Explicit quasi-periodic wave solutions and asymptotic analysis to the supersymmetric Itos equation

Explicit quasi-p erio dic w a v e solutions and asymptotic analysis to the s up ersymmetric Ito’s e quation Engui F an a ∗ , Y. C. Hon b , a. School of Mathematics Sc ie nces, F udan University , Shanghai 200 433, PR China b. Departmen t of Mathematics, City Universit y of Hong Kong , Hong Kong, PR China No ve mber 3, 2018 Abstract: Based on a Riemann theta function and the sup er-Hirota bilinear form, w e prop ose a k ey formula for explicitly constructing quasi-p erio dic wa v e solutions of the sup ers ymmetric Ito’s equation in sup erspace C 2 , 1 Λ . Once a nonlinear equation is written in bilinear forms, then the quasi-p erio dic wa v e s olutions can b e d irectly obtained f r om our form u la. Th e relations b et ween the p erio dic wa v e solutions and the w ell-kno wn soliton solutions are rigorously established. It is sho wn that the quasi-p erio dic w a ve solutions tends to the soliton solutions un der small amplitude limits. 1. In tro d uction The Ito’s equatio n tak es th e form u tt + 6( u x u t ) x + u xxxt = 0 , (1 . 1) whic h w as first prop osed by Ito, and its b ilinear B¨ ac klund tran s formation, Lax represen tation and m ulti-soliton solutions w ere obtained [1] . The other integ r able prop erties of this equation suc h as t h e nonlinear su p erp osition f orm ula, Kac-Mo o dy algebra, bi-Hamilt onian structure ha v e b een further foun d [2]-[5] . Recently , Liu , Hu and Liu prop osed the follo wing su p ersymmetric Its’s equation [6] D t F t + 6( F x ( D t F )) x + D t F xxx = 0 , (1 . 2) and obtained its one-, tw o- and three-soliton solutions, where F = F ( x, t, θ ) is fermionic su - p erfield dep endin g on usual ev en indep enden t v ariable x , t and o dd Grassman v ariable θ . T he differen tial op erator D t = ∂ θ + θ ∂ t is the sup er d eriv ativ e. The bilinear deriv ativ e m etho d develo p ed by Hirota is a p o werful appr oac h for constructing exact solution of nonlinear equ ations[7]–[13] . Based on the Hirota b ilinear form and the Riemann theta functions, Nak am ura pr esen ted an approac h to directly construct a kind of q u asi-p erio dic solutions of n onlinear equation [14 , 15] , where the p erio dic w av e solutions of th e KdV equation ∗ Electronic mail: faneg@fudan.edu.cn. 1 and the Boussinesq equation we re obtained. Th is metho d n ot only conv enien tly obtains p erio dic solutions of a nonlinear equation, but also directly giv es th e explicit r elations among f requen- cies, wa v e-n um b ers, phase sh ifts and amplitudes of th e wa v e. Recen tly , th is metho d is fur ther dev elop ed to in vestig ate th e discrete T o da lattice, (2+1)-dimensional Kadomtsev-P etviash vili equation and Bog o y a vlenskii’s breaking soliton equation[16]-[20] . Our p resen t pap er will considerably improv e the key steps of