τ 2 π i , f − → 1 + e η , g − → 1 − e η , as λ → 0 . In other w ords, the double p erio dic solution (4.9) tends to the one-soliton solution (4.6) u n der a small amplitude limit , that is, u − → u 1 , as λ → 0 . (4 . 11) Pr o of. Here w e will directly use the system (4.8) to analyze asymptotic pr operties of p erio dic solution (4.9). W e explicitly expand th e coefficients of system (4.8) as follo w s ϑ 1 (0 , λ ) = 1 + 2 λ 4 + · · · , ϑ ′′ 1 (0 , λ ) = − 32 π 2 λ 4 + · · · , ϑ ′ 2 (0 , λ ) = − 4 πiλ + 12 π iλ 9 + · · · , ϑ ′′′ 2 (0 , λ ) = 16 π 3 iλ − 48 π 3 iλ 9 + · · · , (4 . 12) Supp ose that the solution of the system (4.8) is of the form ω = ω 0 + ω 1 λ + ω 2 λ 2 + · · · = ω 0 + o ( λ ) , c = c 0 + c 1 λ + c 2 λ 2 + · · · = c 0 + o ( λ ) . (4 . 13) Substituting the expansions (4.12) and (4.13) into the system (4.8) and letting λ − → 0, w e immediately obtain th e follo wing relations c 0 = 0 , − 4 π iω 0 + 16 π 3 iα 2 β = 0 , whic h has a solution c 0 = 0 , w 0 = 4 π 2 α 2 β . (4 . 14) Com bining (4.13) and (4.14) leads to c − → 0 , 2 π iω − → 8 π 3 iα 2 β = − p 2 q , as λ → 0 , 13 or equiv alen tly ˆ ξ = 2 π iξ − π τ = px + q y + 2 π iω t + γ − → px + q y − p 2 q t + γ = η , as λ → 0 . (4 . 15) It remains to iden tify that the p erio dic wa v e (4.9) p ossesses the same form with the one-soliton solution (4.6) under the limit λ → 0. F or this p urp ose, we start to expand the functions f and g in the form f = 1 + λ 2 ( e 2 π iξ + e − 2 π iξ ) + λ 8 ( e 4 π iξ + e − 4 π iξ ) + · · · . g = 1 − λ 2 ( e 2 π iξ + e − 2 π iξ ) + λ 8 ( e 4 π iξ + e − 4 π iξ ) + · · · . By using (4.13)-(4 .15), it follo ws that f = 1 + e ˆ ξ + λ 4 ( e − ˆ ξ + e 2 ˆ ξ ) + λ 12 ( e − 2 ˆ ξ + e 3 ˆ ξ ) + · · · − → 1 + e ˆ ξ − → 1 + e η , as λ → 0; g = 1 − e ˆ ξ + λ 4 ( e 2 ˆ ξ − e − ˆ ξ ) + λ 12 ( e − 2 ˆ ξ − e 3 ˆ ξ ) + · · · − → 1 − e ˆ ξ − → 1 − e η , as λ → 0 . (4 . 16) The expression (4.11) follo ws from (4.16), and th us w e conclude that the double p eriodic solution (4.9) ju s t go es to the one-soliton solution (4.6) as the amplitude λ → 0.  5. The differen tial-difference KdV equation W e consid er differentia l-difference K d V equation d dt  u ( n ) 1 + u ( n )  = u ( n − 1 / 2) − u ( n + 1 / 2) . (5 . 1) Hirota and Hu ha v e found its soliton solutions an d rational solutions [20, 21], among them one-soliton solution reads u 1 ( n ) = (1 + e η + p / 2 )(1 + e η − p / 2 ) (1 + e η ) 2 − 1 , (5 . 2) where η = pn − sin h( p/ 2) t + γ for ev er y p and γ . W e sh all constru ct a p erio dic wa ve solutions to the equation (5.1) b y using Theorem 1. By means of a v ariable transformation u ( n ) = f ( n + 1 / 2) f ( n − 1 / 2) f ( n ) 2 − 1 , (5 . 