Covariant hodograph transformations between nonlocal short pulse models and AKNS (−1) system
aa r X i v : . [ n li n . S I] D ec Covariant hodograph transformations between nonlocal shortpulse models and AKNS( −
1) system
Kui Chen, Shimin Liu, Da-jun Zhang ∗ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
October 25, 2018
Abstract
The paper presents hodograph transformation between nonlocal short pulse models andthe first member in the AKNS negative hierarchy (AKNS( − −
1) that are connected via hodograph transformation are covariantin nonlocal reductions.
Key Words: nonlocal hodograph transformation, AKNS( − In 2004 a short pulse model [1] q xt + q + 12 ( q ) xx = 0 (1)was derived to describe the propagation of short optical pulses in nonlinear media, and fromthen on it has received considerable attention. Recently, a complex short pulse equation [2, 3] q xt + q + 12 ( | q | q x ) x = 0 (2)and its coupled form q i,xt + q i + 12 [( | q | + | q | ) q i,x ] x = 0 , i = 1 , φ x = (cid:18) λ λqλr − λ (cid:19) φ. (4)It was also found the SP equation (1) can be transformed into the sine-Gorden (sG) equationthrough suitable hodograph transformation [9]. Since the sG equation is closely related to thefirst negative order Ablowitz-Kaup-Newell-Suger (AKNS( −
1) for short) system, q xt − q∂ − x ( qr ) t = q, r xt − r∂ − x ( qr ) t = r, (5) ∗ Corresponding author. Email: djzhang@staff.shu.edu.cn −
1) system (e.g. [2,3,11,12]). By hodograph transformations [3,12] the SP type equationsare related to the following system q xt − qs = 0 , s x + ( | q | ) t = 0 . (6)The above system (known as AB system in literature) was first derived by Pedlosky [13] to modelfinite-amplitude baroclinic wave packets evolution in a marginally stable or unstable baroclinicshear flow. Although later Pedlosky’s derivation was pointed out lack of considering sufficientboundary conditions [14], it is still significant in inviscid case (cf. [15]) and it is integrable withan explicit Lax pair and easily transformed to the sG equation when q and s are real [15]. Somecoupled integrable dispersionless systems [16–18] (known as CD or CID systems for short inliterature) proposed by Konno et al can be viewed as generalizations of (6). In fact, either ABor CD systems are exactly the AKNS( −
1) system or its reductions.The purpose of this paper is to describe hodograph transformations between multi-componentnonlocal SP models and AKNS( − − −
1) system, including scalar and matrix (vector) forms of the AKNS( − −
1) systems. Sec.4 contributes conclusion. −
1) and its vector form − Let us recall some results of the AKNS( −
1) system in [20]. Start from the well known AKNSspectral problem [21] (cid:18) ϕ ϕ (cid:19) x = M (cid:18) ϕ ϕ (cid:19) , M = (cid:18) − λ qr λ (cid:19) , (7a)coupled with the time evolution (cid:18) ϕ ϕ (cid:19) t = N (cid:18) ϕ ϕ (cid:19) , N = (cid:18) A BC − A (cid:19) , (7b)where q = q ( x, t ) and r = r ( x, t ) are potential functions, and λ is the spectral parameter. Fromzero-curvature equation M t − N x + [ M, N ] = 0, expanding N suitably into polynomial of λ , onecan derive the AKNS hierarchy (cid:18) qr (cid:19) t = K n = L n (cid:18) − qr (cid:19) , ( n = 1 , , . . . ) , (8)where L is the recursion operator, given by L = σ ( ∂ x − u∂ − x u T σ ) , (9)2ith u = ( q, r ) T , σ = ( ), σ = (cid:0) − (cid:1) , ∂ x = ∂∂x and ∂ − x · = 12 ( Z x −∞ − Z + ∞ x ) · d x. (10)The AKNS hierarchy can be extended to the negative direction by expanding N into polynomialof 1 /λ , and the hierarchy is expressed as (cf. [20]) L n (cid:18) qr (cid:19) t = (cid:18) − qr (cid:19) , ( n = 1 , , · · · ) . (11)When n = 1, the equation in (11) reads q xt − q∂ − x ( qr ) t = q, (12a) r xt − r∂ − x ( qr ) t = r, (12b)denoted by AKNS( −