the ab o ve metho d so as to mak e the metho d m u c h more lucid and str aightforw ard for applying a class of nonlin ear su- p ersymmetric equations. First, the ab o ve metho d will b e generalized into the sup ers ymmetric con text. The quasi-p erio dic solutions of su p ersymmetric equations still seem n ot in vestiga ted to our ac kno wledge. S econd, we a formula that the Riemannn theta functions satisfy a sup er-Hirota bilinear equation. This form u la actually provides us an uniform metho d which can b e used to construct qu asi-p erio dic wa v e solutions of n on lin ear differentia l, difference and sup ersymm etric equations. Once a nonlin ear equation is w ritten in bilinear forms, then the qu asi-p erio dic wa v e solutions of the nonlinear equation can b e ob tained directly b y using the form ula. As illustrative example, we shall constru ct quasi-p erio d ic wa v e solutions to the sup ersymmetric Ito’s equation (1.2). Mo reo ve r , w e also establish the relations b etw een our qu asi-p erio dic w a v e solutions and the soliton solutions that were obtained by Liu and Hu [5] . The organization of this pap er is as follo w s . In section 2, w e br iefly introdu ce a sup er-Hirota bilinear that will b e suitable f or constructing quasi-p erio dic solutions of the equation (1.2). And then introdu ce a general Riemann theta function and p ro vide a key form ula for constructing p erio dic wa v e solutions. In section 3, as app lication of our form u la, we construct one-p erio dic w av e solutions to the equation (1.2). W e furth er pr esent a simple and effectiv e limiting pro cedure to analyze asymptotic b eh a vior of the one-p erio dic wa v e solutions. It is rigorously sh o wn that the qu asi-p erio dic wa v e solutions tends to the known soliton solutions obtained b y Liu and Hu under “small amplitude” limits. A t last, we briefly discuss the conditions on the construction of multi-per io d ic w av e solutio ns of the equation (1.2) in s ection 4. 2. The sup erspace, Hirota bilinear form and the Riemann theta functions T o fix the notations and make our pr esen tation s elf-conta in ed , we br iefly r ecall some prop er ties ab out sup eranalysis and sup er-Hirota bilinear op erators. Th e details ab ou t su p eranalysis refer, for instance, to Vladimiro v’s w ork [22, 23]. A linear space Λ is cal led Z 2 -graded if it represente d as a direct s um of tw o su bspaces Λ = Λ 0 ⊕ Λ , where element s of the sp aces Λ 0 and Λ 1 are homogeneous. W e assu me th at Λ 0 is a subspace consisting of ev en elemen ts and Λ 1 is a sub space consisting of o d d elemen ts. F or the element 2 f ∈ Λ w e denote by f 0 and f 1 its even and o dd comp onen ts. A parit y fun ction is in tro d uced on the Λ, namely , | f | =  0 , if f ∈ Λ 0 , 1 , if f ∈ Λ 1 . W e in tro duce an annihilator of th e set of o d d elemen ts b y setting ⊥ Λ 1 = { λ ∈ Λ : λ Λ 1 = 0 } . A s up