Integrable discretizations for the short wave model of the Camassa-Holm equation

Integrable discr etizations f or the short wa ve model of the Camassa-Ho lm eq uation Bao-Feng F eng 1 ‡ , Ken-ichi Maruno 1 § and Y asuhiro Ohta 2 1 Departmen t of Mathemati cs, The Univ ersity of T e xas-Pa n American, Edinbur g, TX 78541 2 Departmen t of Mathemati cs, Kobe Uni v ersity , Rokko, Kobe 657-85 01, Japan Abstract. The link betwee n the short wave model of the Camassa-Holm equati on (SCHE) and bi linear equation s of the tw o-dimensiona l T od a l attice (2DTL) is clarifie d. The parametric form of N -cuspon solution of the SCHE in Casorati determinant is then giv en. Based on the above finding, inte grable semi-d iscrete and ful l-discret e analogues o f the SCHE are construct ed. T he determinant solution s of both semi-discrete and fully discrete analogue s of the SCHE are also present ed. 21 Nov ember 2018 P A CS numbers: 02.30.Ik, 05.45.Yv , 42 .65.Tg, 42.81.Dp T o be submitted to : J. Phys. A: Math. Gen. 1. In troduction In the present paper , we consider integrable d iscretizations of the nonlinea r p artial differential equation w T X X − 2 κ 2 w X + 2 w X w X X + w w X X X = 0 , (1) which belongs to the Harry-Dym hierarchy [1, 2, 3]. Her e κ is a real parameter and, as sho wn subsequen tly , can be normalized by the scaling transfo rmation when κ 6 = 0 . A conn ection between Eq. (1) an d the sinh-Go rdon equ ation was estab lished in [4]. When κ = 0, Eq.(1) is called the Hunter-Saxton equation and is derived as a model for weakly nonlinear orientation wa ves in massiv e nematic liq uid crystals [5]. The Lax pair a nd bi-Ha miltonian stru cture were discussed by Hunter and Zheng [6]. The dissipative and dispersive weak solutions were discussed in details by the same authors [7, 8]. Equation (1) can be viewed as a short-wave model of the Camassa-Holm equation [9] w T + 2 κ 2 w X − w T X X + 3 ww X = 2 w X w X X + w w X X X . (2) Follo wing the procedur e in [10, 11, 12], we introduce the time and space v ariables ˜ T and ˜ X ˜ T = ε T , ˜ X = ε − 1 X , where ε is a small param eter . Then w is expand ed as w = ε 2 ( w 0 + ε w 1 + · · · ) with w i ( i = 0 , 1 , · · · ) being functions of ˜ T and ˜ X . At the lowest order in ε , we obtain w 0 , ˜ T ˜ X ˜ X − 2 κ 2 w 0 , ˜ X + 2 w 0 , ˜ X w 0 , ˜ X ˜ X + w 0 w 0 , ˜ X ˜ X ˜ X = 0 , (3) ‡ e-mail: feng@utpa.edu § e-mail: kmaruno@utpa.edu Inte grable discr etizatio ns for the short wave model 2 which is exactly Eq .(1) after writing back into the original variables. Based on this fact, Matsuno obtained th e N -cuspon solution of Eq.(1) by taking the short-wave limit o n the N - soliton solution of the Camassa-Holm equation [13, 14]. Note that the parameter κ of Eq.(1) can be nor malized to 1 under the transform ation x = κ X , t = κ T , which leads to w t xx − 2 w x + 2 w x w xx + ww xxx = 0 . (4) W e call Eq.( 4) the sho rt wave m odel of the Camassa-Ho lm equ ation (SCHE). W itho ut loss of gen erality , we will fo cus on Eq. (4) an d its integra ble discretizations, since the solution of Eq.