1) for short. The corresponding (7b) is (cid:18) ϕ ϕ (cid:19) t = N (cid:18) ϕ ϕ (cid:19) , N = − λ (cid:18) ∂ − x ( qr ) t − q t r t − − ∂ − x ( qr ) t (cid:19) . (13)(7a) and (13) constitute a Lax pair of the AKNS( −
1) system (12). If we employ an auxiliaryfunction s ( x, t ) that satisfies s ( x, t ) = ∂ − x ( qr ) t + s , ( s = s ( x, t ) | | x |→∞ = 12 ) , (14)(13) is written as (cid:18) ϕ ϕ (cid:19) t = − λ (cid:18) s − q t r t − s (cid:19) (cid:18) ϕ ϕ (cid:19) . (15)With the help of s , (12) is alternatively written as q xt = 2 qs, r xt = 2 rs, s x = ( qr ) t . (16)In [16–18] Konno et al . switching x and t , considered (15) as a new spectral problem, derivedso called coupled integrable dispersionless (CD or CID for short) systems, i.e. (16) and itsreductions, with x and t switched. The AKNS( −
1) system (12) admits the following local reductions. One is r = σq where σ = ± q xt − σq∂ − x ( q ) t = q, (17)The other is r = σq ∗ where ∗ means complex conjugate, yielding q xt − σq∂ − x ( | q | ) t = q, (18)where | q | = qq ∗ . In terms of (16) the above equations are written as q xt − qs = 0 , s x − σqq t = 0 (19)and q xt − qs = 0 , s x − σ ( | q | ) t = 0 , (20)respectively. (20) was first derived by Pedlosky in 1972 [13] to model finite-amplitude baroclinicwave packets evolution in a marginally stable or unstable baroclinic shear flow. Although3edlosky’s derivation was pointed out lack of considering sufficient boundary condition [14], itis still significant in inviscid case (cf. [15]). Note that with asymptote s it is easy to obtainfrom (20) a normalisation condition (cf. [13, 15]) s − σ | q t | = s .Besides, some nonlocal reductions are r ( x, t ) = − σq ( − x, − t ) which yields (cf. [22]) q xt ( x, t ) − q ( x, t ) s ( x, t ) = 0 , s x ( x, t ) − σ [ q ( x, t ) q ( − x, − t )] t = 0 , (21) r ( x, t ) = − σq ∗ ( − x, − t ) which yields q xt ( x, t ) − q ( x, t ) s ( x, t ) = 0 , s x ( x, t ) − σ [ q ( x, t ) q ∗ ( − x, − t )] t = 0 , (22) r ( x, t ) = − σq ( − x, t ) which yields q xt ( x, t ) − q ( x, t ) s ( x, t ) = 0 , s x ( x, t ) − σ [ q ( x, t ) q ( − x, t )] t = 0 , (23)and r ( x, t ) = − σq ( x, − t ) which yields q xt ( x, t ) − q ( x, t ) s ( x, t ) = 0 , s x ( x, t ) − σ [ q ( x, t ) q ( x, − t )] t = 0 , (24)where σ = ± Here we must note that in the system (16) function s is merely auxiliary, therefore to bilinearizethe AKNS( −
1) system, we always work on (12) rather than (16). With transformation q = gf , r = hf , (25)(12) is bilinearised as [20] D x f · f = − gh, (26a) D x D t g · f = gf, (26b) D x D t h · f = hf, (26c)where we have made use of (26a) to get ∂ − x ( qr ) = f x f , and D is Hirota’s bilinear operatordefined by [23] D mt D nx f ( t, x ) · g ( t, x ) = ( ∂ t − ∂ t ) m ( ∂ x − ∂ x ) n f ( t, x ) g ( t , x ) | t = t,x = x . (27)By the transformation q = g/f (17) is bilinearised as D x f · f = − σg , D x D t g · f = gf, (28)and when σ = −