eralgebra is a Z 2 -graded space Λ = Λ 0 ⊕ Λ in whic h , b esides usu al op erations of addition and multiplicatio n b y num b ers, a pr o duct of element s is defined with the usual d istribution law: a ( αb + β c ) = αab + β ac, ( αb + β c ) a = αba + β ca, where a, b, c ∈ Λ and α, β ∈ C . Moreo v er, a stru cture on Λ is introd uced of an asso ciativ e algebra with a unite e and ev en multiplic ation i.e., the p ro duct of t w o ev en and tw o o d d elemen ts is an ev en elemen t and the pro d uct of an ev en element by an o dd one is an o d d elemen t: | ab | = | a | + | b | mo d (2). A commutativ e sup eralgebra with u nit e = 1 is called a finite-dimensional Grassmann algebra if it con tains a system of antic ommuting generators σ j , j = 1 , · · · , n with the p rop erty: σ j σ k + σ k σ j = 0 , j, k = 1 , 2 , · · · , n , in particular, σ 2 j = 0. The Grassm ann algebra will b e den ote by G n = G n ( σ 1 , · · · , σ n ). The monomials { e 0 , e i = σ j 1 · · · σ j n } , j = ( j 1 < · · · < j n ) form a b asis in the Grassmann algebra G n , dim G n = 2 n . Then it follo ws that any elemen t of G n is a linear com bination of monomials σ j 1 · · · σ j k , j 1 < · · · < j k , that is, f = f 0 + X k ≥ 0 X j 1 < ··· 0 is called the p erio d matrix of the Riemann theta f unction. It is ob vious that the Riemann th eta fu nction (2.3) con verge s absolutely and sup erd ifferen tiable on sup erspace C 2 , 1 Λ . F or the simp licit y , in the case when s = ω = 0, we d enote ϑ ( ξ , τ ) = ϑ ( ξ , 0 , 0 | τ ) . Definition 2. A function f ( ξ ) : C 2 , 1 Λ → C 1 , 0 Λ is said to b e quasi-p erio dic in ξ = αx + ω t + θ σ + δ with fun damen tal p erio ds T , if there exist certain constan ts a, b ∈ Λ 0 , such that f ( ξ + T ) = f ( ξ ) + aξ + b. An example of this is the ordinary W eierstrass zeta function, w h ere ζ ( ξ + ω ) = ζ ( ξ ) + η , for a fixed constan t η when ω is a p erio d of the corresp onding W eierstrass elliptic ℘ function. Prop osition 2. [24] The Riemann theta fu n ction ϑ ( ξ , τ ) defin ed ab o v e h as the p erio dic prop erties ϑ ( ξ + 1 + iτ , τ ) = exp( − 2 π iξ + π τ ) ϑ ( ξ , τ ) . (2 . 4) No w w e tu r n to see the p eriod icit y of the solution (2.4), w e tak e f ( x, t, θ ) in the b ilinear equation (2.2) as f ( x, t, θ ) = ϑ ( ξ , τ ) , where ph ase v ariable ξ = αx + ω t + θ σ + δ . By using (2. 4), it is easy to see that ϑ ′ ξ ( ξ + iτ , τ ) ϑ ( ξ + iτ , τ ) = − 2 π i + ϑ ′ ξ ( ξ , τ ) ϑ ( ξ , τ ) , 6 that is, ∂ ξ ln ϑ ( ξ + iτ , τ ) = − 2 π i + ∂ ξ ln ϑ ( ξ , τ ) . (2 . 5) According to the different ial r elatio n, w e h a v e F ( x, t, θ ) = F ( ξ ) = ∂ − 1 θ F 0 + α∂ ξ ln ϑ ( ξ , τ ) . (2 . 