3) the equation (5.1) is reduced to the bilinear equation  sinh( 1 4 D n ) D t + 2 sinh( 1 4 D n ) sinh( 1 2 D n ) + c  f ( n ) · f ( n ) = 0 , (5 . 4) where c is a constant . 14 No w we tak e into accoun t the p erio dicit y of the s olution (5.3), in which we tak e f ( n ) = ϑ ( ξ , τ ) , wh ere p hase v ariable ξ = ν n + ω t + σ. Then solution (5.3) is wr itten as u ( ξ ) ≡ u ( n ) = ϑ ( ξ + 1 2 ν, τ ) ϑ ( ξ − 1 2 ν, τ ) ϑ ( ξ , τ ) 2 − 1 . (5 . 5) By means of Prop osition 2, it is easy to d educe th at u n is a double p er io dic fu nction with t w o fund amental p eriod s 1 and iτ . Substituting (5.5) into (5.4) and using formula (2.10) leads to a linear system sinh( 1 4 D n ) ϑ ′ 1 (0 , λ ) ω + ϑ 1 (0 , λ ) c + sin h( 1 4 D n ) sinh( 1 2 D n ) ϑ 1 (0 , λ ) = 0 , sinh( 1 4 D n ) ϑ ′ 2 (0 , λ ) ω + ϑ 2 (0 , λ ) c + sin h( 1 4 D n ) sinh( 1 2 D n ) ϑ 2 (0 , λ ) = 0 , (5 . 6) where ϑ 1 ( ξ , λ ) and ϑ 2 ( ξ , λ ) are the s ame as those in (3.7) with ξ = ν n + ω t + σ. By using the solution ω and c of system (5.6), a p erio dic w a ve solution is obtained by (5.5). In the follo wing, we fu rther consid er asymptotic pr op erties of th e d ouble p erio dic wa v e solution. The relation b et w een the p erio dic w a v e solution (5.5) and the one-solito n solution (5.2) can b e established as follo ws . Theorem 5. Supp ose th at th e ve ctor ( ω , c ) T is a solution of the system (5.6). In the p erio dic wa ve solution (5.5), we c ho ose parameters as ν = p 2 π i , σ = γ + π τ 2 π i , (5 . 7) where the p an d γ are the same as those in (5.2). Th en w e h a ve the follo wing asymp totic prop erties c − → 0 , ξ − → η + π τ 2 π i , ϑ ( ξ , τ ) − → 1 + e η , as λ → 0 . In other w ords, the p erio dic solution (5.5) tend s to the one-soliton solution (5.2) un der a small amplitude limit , that is, u ( n ) − → u 1 ( n ) , as λ → 0 . (5 . 8) Pr o of. Here w e will directly use the system (5.6) to analyze asymptotic pr operties of p erio dic solution (5.5). W e explicitly expand th e coefficients of system (5.6) as follo w s ϑ 1 (0 , λ ) = 1 + 2 λ 4 + · · · , sinh( 1 4 D n ) ϑ ′ 1 (0 , λ ) = 8 π i sinh ( iπν ) λ 4 + · · · , sinh( 1 4 D n ) sinh( 1 2 D n ) ϑ 1 (0 , λ ) = 2 sin h ( iπν ) sin h(2 iπν ) λ 4 + · · · , ϑ 2 (0 , λ ) = 2 λ + 2 λ 9 + · · · , sin h( 1 4 D n ) ϑ ′ 2 (0 , λ ) = 4 π i sinh ( iπν / 2) λ + · · · , sinh( 1 4 D n ) sinh( 1 2 D n ) ϑ 2 (0 , λ ) = 2 sin h ( iπν ) sin h( iπ ν / 2) λ + · · · . (5 . 9) 15 Supp ose that the solution of the system (5.6) is of the form ω = ω 0 + ω 1 λ + ω 2 λ 2 + · · · = ω 0 + o ( λ ) , c = c 0 + c 1 λ + c 2 λ 2 + · · · = c 0 + o ( λ ) . (5 . 