(1) with arbitrary nonzero κ , its integrable discretizations and the correspond ing solutions can be recovered throug h the above transform ation. The rem inder o f the pre sent p aper is organized as follows. In section 2, we reveal a connectio n betwee n the SCHE and th e b ilinear form two-d imensional T oda-lattice (2 DTL) equations. The p arametric form of N -cu spon so lution expressed by the Casorti determinan t is given, which is con sistent with th e solution given in [13]. Based on this fact, we prop ose an integrable semi-discrete analogue of the SCHE in section 3, an d further its integrable full- discrete analogu e in s ection 4. The con cluding remark is gi ven in section 5. 2. The connection with 2DTL equatio ns, and N -cuspon solutio n in determinant f orm 2.1. The link of the SCHE with the two-r eduction of 2DTL equation s In this section, we will show that the SCHE can b e derived from the bilinea r for m of two- dimensiona l T oda lattice (2DTL) equations −  1 2 D − 1 D 1 − 1  τ n · τ n = τ n + 1 τ n − 1 , (5) where D x is the Hirota D -deriv ativ e defined as D n x f · g =  ∂ ∂ x − ∂ ∂ y  n f ( x ) g ( y ) | y = x , and D − 1 and D 1 represent the Hirota D derivati ves with respect to variables x − 1 and x 1 , respectively . It is sho wn that the N -soliton solution of the 2DTL equations (5) can be e x pressed as the Casorati determin ant [16, 17] τ n =    ψ ( n + j − 1 ) i ( x 1 , x − 1 )    1 ≤ i , j ≤ N =          ψ ( n ) 1 ψ ( n + 1 ) 1 · · · ψ ( n + N − 1 ) 1 ψ ( n ) 2 ψ ( n + 1 ) 2 · · · ψ ( n + N − 1 ) 2 . . . . . . . . . . . . ψ ( n ) N ψ ( n + 1 ) N · · · ψ ( n + N − 1 ) N          , (6) with ψ ( n ) i satisfying the following dispersion relations: ∂ψ ( n ) i ∂ x − 1 = ψ ( n − 1 ) i , ∂ψ ( n ) i ∂ x 1 = ψ ( n + 1 ) i . A particular choice of ψ ( n ) i ψ ( n ) i = a i , 1 p n i e p i − 1 x − 1 + p i x 1 + η 0 i + a i , 2 q n i e q i − 1 x − 1 + q i x 1 + η ′ 0 i , (7) Inte grable discr etizatio ns for the short wave model 3 automatically satisfies the above dispersion relations. Applying the tw o-red uction τ n − 1 = ( ∏ N i = 1 p 2 i ) − 1 τ n + 1 , i.e., enforcing p i = − q i , i = 1 , · · · , N , we get −  1 2 D − 1 D 1 − 1  τ n · τ n = τ 2 n + 1 , (8) where the g auge tr ansformatio n τ n → ( ∏ N i = 1 p i ) n τ n is used. Letting τ 0 = f , τ 1 = g and x − 1 = s , x 1 = y , the above bilinear equation (8) takes the following form: −  1 2 D s D y − 1  f · f = g 2 , (9) −  1 2 D s D y − 1  g · g = f 2 . (10) Introd ucing u = g / f , Eqs.(9) and (10) can be con verte d into − ( ln f ) ys + 1 = u 2 , (11) − ( ln g ) ys + 1 = u − 2 . (12) Subtracting Eq.(1 2 ) from Eq.(1 1), one obtains ρ 2 ( ln ρ ) ys + 1 = ρ 2 , (13) by letting ρ = u 2 . Introd ucing the depen dent v ar iable transformation w = − 2 ( ln g ) ss , it then follows 1 2 w y = − ρ s ρ 2 , or ( ln ρ ) s = − ρ 2 w y , (14) by differentiating Eq.(12 ) with respect to s . In view of Eq.(14), Eq.(13) becomes − ρ 2  ρ 2 w y  y + 1 = ρ 2 . (15) Introd ucing the hod ograph transform ation  x = 2 y − 2 ( ln g ) s , t = s , and referrin g to Eq.(12), we have ∂ x ∂ y = 2 − 2 ( ln g ) ys = 2 ρ , ∂ x ∂ s = − 2 ( ln g ) ss = w , which implies    ∂ y = 2 ρ ∂ x , ∂ s = ∂ t + w ∂ x . Inte grable discr etizatio ns for the short wave model 4 Thus, Eqs.