6) The equations (2.5 ) and (2.6) demons tr ate that F ( ξ + 1 + iτ ) = ∂ − 1 θ F 0 + α∂ ξ ln ϑ ( ξ + 1 + iτ , τ ) = − 2 π iα + F ( ξ ) . Therefore the solution F ( ξ ) is a quasi-p erio dic fun ction with t wo fun damen tal p erio d s 1 and iτ . In follo w in g, we establish uniform formula on the Riemannn theta function, whic h will pla y a key role in the construction of the p erio d ic wa v e solutions. Prop osition 2. [21] Supp ose th at f ( x, t, θ ) , g ( x, t, θ ) are su p er d ifferen tiable on s p ace C 2 , 1 Λ . Then the Hirota bilinear op erators D x , D t and sup er-Hirota bilinear op erator S x ha ve prop er ties S 2 N x f · g = D N x f · g , D m x D n t e ξ 1 · e ξ 2 = ( α 1 − α 2 ) m ( ω 1 − ω 2 ) n e ξ 1 + ξ 2 , S x e ξ 1 · e ξ 2 = [ σ 1 − σ 2 + θ ( α 1 − α 2 )] e ξ 1 + ξ 2 , (2 . 7) where ξ j = α j x + ω j t + θ σ j + δ j , j = 1 , 2. More generally , w e ha v e F ( S x , D x , D t ) e ξ 1 · e ξ 2 = F ( σ 1 − σ 2 + θ ( α 1 − α 2 ) , α 1 − α 2 , ω 1 − ω 2 ) e ξ 1 + ξ 2 , (2 . 8) where G ( S t , D x , D t ) is a p olynomial ab out S t , D x and D t . This pr op erties will b e utilized later to explore the quasi-p erio dic w a ve solutions of th e equation (1.2). Prop osition 3. The Hirota b ilinear op erators D x , D t and sup er-Hirota bilinear op erator S x ha ve prop erties wh en they act on the Riemann theta functions D x ϑ ( ξ , ε ′ , s ′ | τ ) · ϑ ( ξ , ε, s | τ ) = X µ =0 , 1 ∂ x ϑ (2 ξ , ε ′ − ε, ( s ′ − s − µ ) / 2 | 2 τ ) | ξ =0 ϑ (2 ξ , ε ′ + ε, ( s ′ + s + µ ) / 2 | 2 τ ) , (2 . 9) S t ϑ ( ξ , ε ′ , s ′ | τ ) · ϑ ( ξ , ε, s | τ ) = X µ =0 , 1 D t ϑ (2 ξ , ε ′ − ε, ( s ′ − s − µ ) / 2 | 2 τ ) | ξ =0 ϑ (2 ξ , ε ′ + ε, ( s ′ + s + µ ) / 2 | 2 τ ) , (2 . 10) where P µ =0 , 1 indicates sum with resp ectiv e to µ = 0 , 1. In general, for a p olynomial op erator G ( S t , D x , D t ) ab out S t , D x and D t , we ha ve G ( S t , D x , D t ) ϑ ( ξ , τ ) · ϑ ( ξ , τ ) = X µ =0 , 1 C ( α, ω , σ, µ ) ϑ (2 ξ , µ/ 2 | 2 τ ) , (2 . 11) where ξ = αx + ω t + θ σ + γ . C ( α, ω , σ, µ | τ ) = X n ∈ Z G { 4 π i ( n − µ / 2) α, 4 π i ( n − µ / 2) ω, 4 π i ( n − µ / 2)( σ + θ ω ) } × exp  − 2 π τ ( n − µ/ 2) 2  . (2 . 12) 7 Pr o of. B y using (2.7), w e ha v e Γ = S t ϑ ( ξ , ε ′ , s ′ | τ ) · ϑ ( ξ , ε, s | τ ) = X m ′ ,m ∈ Z S t exp { 2 π i ( m ′ + s ′ )( ξ + ε ′ ) − π ( m ′ + s ′ ) 2 τ } · exp { 2 π i ( m + s )( ξ + ε ) − π ( m + s ) 2 τ } , = X m ′ ,m ∈ Z 2 π i ( σ + θ ω )( m ′ − m + s ′ − s ) exp  2 π i ( m ′ + m + s ′ + s ) ξ − 2 π i [( m ′ + s ′ ) ε ′ + ( m + s ) ε ] − π τ [( m ′ + s ′ ) 2 + ( m + s ) 2 ]  m = l ′ − m ′ = X l ′ ,m ′ ∈ Z 2 π i ( σ + θ ω )(2 m ′ − l ′ + s ′ − s ) exp  2 π i ( l ′ + s ′ + s ) ξ − 2 π i [( m ′ + s ′ ) ε ′ +( l ′ − m ′ + s ) ε ] − π [( m ′ + s ′ ) 2 + ( l ′ − m ′ + s ) 2 ] τ  l ′ =2 l + µ = X µ =0 , 1 X l,m ′ ∈ Z 2 π i ( σ + θ ω )(2 m ′ − 2 l + s ′ − s − µ ) exp { 4 πiξ [ l + ( s ′ + s + µ ) / 2] − 2 π i [( m ′ + s ′ ) ε ′ − ( m ′ − 2 l − s − µ ) ε ] − π [( m ′ + s ′ ) 2 + ( m ′ − 2 l − s − µ ) 2 ] τ } Let m ′ = n + l , and using the relations n + l + s ′ = [ n + ( s ′ − s − µ ) / 2] + [ l + ( s ′ + s + µ ) / 2] , n − l − s − µ = [ n + ( s ′ − s − µ ) / 2] − [ l + ( s ′ + s + µ ) / 2] , w e finally obtain that Γ = X µ =0 , 1 " X n ∈ Z 4 π i ( σ + θω )[ n + ( s ′ − s − µ ) / 2] exp {− 2 π i [ n + ( s ′ − s − µ ) / 2]( ε ′ − ε ) − 2 π τ [ n + ( s ′ − s − µ ) / 2] 2 } # × " X l ∈ Z exp { 2 π i [ l + ( s ′ + s + µ ) / 2](2 ξ + ε ′ + ε ) − 2 π τ [ l + ( s ′ + s + µ ) / 2 ] 2 # = X µ =0 , 1 D t ϑ (2 ξ , ε ′ − ε, ( s ′ − s − µ ) / 2 | 2 τ ) | ξ =0 ϑ (2 ξ , ε ′ + ε, ( s ′ + s + µ ) / 2 | 2 τ ) . In a similar w a y , we can prov e the form u lae (2.9). As a sp ecial case when ε = s = 0 of the Riemann theta function (2.3), by using (2.9) an (2.10), we can p ro ve th e form u la (2.11) .  F r om the formulae (2.11) and (2.1 2), it is seen that if the f ollo wing equations are satisfied C ( α, ω , σ, µ | τ ) = 0 , for µ = 0 , 1, then ϑ ( ξ , τ ) is a solution of the bilinear equation G ( S t , D x , D t ) ϑ ( ξ , τ ) · ϑ ( ξ , τ ) = 0 . 3. Quasi-p erio dic w a v es and asymptotic prop erties In th is s ection, w e consider p erio d ic w av e solutions of the equation (1.2). As a simple case of the theta function (2.3) wh en N = 1 , s = 0, we tak e f ( x, t, θ ) as f ( x, t, θ ) = ϑ ( ξ , τ ) = X n ∈ Z exp(2 π inξ − π n 2 τ ) , (3 . 1) 8 where the p hase v ariable ξ = αx + ω t + θ σ + δ , and the p arameter τ > 0. T o let th e Riemann th eta function (3.1) b e a solution of the bilinear equation (2.2), according to the form ula (2.11), the follo win g equatio n s only need to b e satisfied X n ∈ Z  − 16 π 2 ( n − µ/ 2) 2 ( σ + θ ω ) ω + 256 π 4 ( n − µ/ 2) 4 ( σ + θ ω ) α 3 − 48 π 2 ( n − µ/ 2) 2 α 2 F 0 + c  exp[ − 2 π ( n − µ/ 2) 2 τ ] = 0 , µ = 0 , 1 . (3 . 2) W e in tro duce the notatio ns b y λ = e − π τ / 2 , ϑ 1 ( ξ , λ ) = ϑ (2 ξ , 2 τ ) = X n ∈ Z λ 4 n 2 exp(4 iπ nξ ) , ϑ 2 ( ξ , λ ) = ϑ (2 ξ , 0 , − 1 / 2 , 2 τ ) = X n ∈ Z λ (2 n − 1) 2 exp[2 iπ (2 n − 1) ξ ] . (3 . 3) By us ing form ula (3.3), the equ ation (3.2) can b e written as a linear system θ ϑ ′′ 1 ω 2 + ( σ ϑ ′′ 1 + α 3 θ ϑ (4) 1 ) ω + ϑ 1 c + σ α 3 ϑ (4) 1 + 3 α 2 F 0 ϑ ′′ 1 = 0 , θ ϑ ′′ 2 ω 2 + ( σ ϑ ′′ 2 + α 3 θ ϑ (4) 2 ) ω + ϑ 2 c + σ α 3 ϑ (4) 2 + 3 α 2 F 0 ϑ ′′ 2 = 0 , (3 . 4) where ω ∈ Λ 0 is even and c, F 0 : C 2 , 1 Λ → C 0 , 1 Λ are o dd . In addition, w e ha ve denoted deriv ativ es of ϑ j ( ξ , λ ) at ξ = 0 b y simple notations ϑ ( k ) j = ϑ ( k ) j (0 , λ ) = d k ϑ j ( ξ , λ ) dξ k | ξ =0 , j = 1 , 2 , k = 0 , 1 , 2 , · · · Moreo v er, these functions are ind ep endent of Gr assmann v ariable θ and σ . W e sh o w there existence real solutions to the system (3.4). S ince c = c ( θ , t ) and F = F 0 ( θ ) are fu nction of Grassmann v ariable θ , we can expand them in the form c = c 1 + c 2 θ , F 0 = f 1 + f 2 θ , (3 . 5) where c 1 , f 1 ∈ Λ 1 are o dd and c 2 , f 2 ∈ Λ 0 are eve n . Sub stituting (3.5) into (3.4) leads to ( σ ϑ ′′ 1 ω + ϑ 1 c 1 + σ α 3 ϑ (4) 1 + 3 α 2 ϑ ′′ 1 f 1 ) + θ (3 α 2 ϑ ′′ 1 f 2 + ϑ 1 c 2 + ϑ ′′ 1 ω 2 + α 3 ϑ (4) 1 ω ) = 0 , ( σ ϑ ′′ 2 ω + ϑ 2 c 1 + σ α 3 ϑ (4) 2 + 3 α 2 ϑ ′′ 2 f 1 ) + θ (3 α 2 ϑ ′′ 2 f 2 + ϑ 2 c 2 + ϑ ′′ 2 ω 2 + α 3 ϑ (4) 2 ω ) = 0 , (3 . 