10) Substituting th e expansions (5.9) and (5.10) into the sys tem (5.6) and letting λ − → 0, w e immediately obtain th e follo wing relations c 0 = 0 , 4 π i sinh( iπ ν / 2) ω 0 + 2 sinh( iπ ν / 2) sinh( iπ ν ) = 0 , whic h implies c 0 = 0 , w 0 = − 1 2 π i sinh( iπ ν ) . (5 . 11) Com bining (5.9) and (5.10) leads to c − → 0 , 2 π iω − → − sinh( iπ ν ) = − sin h( p/ 2 ) , as λ → 0 , or equiv alen tly ˆ ξ = 2 π iξ − π τ = pn + 2 π iω t + γ − → pn − sinh( p/ 2) t + γ = η , as λ → 0 . (5 . 12) It remains to consider asymp toti c pr op er ties of the p erio dic wa ve solution (5.5) under the limit λ → 0. By expanding the Riemann theta fun ctio n ϑ ( ξ , τ ), it follo w s th at ϑ ( ξ , τ ) = 1 + e ˆ ξ + λ 4 ( e − ˆ ξ + e 2 ˆ ξ ) + λ 12 ( e − 2 ˆ ξ + e 3 ˆ ξ ) + · · · − → 1 + e ˆ ξ − → 1 + e η , as λ → 0 , whic h together with (5.5 ) lead to (5.8). Therefore we conclud e that the p erio dic solutio n (5.5) just go es to the one-soliton solution (5.2) as th e amplitude λ → 0.  Ac kno wledgmen t The wo rk describ ed in this p ap er was supp orted by gran ts from Researc h Grants Coun- cil contract s HKU, the National Science F oundation of Ch ina (No.1097 1031), Shanghai Shuguang T rac king Pro j ect (No .08GG01 ) and Inn ov atio n Program of Shanghai Municipal Education Commission (No.10ZZ131). Referen ces [1] R. Hiro ta and J. Satsuma: Pr og. Theor. Phys. 57 (1977 ) 797. [2] R. Hiro ta: D ir e ct metho ds in soliton the ory (Spring er-verlag, Berlin, 200 4). [3] X. B. Hu, C X Li, J. J. C. Nimmo and G. F. Y u, J. Ph ys. A, 38 (2005) 195. [4] R Hiro ta and Y. Ohta: J. Phys. So c. J pn. 60 (1991 ) 798. [5] K W Chow, C K Lam, K Nakkeeran and B Malmed. J. Phys. So c. Jpn. 77 (2008 ), 05400 1 16 [6] K . Sawada and T K otera: Pr og. Theor . Phys. 51 (1 974) 1355 . [7] A. Nak amura, J. Phys. So c. J pn. 47 , 1701 -1705 (1979 ). [8] A. Nak amura, J. Phys. So c. J pn. 48 (1980 ), 136 5. [9] H. H. Dai, E. G. F an and X. G. Geng, a rxiv.org/ pdf/nlin/06 02015 [10] Y. Z ha ng, L. Y. Y e, Y. N. Lv a nd H. Q . Zhao, J. P h ys A, 40 (2007 ), 5539 . [11] Y. C. Hon, E. G. F a n and Z. Y. Qin, Moder n Phys Lett B, 22 (20 08), 547. [12] E. G. F an a nd Y. C. Hon, Phys Rev E, 78 (2008 ), 03 6607. [13] E. G. F an, J . Phys A, 42 (20 0 9), 0 95206. [14] W. X. Ma, R. G. Z ho u, J. Math. P hys, 24 (2009 ), 167 7. [15] H. M. F ark as and I. Kra, Rie mann Surfac es , New Y ork, Spr inger-V erlag, 1992 . [16] P . A. Cla rkson a nd E. L. Mansfield, Nonlinearity , 7 (199 94), 975 . [17] R. Hir ota a nd J. Satsuma, J. Phys. So c. Jpn. 40 (197 6 ), 61 1. [18] S. J . Y u, K . T o da, N. Sasa and T. F ukuy ama, J. Phys A, 31 (1998 ), 333 7. [19] O. I. B ogoy a venskii, Math. USSR Izv. 36 (1 991), 12 9. [20] R Hir ota a nd J. Satsuma, Prog r. Theor. Phys. Suppl. 64 (19 7 6), 64. [21] X. B. Hu and P . A. Cla rkson, J . Phys. A 28 (19 95) 500 9.