(14) and (15) can be cast into ( ( ∂ t + w ∂ x ) ln ρ = − w x , − w xx + 1 = ρ 2 . (16) By eliminating ρ , we arrive at ( ∂ t + w ∂ x ) ln ( − w xx + 1 ) = − 2 w x , or ( ∂ t + w ∂ x ) w xx − 2 w x ( 1 − w xx ) = 0 , which is actually the SCHE (4). 2.2. The N -cusp on solution of the SCHE Based on the link of the SCHE with th e two-reduc tion of 2DT L equa tions, the N - cuspon solution of the SCHE (4) is giv en as follows: w = − 2 ( ln g ) ss ,  x = 2 y − 2 ( ln g ) s , t = s , g =    ψ ( j ) i ( y , s )    1 ≤ i , j ≤ N , ψ ( j ) i = a i , 1 p j i e p i − 1 s + p i y + η 0 i + a i , 2 ( − p i ) j e − p i − 1 s − p i y + η ′ 0 i . (17) Moreover , the N -cuspon solution of the SCHE (1) with non-zer o κ is giv en as follows: w ( y , T ) = − 2 ( ln g ) ss , (18)  X = 2 y κ − 2 κ ( ln g ) s , T = s κ , (19) where g =    ψ ( j ) i ( y , s )    1 ≤ i , j ≤ N , with ψ ( n ) i = a i , 1 p n i e p i y + s / p i + η i 0 + a i , 2 ( − p i ) n e − p i y − s / p i + η ′ i 0 . W e remark here that to assure the regularity of the solution , the τ -functio n is r equired to b e positive d efinite. In what follows, we list the one-cuspo n and two-cu spon solution s. For N = 1, the τ -function is g = 1 + e 2 p 1 ( y + κ T / p 2 1 + y 0 ) , by choosing a 1 , 1 / a 1 , 2 = − 1, which yields the one-cuspo n solution w ( y , T ) = − 2 p 2 1 sech 2  p 1 ( y + κ T / p 2 1 + y 0 )  , X = 2 y κ − 2 κ p 1  1 + tanh  p 1 ( y + κ T / p 2 1 + y 0 )  . The profiles of one-cu spon with κ = 1 . 0 and κ = 0 . 1 are plotted in Fig. 1. Inte grable discr etizatio ns for the short wave model 5 −50 0 50 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 X w(X,t) −50 0 50 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 X w(X,t) (a) (b) Figure 1. Plots for one-cuspon solution for p 1 = √ 2 and dif ferent κ : (a) κ = 1 . 0; (b) κ = 0 . 1. The τ -function corresp onding to the two-cuspon solution is g = 1 + e θ 1 + e θ 2 +  p 1 − p 2 p 1 − p 2  2 e θ 1 + θ 2 , with θ i = 2 p i ( y + κ T / p 2 i + y i 0 ) , i = 1 , 2 . Here a 1 , 1 / a 1 , 2 = − 1 and a 2 , 1 / a 2 , 2 = 1 are chosen to assure the regularity of the solution. 3. Integrable semi-discretization of the SCHE Based on the link of the SCHE with the two-reduction of 2DTL eq uations clar ified in th e previous section, we attempt to construct the integrable semi-discrete analogue of the SCHE. Consider a Casorati determin ant τ n ( k ) =    ψ ( n + j − 1 ) i ( k )    1 ≤ i , j ≤ N =          ψ ( n ) 1 ( k ) ψ ( n + 1 ) 1 ( k ) · · · ψ ( n + N − 1 ) 1 ( k ) ψ ( n ) 2 ( k ) ψ ( n + 1 ) 2 ( k ) · · · ψ ( n + N − 1 ) 2 ( k ) . . . . . . . . . . . . ψ ( n ) N ( k ) ψ ( n + 1 ) N ( k ) · · · ψ ( n + N − 1 ) N ( k )          , with ψ ( n ) i satisfies the following dispersion relations ∆ k ψ ( n ) i = ψ ( n + 1 ) i , (20) ∂ s ψ ( n ) i = ψ ( n − 1 ) i , ( 21) where ∆ k is defined as ∆ k ψ ( k ) = ψ ( k ) − ψ ( k − 1 ) a . I n particular, we can choose ψ ( n ) i as ψ ( n ) i ( k ) = p n i ( 1 − a p i ) − k e ξ i + q n i ( 1 − aq i ) − k e η i , ξ i = 1 p i s + ξ i 0 , η i = 1 q i s + η i 0 , Inte grable discr etizatio ns for the short wave model 6 