6) where ω , c 1 , c 2 , f 1 and f 2 are parameters to b e d etermined. Since θ is a Grassmann v ariable, the system (3.6) will b e satisfied pro vided that σ ϑ ′′ 1 ω + ϑ 1 c 1 + σ α 3 ϑ (4) 1 + 3 α 2 ϑ ′′ 1 f 1 = 0 , σ ϑ ′′ 2 ω + ϑ 2 c 1 + σ α 3 ϑ (4) 2 + 3 α 2 ϑ ′′ 2 f 1 = 0 (3 . 7) and 3 α 2 ϑ ′′ 1 f 2 + ϑ 1 c 2 + ϑ ′′ 1 ω 2 + α 3 ϑ (4) 1 ω = 0 , 3 α 2 ϑ ′′ 2 f 2 + ϑ 2 c 2 + ϑ ′′ 2 ω 2 + α 3 ϑ (4) 2 ω = 0 . (3 . 8) 9 In the systems (3.7) and (3.8), it is ob vious that vecto r s ( ϑ 1 , ϑ 2 ) T and ( ϑ ′′ 1 , ϑ ′′ 2 ) T are linear indep end en t, and ( ϑ (4) 1 , ϑ (4) 2 ) T 6 = 0. Therefore the system (3.7) admits a solution ω = − 3 β α 2 + ( ϑ (4) 2 ϑ 1 − ϑ (4) 1 ϑ 2 ) α 3 ϑ ′′ 1 ϑ 2 − ϑ ′′ 2 ϑ 1 ∈ Λ 0 , c 1 = ( ϑ (4) 2 ϑ ′′ 1 − ϑ (4) 1 ϑ ′′ 2 ) α 3 σ ϑ ′′ 1 ϑ 2 − ϑ ′′ 2 ϑ 1 ∈ Λ 1 , (3 . 9) here we h a ve tak en f 1 = β σ, β ∈ R for s implicit y , and other parameters α, τ , σ , β are free. By us ing (3.9) and solving system (3.8), w e obtain that f 2 = − β ω ∈ Λ 0 , c 2 = ( ϑ (4) 1 ϑ ′′ 2 − ϑ (4) 2 ϑ ′′ 1 ) α 3 ω ϑ ′′ 1 ϑ 2 − ϑ ′′ 2 ϑ 1 ∈ Λ 0 . (3 . 10) Noting that R θ dθ = 1 and R dθ = 0, w e ha v e ∂ − 1 F 0 = Z ( β σ − β ω θ ) dθ = − β ω . In this w a y , w e indeed can get an explicit p erio dic wa v e solution of the equation (1.12) F = − β ω + ∂ x ln ϑ ( ξ , τ ) , (3 . 11) with the theta function ϑ ( ξ , τ ) giv en by (3.1) and parameters ω , c 1 , c 2 b y (3.9) and (3.10), wh ile other parameters α, σ, τ , δ, β are f ree. Among th em, the three parameters α, σ and τ completely dominate a p erio dic wa v e. I n sum mary , the p erio dic w av e (3.11) is real-v alued and b ounded for all complex v ariables ( x, t, θ ). It is one-dimensional, i.e. th ere is a single phase v ariable ξ , and has t wo fu ndament al p erio ds 1 and iτ in p hase v ariable ξ . In the follo wing, w e further consider asymptotic pr op erties of the p erio d ic wa v e solution. In terestingly , the relation b et ween the one-p erio dic wa v e solution (3.11) and the one-sup er soliton solution (1.5) can b e established as follo ws. Theorem 1. Su pp ose that th e ω ∈ Λ 0 and c ∈ Λ 1 are given given by (3.5), (3.9) and (3.10). F or the one-p er io d ic w av e solutio n (3. 11), w e let α = k 2 π i , σ = ζ 2 π i , δ = γ + π τ 2 π i , (3 . 12) where the k , ζ and γ are th e same as those in (1.5). Then we hav e the follo wing asymptotic prop erties c − → 0 , ξ − → η + π τ 2 π i , ϑ ( ξ , τ ) − → 1 + e η , as λ → 0 . (3 . 13) In other words, th e p erio dic solution (3.11) tends to th e one-soliton solution (1.5) und er a small amplitude limit , that is, F − → F 1 , as λ → 0 . (3 . 14) Pr o of. Here w e w ill directly u se the sy s tem (3.4) to analyze asymp totic pr op erties of one- p erio dic solution, w hic h is more simple and effectiv e than our original metho d by solving the system [16]-[20] . Since the co efficients of system (3.4) are p o wer series ab out λ , its solution ( ω , c ) T also should b e a series ab out λ . 