1, taking q = gf , f ∗ = f, (29)(18) is bilinearised as D x f · f = 2 gg ∗ , D x D t g · f = gf. (30)The above two bilinear forms can be reduced from (26) (cf. [20]) and for (28) the two cases of σ = ± g → ig .However, for (18) with σ = 1, one may consider the following bilinear form, still through(29), D x f · f = − gg ∗ + λf , (31a) D x D t g · f = gf, (31b)4here λ >
0. Employing standard procedure of Hirota’s methods, expanding f, g as f ( x, t ) = 1 + ∞ X n =1 ε n f ( n ) , g ( x, t ) = g (0) (1 + ∞ X n =1 ǫ n g ( n ) ) , by calculation we give the following multi-soliton expression, g = g X µ =0 , exp (cid:20) N X j =1 µ j ( ξ j + 2 iθ j ) + N X ≤ i 1) and vector form The AKNS spectral problem (7) can be extended to matrix form. Starting from φ x = M φ, M = (cid:18) − ηI n × n QR T ηI m × m (cid:19) , (33a) φ t = N φ, N = (cid:18) A BC T D (cid:19) , (33b)where potential matrices Q, R ∈ C n × m [ x, t ], I n × n is the n th-order unit matrix, A, B, C, D arerespectively n × n , n × m , n × m and m × m undetermined matrices, similar to the scalar case,one can derive matrix AKNS hierarchy [25], (cid:18) QR (cid:19) t = L n (cid:18) − QR (cid:19) , n = 1 , , · · · , (34)where L is the recursive operator defined by L (cid:18) AB (cid:19) = (cid:18) − A x + [ ∂ − x AR T + ∂ − x QB T ] Q + Q [ ∂ − x R T A + ∂ − x B T Q ] B x − [ ∂ − x RA T + ∂ − x BQ T ] R − R [ ∂ − x A T R + ∂ − x Q T B ] (cid:19) , (35)in which A, B ∈ C n × m , and ∂ − x is defined as (10). If expanding ( B, C ) into a polynomial of1 /η , one can obtain negative matrix AKNS hierarchy L n (cid:18) QR (cid:19) t = (cid:18) − QR (cid:19) , n = 1 , , · · · , (36)5he first member of which is Q xt − [ ∂ − x ( QR T ) t ] Q − Q [ ∂ − x ( R T Q ) t ] = Q, (37a) R xt − [ ∂ − x ( RQ T ) t ] R − R [ ∂ − x ( Q T R ) t ] = R. (37b)Its Lax pair is provided by (33) with N = − η (cid:18) ∂ − x ( QR T ) t + I n × n − Q t R Tt − ∂ − x ( R T Q ) t − I m × m (cid:19) . (38)(37) is a 2 nm -component system. To get vector-type reductions, we employ an eleganttechnique which is first introduced by Tsuchida and Wadati [26]. Suppose q and r are s -ordervectors q = ( q , q , · · · , q s ) T , r = ( r , r , · · · , r s ) T , (39)and q i , r i are functions of ( x, t ). Construct a sequence of matrices Q ( j ) , R ( j ) as the following, Q (1) = R (1) T = (cid:18) q r (cid:19) , (40a) Q ( j +1) = R ( j +1) T = Q ( j ) q j +1 I j × j r j +1 I j × j − R ( j ) T ! . (40b)Note that each Q ( j ) and R ( j ) are 2 j × j matrices, and QR T = R T Q = Q T R = RQ T = q T r = s X j =1 q j r j I s × s , (41)where we have taken Q = Q ( s ) and R = R ( s ) . Then, with Q and R defined above and makinguse of expression (41), system (37) is written as q xt − q ∂ − x ( q T r ) t = q , (42a) r xt − r ∂ − x ( q T r ) t = r , (42b)which is a vector version of AKNS( − q xt = 2 ρ q , (43a) r xt = 2 ρ r , (43b) ρ x − ( q T r ) t = 0 , (43c)where ρ = ∂ − x ( q T r ) t + ρ , ( ρ = ρ ( x, t ) | ( | x |→ + ∞ ) = 12 ) . (44)Note that ρ is a scalar and (43c) is a conservation law for (42). As in Sec.2.1.2, we can have similar reductions either on (37) or (42). Consider r ( x, t ) = W q ( x, t ) , (45a)where W = W T ∈ R s × s and det[ W ] = 0. As discussion in [27], since W can be diagonalisedwe can always normalized W to be W = diag { λ , λ , · · · , λ s } , (45b) There are more