which au tomatically satisfies the dispersion relation s ( 20) and ( 21). The ab ove Casorati determinan t satisfies the bilinea r form of the semi-d iscrete 2DTL equatio n (the B ¨ acklund transform ation of the bilinear equation of the 2DTL equation ) [17, 18]  1 a D s − 1  τ n ( k + 1 ) · τ n ( k ) + τ n + 1 ( k + 1 ) τ n − 1 ( k ) = 0 . (22) Applying a two-redu ction conditio n p i = − q i , i = 1 , · · · , N , wh ich imp lies τ n − 1 ≎ τ n + 1 , we obtain −  1 a D s − 1  f k + 1 · f k = g k + 1 g k , (23) −  1 a D s − 1  g k + 1 · g k = f k + 1 g k , (24) by letting τ 0 ( k ) = f k , τ 1 ( k ) = g k . Letting u k = g k / f k , Eqs.(23) and (24) are equiv alent to − 1 a  ln f k + 1 f k  s + 1 = u k + 1 u k , (25) − 1 a  ln g k + 1 g k  s + 1 = u − 1 k + 1 u − 1 k . (26) Subtracting Eq.(2 6 ) from Eq.(2 5), one obtains u k + 1 u k a  ln u k + 1 u k  s + 1 = u 2 k + 1 u 2 k . (27) Introd ucing the discrete analog ue of hodograph transformation x k = 2 ka − 2 ( ln g k ) s , and δ k = x k + 1 − x k = 2 a − 2  ln g k + 1 g k  s . It then follows from Eq.(26) δ k = 2 a u k + 1 u k , or ρ k + 1 ρ k = 4 a 2 δ 2 k , (28) by assuming ρ k = u 2 k . Introd ucing the depen dent v ar iable transformation w k = − 2 ( ln g k ) ss , Eq.(27) becomes 1 δ k  ln ρ k + 1 ρ k  s + 1 − 4 a 2 δ 2 k = 0 . (29) Differentiating Eq.(26) with respect to s , we have 1 2 a ( w k + 1 − w k ) = − 1 u k + 1 u k ( ln u k + 1 u k ) s = − 1 2 u k + 1 u k ( ln ρ k + 1 ρ k ) s , Inte grable discr etizatio ns for the short wave model 7 or ( ln ρ k + 1 ρ k ) s = − 2 δ k ( w k + 1 − w k ) . (30) Eliminating ρ k and ρ k + 1 from Eqs.(29) and (30), we obtain 1 δ k ( w k + 1 − w k ) − 1 δ k − 1 ( w k − w k − 1 ) = 1 2 ( δ k + δ k − 1 ) − 2 a 2  1 δ k + 1 δ k − 1  , (31) or ∆ 2 w k = 1 δ k M  δ k − 4 a 2 δ k  , (32) by defining a difference operator ∆ and an av erage operator M as follows ∆ F k = F k + 1 − F k δ k , M F k = F k + 1 + F k 2 . Furthermo re, a substitution of Eq.(2 8 ) into Eq. (30) leads to d δ k d s = w k + 1 − w k . (33) Equation s (31) and (33 ) constitute the semi-discre te analogue of t he SCHE. Next, let u s sho w t hat in the con tinuou s limit, a → 0 ( δ k → 0), the proposed s emi-discrete SCHE recovers the continuo us SCHE. T o this end, Eqs.(31) and (33) are rewritten as      − 2 δ k + δ k − 1 ( ∆ w k − ∆ w k − 1 ) + 1 = 4 a 2 δ k δ k − 1 , ∂ s δ k = w k + 1 − w k . By taking logarithm ic deriv ative of the first equation, we get ∂ s  − 2 δ k + δ k − 1 ( ∆ w k − ∆ w k − 1 ) + 1  − 2 δ k + δ k − 1 ( ∆ w k − ∆ w k − 1 ) + 1 = − ∂ s δ k δ k − ∂ s δ k − 1 δ k − 1 . The depend ent v ariable w is re garded as a function of x and t , where x is the space coordinate of the k -th lattice po int and t is the time, defined by x k = x 0 + k − 1 ∑ j = 0 δ j , t = s . In the continuo us limit, a → 0 ( δ k → 0), we have ∂ s δ k δ k = w k + 1 − w k δ k → w x , ∂ s δ k − 1 δ k − 1 = w k − w k − 1 δ k − 1 → w x , 2 δ k + δ k − 1 ( ∆ w k − ∆ w k − 1 ) → w xx , ∂ x k ∂ s = ∂ x 0 ∂ s + k − 1 ∑ j = 0 ∂δ j ∂ s = ∂ x 0 ∂ s + k − 1 ∑ j = 0 ( w j + 1 − w j ) → w , Inte grable discr etizatio ns for the short wave model 8 ∂ s = ∂ t + ∂ x ∂ s ∂ x → ∂ t + w ∂ x , where the or igin of space coordin ate x 0 is ta ken so that ∂ x 0 ∂ s cancels w 0 . Thus the above semi-discrete SCHE conver ges to ( ∂ t + w ∂ x )( − w xx + 1 ) − w xx + 1 = − 2 w x , or ( ∂ t + w ∂ x ) w xx = 2 w x ( − w xx + 1 ) , (34) which is nothing but the SCHE (4). In summary , the semi-discre te an alogue of the SCHE and its determ inant solu tion ar e giv en as follows: The semi-discrete analogue of the SCHE          1 δ k ( w k + 1 − w k ) − 1 δ k − 1 ( w k − w k − 1 ) = 1 2 ( δ k + δ k − 1 ) − 2 a 2  1 δ k + 1 δ k − 1  , d δ k d t = w k + 1 − w k . (35) The determinant solution of the semi-discrete SCHE w k = − 2 ( ln g k ) ss , δ k = x k + 1 − x k = 2 a f k + 1 f k g k + 1 g k ,  x k = 2 ka − 2 ( ln g k ) s , t = s , g k =    ψ ( j ) i ( k )    1 ≤ i , j ≤ N , f k =    ψ ( j − 1 ) i ( k )    1 ≤ i , j ≤ N , ψ ( j ) i ( k ) = a i , 1 p j i ( 1 − a p i ) − k e p i − 1 s + η 0 i + a i , 2 ( − p i ) j ( 1 + a p i ) − k e − p i − 1 s + η ′ 0 i . (36) Introd ucing new indepen dent variables X k = x k / κ an d T = t / κ , we can include the parameter κ in the semi-discrete SCHE (35)          1 δ k ( w k + 1 − w k ) − 1 δ k − 1 ( w k − w k − 1 ) = 1 2 κ 2 ( δ k + δ k − 1 ) − 2 a 2  1 δ k + 1 δ k − 1  , d δ k d T = w k + 1 − w k , (37) where δ k = X k + 1 − X k and s = κ T . This is the semi-discr ete analogue of the SCHE (1). The N -cuspon solution of the semi-d iscrete SCHE (37) with the parameter κ is gi ven by w k = − 2 ( ln g k ) ss , δ k = X k + 1 − X k = 2 a κ f k + 1 f k g k + 1 g k ,  X k = 2 ka κ − 2 κ ( ln g k ) s , T = s κ , g k =    ψ ( j ) i ( k )    1 ≤ i , j ≤ N , f k =    ψ ( j − 1 ) i ( k )    1 ≤ i , j ≤ N , ψ ( j ) i ( k ) = a i , 1 p j i ( 1 − a p i ) − k e p i − 1 s + η 0 i + a i , 2 ( − p i ) j ( 1 + a p i ) − k e − p i − 1 s + η ′ 0 i . (38) Inte grable discr etizatio ns for the short wave model 9 4. Full-discretization of the SCHE In mu ch the same way o f findin g the semi-discrete analo gue o f the SCHE, we seek for its full-discrete analogu e and in the proce ss we arri ve at its N - cuspon solution. Consider the following Casorati determinant τ n ( k , l ) =    ψ ( n + j − 1 ) i ( k , l )    1 ≤ i , j ≤ N , (39) where ψ ( n ) i ( k , l ) = a i , 1 p n i ( 1 − a p i ) − k  1 − b p i − 1  − l e ξ i + a i , 2 q n i ( 1 − aq i ) − k  1 − bq i − 1  − l e η i , with ξ i = p i − 1 s + ξ i 0 , η i = q i − 1 s + η i 0 . It is known that the abov e determinant satis fies bilinea r equations [18]  1 a D s − 1  τ n ( k + 1 , l ) · τ n ( k , l ) + τ n + 1 ( k + 1 , l ) τ n − 1 ( k , l ) = 0 , (40) and ( bD s − 1 ) τ n ( k , l + 1 ) · τ n + 1 ( k , l ) + τ n ( k , l ) τ n + 1 ( k , l + 1 ) = 0 . (41) Here a , b are mesh sizes f or space and time v ariables, respectiv ely . Applying the tw o-red uction τ n − 1 = ( ∏ N i = 1 p 2 i ) − 1 τ n + 1 , i.e., enforcing p i = − q i , i = 1 , · · · , N , and letting τ 0 ( k , l ) = f k , l , τ 1 ( k , l ) = g k , l , the above bilinear equ ations take the following form:  1 a D s − 1  f k + 1 , l · f k , l + g k + 1 , l g k , l = 0 , (42)  1 a D s − 1  g k + 1 , l · g k , l + f k + 1 , l f k , l = 0 , (43) ( bD s − 1 ) f k , l + 1 · g k , l + f k , l g k , l + 1 = 0 , (44) ( bD s − 1 ) g k , l + 1 · f k , l + g k , l f k , l + 1 = 0 , (45) where the gauge transforma tion τ n → ( ∏ N i = 1 p i ) n τ n is used. It is readily shown that the above equations are equiv alent to 1 a  ln f k + 1 , l f k , l  s = 1 − g k + 1 , l g k , l f k + 1 , l f k , l , (46) 1 a  ln g k + 1 , l g k , l  s = 1 − f k + 1 , l f k , l g k + 1 , l g k , l , ( 47) b  ln f k , l + 1 g k , l  s = 1 − f k , l g k , l + 1 f k , l + 1 g k , l , (48) b  ln g k , l + 1 f k , l  s = 1 − g k , l f k , l + 1 g k , l + 1 f k , l . (49) W e introd uce a dependent v ar iable transformation w k , l = − 2 ( ln g k , l ) ss , (50) and a discrete hodog raph transform ation x k , l = 2 ka − 2 ( ln g k , l ) s , (51) Inte grable discr etizatio ns for the short wave model 10 then the mesh δ k , l = x k + 1 , l − x k , l = 2 a − 2  ln g k + 1 , l g k , l  s (52) is naturally defined. It then follows  ln g k + 1 , l g k − 1 , l  s = 2 a − 1 2 ( δ k , l + δ k − 1 , l ) . (53) In view of Eq.(47), one obtains f k + 1 , l f k , l g k + 1 , l g k , l = δ k , l 2 a . (54) A substitution into Eq.(46) yields  ln f k + 1 , l f k , l  s = a − 2 a 2 δ k , l , (55) it then follows  ln f k + 1 , l f k − 1 , l  s = 2 a − 2 a 2  1 δ k , l + 1 δ k − 1 , l  . (56) Starting from an alternative form of Eq.(47) 2 a − 2  ln g k + 1 , l g k , l  s = 2 a f k + 1 , l f k , l g k + 1 , l g k , l , (57) we obtain w k + 1 , l − w k , l δ k , l = − 2  ln g k + 1 , l g k , l  ss 2 a − 2  ln g k + 1 , l g k , l  s =  ln f k + 1 , l f k , l g k + 1 , l g k , l  s , (58) by taking logarithmic deriv ative with respect to s . A sh ift from k to k − 1 gives w k , l − w k − 1 , l δ k − 1 , l =  ln f k , l f k − 1 , l g k , l g k − 1 , l  s . (59) Subtracting Eq.(5 9 ) from Eq.(5 8), we obtain w k + 1 , l − w k , l δ k , l − w k , l − w k − 1 , l δ k − 1 , l =  ln f k + 1 , l f k − 1 , l  s −  ln g k + 1 , l g k − 1 , l  s . (60) By using the relations (53) and (56), we finally arrive at w k + 1 , l − w k , l δ k , l − w k , l − w k − 1 , l δ k − 1 , l − 1 2 ( δ k , l + δ k − 1 , l ) + 2 a 2  1 δ k , l + 1 δ k − 1 , l  = 0 . (61) Similar to Eq.( 32), Eq.(61 ) constitutes the first eq uation o f the full-discretiza tion of the SCHE, which can be cast into a simpler form: ∆ 2 w k , l = 1 δ k , l M  δ k , l − 4 a 2 δ k , l  . (62) Next, we seek f or the second equation o f the full-discretization. Recalling (46)–(49), one could obtain x k + 1 , l + 1 − x k , l + 1 x k + 1 , l − x k , l = 2 a − 2  ln g k + 1 , l + 1 g k , l + 1  s 2 a − 2  ln g k + 1 , l g k , l  s =  ln g k + 1 , l + 1 f k + 1 , l  s − 1 b  ln f k , l + 1 g k , l  s − 1 b , (63) Inte grable discr etizatio ns for the short wave model 11 here a shift from l to l + 1 in (47) and a shift from k to k + 1 in (49) are employed . From Eqs.(50), (55) and (58), one can find the following tw o re lations  ln g k + 1 , l + 1 f k + 1 , l  s = − w k + 1 , l − w k , l − 2 a 2 2 δ k , l + 1 4 ( x k + 1 , l + x k , l − 2 x k + 1 , l + 1 ) , (64)  ln f k , l + 1 g k , l  s = w k + 1 , l + 1 − w k , l + 1 + 2 a 2 2 δ k , l + 1 − 1 4 ( x k + 1 , l + 1 + x k , l + 1 − 2 x k , l ) , (65) after some tedio us a lgebraic man ipulations. Su bstituting these two re lations into (63), we finally obtain the second equation of the fully discrete analogue of the SCHE δ k , l + 1 − δ k , l b + 1 4 δ k , l + 1 ( x k + 1 , l + 1 + x k , l + 1 − 2 x k , l ) + 1 4 δ k , l ( x k + 1 , l + x k , l − 2 x k + 1 , l + 1 ) = 1 2 ( w k + 1 , l + 1 + w k + 1 , l − w k , l + 1 − w k , l ) . (66) T aking the continu ous limit b → 0 in time, we have δ k , l + 1 − δ k , l b → d δ k d s , δ k , l + 1 ( x k + 1 , l + 1 + x k , l + 1 − 2 x k , l ) → 0 , δ k , l + 1 δ k , l ( x k + 1 , l + x k , l − 2 x k + 1 , l + 1 ) → 0 , and 1 2 ( w k + 1 , l + 1 + w k + 1 , l − w k , l + 1 − w k , l ) → w k + 1 − w k . Therefo re, one recovers exactly the second equatio n of the semi-discrete SC HE (33). In summary , the fully discr ete an alogue of the SCHE and its deter minant so lution a re giv en as follows: The fully discrete analogue of the SCHE                    w k + 1 , l − w k , l δ k , l − w k , l − w k − 1 , l δ k − 1 , l − 1 2 ( δ k , l + δ k − 1 , l ) + 2 a 2  1 δ k , l + 1 δ k − 1 , l  = 0 , δ k , l + 1 − δ k , l b + 1 4 δ k , l + 1 ( x k + 1 , l + 1 + x k , l + 1 − 2 x k , l ) + 1 4 δ k , l ( x k + 1 , l + x k , l − 2 x k + 1 , l + 1 ) = 1 2 ( w k + 1 , l + 1 + w k + 1 , l − w k , l + 1 − w k , l ) . (67) The determinant solution of the fully discrete SCHE w k , l = − 2 ( ln g k , l ) ss = − 2 ¯ h k , l g k , l − h 2 k , l g 2 k , l , x k , l = 2 ka − 2 ( ln g k , l ) s = 2 ka − 2 h k , l g k , l , δ k , l = x k + 1 , l − x k , l = 2 a f k + 1 , l f k , l g k + 1 , l g k , l , Inte grable discr etizatio ns for the short wave model 12 g k , l =    ψ ( j ) i ( k , l )    1 ≤ i , j ≤ N , f k , l =    ψ ( j − 1 ) i ( k , l )    1 ≤ i , j ≤ N , h k , l = ∂ g k , l ∂ s =          ψ ( 0 ) 1 ( k , l ) ψ ( 2 ) 1 ( k , l ) ψ ( 3 ) 1 ( k , l ) · · · ψ ( N ) 1 ( k , l ) ψ ( 0 ) 2 ( k , l ) ψ ( 2 ) 2 ( k , l ) ψ ( 3 ) 2 ( k , l ) · · · ψ ( N ) 2 ( k , l ) . . . . . . . . . . . . . . . ψ ( 0 ) N ( k , l ) ψ ( 2 ) N ( k , l ) ψ ( 3 ) N ( k , l ) · · · ψ ( N ) N ( k , l )          , ¯ h k , l = ∂ 2 g k , l ∂ s 2 =          ψ ( − 1 ) 1 ( k , l ) ψ ( 2 ) 1 ( k , l ) ψ ( 3 ) 1 ( k , l ) · · · ψ ( N ) 1 ( k , l ) ψ ( − 1 ) 2 ( k , l ) ψ ( 2 ) 2 ( k , l ) ψ ( 3 ) 2 ( k , l ) · · · ψ ( N ) 2 ( k , l ) . . . . . . . . . . . . . . . ψ ( − 1 ) N ( k , l ) ψ ( 2 ) N ( k , l ) ψ ( 3 ) N ( k , l ) · · · ψ ( N ) N ( k , l )          +          ψ ( 0 ) 1 ( k , l ) ψ ( 1 ) 1 ( k , l ) ψ ( 3 ) 1 ( k , l ) · · · ψ ( N ) 1 ( k , l ) ψ ( 0 ) 2 ( k , l ) ψ ( 1 ) 2 ( k , l ) ψ ( 3 ) 2 ( k , l ) · · · ψ ( N ) 2 ( k , l ) . . . . . . . . . . . . . . . ψ ( 0 ) N ( k , l ) ψ ( 1 ) N ( k , l ) ψ ( 3 ) N ( k , l ) · · · ψ ( N ) N ( k , l )          , ψ ( j ) i ( k , l ) = a i , 1 p j i ( 1 − a p i ) − k  1 − b p i − 1  − l e ξ i + a i , 2 ( − p i ) j ( 1 + a p i ) − k  1 + b p i − 1  − l e η i , ξ i = p i − 1 s + ξ i 0 , η i = − p i − 1 s + η i 0 . (68) Note that s is an auxiliary par ameter . By virtue o f s , h k , l and ¯ h k , l can be expressed as h k , l = ∂ s g k , l and ¯ h k , l = ∂ 2 s g k , l , respecti vely , because the auxiliary param eter s