10 W e explicitly expand the coefficients of system (3.4) 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 + 2 λ 8 + · · · ϑ ′′ 2 (0 , λ ) = − 8 π 2 − 72 π 2 λ 8 + · · · , ϑ (4) 2 (0 , λ ) = 32 π 4 + 2592 π 4 λ 8 + · · · . (3 . 15) Let the s olution of the s ystem (3.4) b e in the form ω = ω 0 + ω 1 λ + ω 2 λ 2 + · · · = ω 0 + o ( λ ) , c = b 0 + b 1 λ + b 2 λ 2 + · · · = b 0 + o ( λ ) , (3 . 16) where ω j ∈ Λ 0 , b j ∈ Λ 1 , j = 0 , 1 , 2 · · · Substituting the expansions (3.11) and (3.12) in to the system (3.5) and letting λ − → 0, w e immediately obtain the follo wing relations b 0 = 0 , − 8 π 2 σ ω 0 + 2 b 0 + 32 π 4 σ α 3 = 0 , whic h has a solution b 0 = 0 , w 0 = 4 π 2 α 3 . Then from the relations (3.12) and (3.16), w e hav e c − → 0 , 2 π iω − → 8 π 3 iα 3 = − k 3 , as λ → 0 , and thus ˆ ξ = 2 π iξ − π τ = k x + 2 π iω t + θ ζ + γ − → k x − k 3 t + θ ζ + γ = η , as λ → 0 , (3 . 17) It remains to sh o w th at the one-p erio d ic w av e (3.11) p ossesses th e same form w ith th e one- soliton s olution (1.5) u nder the limit λ → 0. F or this pur p ose, we fi rst expand the Riemann theta fun ction ϑ ( ξ , τ ) in the f orm ϑ ( ξ , τ ) = 1 + λ 2 ( e 2 π iξ + e − 2 π iξ ) + λ 8 ( e 4 π iξ + e − 4 π iξ ) + · · · . By us ing the (3.12) and (3.17), it follo ws that 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 implies (3.13) and (3.14) . Therefore we conclude that the one-p erio dic solution (3.11) just go es to the one-soliton solution (1.5) as the amplitude λ → 0.  11 4. Discussion on the conditions of N -p erio dic w a v e solutions In this section, w e consider condition for N -p erio d ic w a ve solutions of th e equatio n (1.2). The theta fun ction is tak en the form ϑ ( ξ , τ ) = ϑ ( ξ 1 , · · · , ξ N , τ ) = X n ∈ Z N exp { 2 π i < ξ , n > − π < τ n , n > } , (4 . 1) where n = ( n 1 , · · · , n N ) T ∈ Z N , ξ = ( ξ 1 , · · · , ξ N ) T ∈ C N , ξ i = α j x + ω j t + θ σ j + δ j , j = 1 , · · · , N , τ is a N × N sy m metric p ositiv e defin ite matrix. T o mak e the theta fu nction (4.1) satisfy the bilinear equ ation (2.2), we obtain that according to the form ula (2.11) X µ =0 , 1 ∞ X n 1 , ··· ,n N = −∞ G    4 π i N X j = 1 ( n j − µ j / 2) α j , 4 π i N X j = 1 ( n j − µ j / 2) ω j , 4 π i N X j = 1 ( n j − µ j / 2)( σ j + θ ω j )    × exp   − 2 π N X j,k =1 ( n j − µ j / 2) τ j k ( n k − µ k / 2)   = 0 . (4 . 