works o n elements of the above determinant by ∂ s ψ ( n ) i ( k , l ) = ψ ( n − 1 ) i ( k , l ) . Introd ucing new ind ependen t variables X k , l = x k , l / κ and ˜ b = b / κ , we can includ e th e parameter κ in the full-discre te SC HE (67):                    w k + 1 , l − w k , l δ k , l − w k , l − w k − 1 , l δ k − 1 , l − 1 2 κ 2 ( δ k , l + δ k − 1 , l ) + 2 a 2  1 δ k , l + 1 δ k − 1 , l  = 0 , δ k , l + 1 − δ k , l ˜ b + 1 4 κ 2 δ k , l + 1 ( X k + 1 , l + 1 + X k , l + 1 − 2 X k , l ) + 1 4 κ 2 δ k , l ( X k + 1 , l + X k , l − 2 X k + 1 , l + 1 ) = 1 2 ( w k + 1 , l + 1 + w k + 1 , l − w k , l + 1 − w k , l ) . (69) Similarly , the N -cuspon solution of the full-discre te SCHE (69) with the para meter κ is giv en as follows: w k , l = − 2 ( ln g k , l ) ss = − 2 ¯ h k , l g k , l − h 2 k , l g 2 k , l , X k , l = 2 ka κ − 2 κ ( ln g k , l ) s = 2 ka κ − 2 κ h k , l g k , l , δ k , l = X k + 1 , l − X k , l = 2 a κ f k + 1 , l f k , l g k + 1 , l g k , l , g k , l =    ψ ( j ) i ( k , l )    1 ≤ i , j ≤ N , f k , l =    ψ ( j − 1 ) i ( k , l )    1 ≤ i , j ≤ N , Inte grable discr etizatio ns for the short wave model 13 h k , l = ∂ g k , l ∂ s = 1 κ          ψ ( 0 ) 1 ( k , l ) ψ ( 2 ) 1 ( k , l ) ψ ( 3 ) 1 ( k , l ) · · · ψ ( N ) 1 ( k , l ) ψ ( 0 ) 2 ( k , l ) ψ ( 2 ) 2 ( k , l ) ψ ( 3 ) 2 ( k , l ) · · · ψ ( N ) 2 ( k , l ) . . . . . . . . . . . . . . . ψ ( 0 ) N ( k , l ) ψ ( 2 ) N ( k , l ) ψ ( 3 ) N ( k , l ) · · · ψ ( N ) N ( k , l )          , ¯ h k , l = ∂ 2 g k , l ∂ s 2 = 1 κ 2          ψ ( − 1 ) 1 ( k , l ) ψ ( 2 ) 1 ( k , l ) ψ ( 3 ) 1 ( k , l ) · · · ψ ( N ) 1 ( k , l ) ψ ( − 1 ) 2 ( k , l ) ψ ( 2 ) 2 ( k , l ) ψ ( 3 ) 2 ( k , l ) · · · ψ ( N ) 2 ( k , l ) . . . . . . . . . . . . . . . ψ ( − 1 ) N ( k , l ) ψ ( 2 ) N ( k , l ) ψ ( 3 ) N ( k , l ) · · · ψ ( N ) N ( k , l )          + 1 κ 2          ψ ( 0 ) 1 ( k , l ) ψ ( 1 ) 1 ( k , l ) ψ ( 3 ) 1 ( k , l ) · · · ψ ( N ) 1 ( k , l ) ψ ( 0 ) 2 ( k , l ) ψ ( 1 ) 2 ( k , l ) ψ ( 3 ) 2 ( k , l ) · · · ψ ( N ) 2 ( k , l ) . . . . . . . . . . . . . . . ψ ( 0 ) N ( k , l ) ψ ( 1 ) N ( k , l ) ψ ( 3 ) N ( k , l ) · · · ψ ( N ) N ( k , l )          , ψ ( j ) i ( k , l ) = a i , 1 p j i ( 1 − a p i ) − k  1 − b p i − 1  − l e ξ i + a i , 2 ( − p i ) j ( 1 + a p i ) − k  1 + b p i − 1  − l e η i , ξ i = p i − 1 s + ξ i 0 , η i = − p i − 1 s + η i 0 . (70) 5. Concluding remarks In the present p aper, bilinear equations and th e determinant solu tion of the SCHE are o btained from the two-red uction of 2 DTL equa tions. Based on this fact, in tegrable semi-an d f ull- discrete analog ues of the SCHE are constructed. The N -soliton solutions of both continu ous and d iscrete SCHEs ar e for mulated in the form of the Casorati determin ant. Note that the short pulse equation was also obtained from the two-reduction of the 2DTL equation [19]. Finally , we remark that the presen t paper is one of our series of work in an attemp t of obtaining in tegrable discrete analo gues for a class of integrable nonlien ar PDEs whose solutions possess singularities such as peakon, cuspon or loop soliton solutions. 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