2) No w w e consider the n umb er of equation and some u n kno wn parameters. Obvio usly , in the case of su p ersymmetric equations, the num b er of constrain t equations of the t yp e (4.2) is 2 N +1 , whic h is t wo times of th e constraint equations n eeded in the case of ordinary equations [16]-[20] . On the other hand we h a v e parameters τ ij = τ j i , c 1 , c 2 , f 1 , f 2 , α i , ω i , whose total n u m b er is 1 2 N ( N + 1) + 2 N + 4. Among th em, 2 N p arameters τ ii , ω i are take n to b e th e giv en p arameters related to the amplitudes and wa v e n um b ers (or frequencies) of N -p erio dic w a ves; 1 2 N ( N + 1) parameters τ ij implicitly app ear in series form, wh ic h is general can n ot to b e solv ed explicit. Hence, the num b er of the explicit unk n o wn p arameters is only N + 4. The num b er of equations is larger than the un kno wn p arameters in the case when N ≥ 2. In this pap er, w e consider one-p erio dic w a v e solution of the equation (1.2), wh ic h b elongs to the cases w hen N = 1. T here are still certain difficulties in the calculation for the case N ≥ 2, which will b e considered in ou r future wo rk. Ac kno wledgmen t I w ould lik e to expr ess my sp ecial thanks to th e referee for constructiv e su ggestions w hic h ha v e b een follo we d in the present imp ro ved version of the pap er. The w ork d escrib ed in this pap er was supp orted b y grants fr om the Researc h Grant s C ou n cil of Hong Kong (No.9041 473), the National Science F oundation of China (No.1097103 1), S hanghai Shuguang T rac kin g P ro ject (No.08GG 01) and In n o v ation Program of Shanghai Mun icipal Education Commiss ion (No.1 0ZZ 131). 12 References [1] M. Ito, J. P hys. So c. Jpn. 49 (1980 ) 771 [2] X. B. Hu and Y L i: J. Phys. A 24 (1991) 1979 . [3] M. Jimb o and T. Miwa : Pub l. RIMS Ky oto Univ. 19 (1983) 943. [4] V. G. Drinfeld and V. Sok olo v: J. S o v. Math. 30 (1985) 197 5. [5] Q. P . Liu: Phys. Lett. A 277 (2000), 31 . [6] S. Q. Liu , X. B. Hu and Q . P . Liu , J. Phys. So c. Jp n 75 (2006), 064 004 [7] R. Hirota and J. Satsuma: Prog. Theor. Ph ys. 57 (197 7) 797. [8] R. Hirota: Dir e ct metho ds in soliton the ory (Springer-v erlag, Berlin, 2004) . [9] X. B. Hu: P . A. Clarkson, J . Phys. A 28 (1995) 5009. [10] X. B. Hu : C X Li, J. J. C. Nimm o, G. F. Y u, J. Phys. A, 38 (2005) 195. [11] R Hirota and Y. Oht a: J. Ph ys. S o c. Jpn. 60 (1991) 798 . [12] D. J. Zh ang: J . Phys. So c. Jpn . 71 (2002) 2649 . [13] K Sa wada and T Kotera: Prog. Theor. Ph ys. 51 (1974) 1355. [14] A. Nak am ura, J. Phys. So c. Jpn. 47 , 1701-1705 (1979 ). [15] A. Nak am ura, J. Phys. So c. Jpn. 48 (1980 ), 1365. [16] H. H. Dai, E. G. F an and X. G. Geng, arxiv.org/p df/nlin/0602015 [17] Y. C. Hon and E. G. F an, Mod ern Phys Lett B, 22 (2008), 547 . [18] E. G. F an and Y. C . Hon, Phys Rev E, 78 (2008), 036607. [19] W. X. Ma, R. G. Zh ou and L. Gao, Mo dern Phys. Lett. A, 21 (2009), 1677. [20] E. G. F an, J. Phys A, 42 (2009) , 095206. [21] A. S. C arstea, Nonlinearit y 13 (2000), 164 5. [22] V. S. Vladimirov, Theor. Mat h. P h ys. 59 (1984) , 3. [23] V. S. Vladimirov, Theor. Mat h. P h ys. 60 (1984) , 169. [24] D. Mumford, T ata L e ctur es on Theta II Progress in Mathmatics, V ol. 43 ( Boston: Birkh¨ auser, 1984). 13