cases for choice of W but here we only consider symmetric nonsingular matrix W which hasreal nonzero eigenvalues. λ j ∈ {− , } and λ ≤ λ ≤ · · · ≤ λ s . Thus, the canonical form of the reduced realequations from (42) is q xt = 2 ρ q , ρ x − s X j =1 λ j ( q j ) t = 0 . (46)Complex reduction is available by taking r ( x, t ) = H q ∗ ( x, t ) , H = H † , det[ W ] = 0 , (47a)i.e. H is a nonsingular Hermitian matrix. As discussion in [27], we only need to consider H = diag { λ , λ , · · · , λ s } , (47b)where λ j ∈ {− , } and λ ≤ λ ≤ · · · ≤ λ s . The canonical form of thr reduced complexequations from (42) read q xt = 2 ρ q , ρ x − s X j =1 λ j ( | q j | ) t = 0 . (48)The cases of λ j ≡ λ j ≡ − q xt = 2 ρ q , ρ x ( x, t ) − s X j =1 λ j ( q j ( x, t ) q j ( − x, − t )) t = 0 (49)with reverse-( x, t ) reduction r ( x, t ) = W q ( − x, − t ) , (50)to q xt = 2 ρ q , ρ x ( x, t ) − s X j =1 λ j ( q j ( x, t ) q ( − x, t )) t = 0 (51)with reverse- x reduction r ( x, t ) = W q ( − x, t ) , (52)to q xt = 2 ρ q , ρ x ( x, t ) − s X j =1 λ j ( q j ( x, t ) q ( x, − t )) t = 0 (53)with reverse- t reduction r ( x, t ) = W q ( x, − t ) , (54)where W = W T but we take the form (45b) with λ j ∈ {− , } and λ ≤ λ ≤ · · · ≤ λ s .For nonlocal complex reductions, (43) can be respectively reduced to (only consider canonicalcases) q xt = 2 ρ q , ρ x ( x, t ) − s X j =1 λ j ( q j ( x, t ) q ∗ j ( − x, − t )) t = 0 (55)with reverse-( x, t ) reduction r ( x, t ) = H q ∗ ( − x, − t ) , (56)to q xt = 2 ρ q , ρ x ( x, t ) − s X j =1 λ j ( q j ( x, t ) q ∗ j ( − x, t )) t = 0 (57)with reverse- x reduction r ( x, t ) = H q ∗ ( − x, t ) , (58)7o q xt = 2 ρ q , ρ x ( x, t ) − s X j =1 λ j ( q j ( x, t ) q ∗ j ( x, − t )) t = 0 (59)with reverse- t reduction r ( x, t ) = H q ∗ ( x, − t ) , (60)where H = H † but we take the form (47b) with λ j ∈ {− , } and λ ≤ λ ≤ · · · ≤ λ s .Note that mixed-local-nonlocal reduction introduced in a recent paper [28] could be alsoavailable for vector AKNS( − 1) system. We present bilinear forms of the elementary vector AKNS( − 1) (42). Through transformation q i = g i f , r i = h i f , ( i = 1 , , · · · , s ) , (61)(42) can be rewritten as the following bilinear equations D x f · f = − s X j =1 g j h j , (62a) D x D t h i · f = h i f, D x D t g i · f = g i f, ( i = 1 , , · · · , s ) . (62b)For the reduced real system (46), under (61) it has a bilinear form D x f · f = − s X j =1 λ j g j , (63a) D x D t g i · f = g i f, ( i = 1 , , · · · , s ) , (63b)where λ j ∈ { , − } . Note that this system has been derived in [12].For the reduced complex system (48), through transformation (61) where f is real but g i are complex, we have a bilinear form D x f · f = − s X j =1 λ j | g j | + λf , (64a) D x D t g i · f = g i f, ( i = 1 , , · · · , s ) , (64b)where λ j ∈ { , − } . When λ j ≡ − λ = 0, solutions can be presented in terms ofPfaffian [3]. Other cases are left for further investigation. One special example can be foundin [6] with an alternative bilinear form. It is known that the scalar SP equations are linked to the AKNS( − 1) system (cf. [3, 12]) byhodograph transformations. In the following we extend these links to vector and nonlocal cases. The conservation law (43c) for the vector AKNS( − 1) (42) allows us to introduce a new variable z = z ( x, t ) which is defined throughd z ( x, t ) = ρ d t + q T r d x. (65a)8n addition, a second variable y = y ( x, t ) is introduced throughd y ( x, t ) = − d x. (65b)(65a) and (65b) define a hodograph transformation which is then written as z t = ρ, z x = q T r , y x = − , y t = 0 . (66)With the new coordinate ( y, z ) let us define new potentials { u , v } by u = ( u , u , · · · , u s ) T = u ( y, z ) = u ( y ( x, t ) , z ( x, t )) = q ( x, t ) , (67a) v = ( v , v , · · · , v s ) T = v ( y, z ) = v ( y ( x, t ) , z ( x, t )) = r ( x, t ) . (67b)It then arises from the AKNS( − 1) (43) that u yz + 2 u − [( u T v ) u z ] z = 0 , v yz + 2 v − [( u T v ) v z ] z = 0 . (68)This is an unreduced vector SP-type system, equivalent to the matrix form U yz + 2 U − ( U U z ) z = 0 , (69)which is integrable with a Lax pair (cf. Eqs.(49,50) in [3])Φ z = (cid:18) λI λU z − λU z − λI (cid:19) Φ , (70a)Φ y = (cid:18) λU − I λ λU U z + U − λU U z + U − λU + I λ (cid:19) Φ , (70b)where the size of matrices U and I is 2 s × s , and U is constructed by using u , v as (40) andholds the property U = u T v I .With respect to reductions of (68), considering v ( y, z ) = W u ( y, z ) (71)where W = W T ∈ R s × s but can be normalized to (45b) with λ j ∈ {− , } and λ ≤ λ ≤ · · · ≤ λ s , then (68) is reduced to u yz + 2 u − (cid:18) u z s X j =1 λ j u j (cid:19) z = , (72)which is the vector SP equation. Note that if one considers a general symmetric matrix W =( w ij ) s × s with w ii = 0, (68) gives u i,yz + 2 u i − (cid:18) u i,z X ≤ j 1) system (68) admits the following two nonlocal reductions. One isreverse-( y, z ) reduction v ( y, z ) = W u ( − y, − z ) , ( W given by (45b)) , (75)which yields a nonlocal vector SP system u yz ( y, z ) + 2 u ( y, z ) − (cid:16) u z ( y, z ) s X j =1 λ j u j ( y, z ) u j ( − y, − z ) (cid:17) z = 0 . (76)The other is reverse-( y, z ) complex reduction v ( y, z ) = H u ∗ ( − y, − z ) , ( H given by (47b)) , (77)which gives rise to a nonlocal vector complex SP system u yz ( y, z ) + 2 u ( y, z ) − (cid:16) u z ( y, z ) s X j =1 λ j u j ( y, z ) u ∗ j ( − y, − z ) (cid:17) z = 0 . (78)Note that in the above reductions it originally admits W = W T and H = H † but we onlypresent normalized cases.Replaced y by − iy for (68) we get an alternative vector AKNS( − 1) system i u yz + 2 u − [( u T v ) u z ] z = 0 , i v yz + 2 v − [( u T v ) v z ] z = 0 . (79)It admits two more nonlocal reductions: v ( y, z ) = H u ∗ ( − y, z ) , ( H = H † ) , (80a) v ( y, z ) = H u ∗ ( y, − z ) , ( H = H † ) , (80b)which respectively lead to reduced vector equations: i u yz ( y, z ) + 2 u ( y, z ) − (cid:18) u z ( y, z ) s X j =1 λ j u j ( y, z ) u ∗ j ( − y, z ) (cid:19) z = 0 , (81a) i u yz ( y, z ) + 2 u ( y, z ) − (cid:18) u z ( y, z ) s X j =1 λ j u j ( y, z ) u ∗ j ( y, − z ) (cid:19) z = 0 , (81b)where W and H are already normalized to diagonal cases. Scalar cases of nonlocal complex SPsystems obtained in this subsection were also presented in [29]. Next let us investigate hodograph links between the nonlocal SP system and the nonlocalAKNS( − 1) system. Since there are independent variables and integrating operation involved inhodograph transformations, one has to carefully analyze changes of these variables in nonlocalreductions.First, we specify integration operator ∂ − x defined in (10). From the hodograph transforma-tion (66) we have y ( x, t ) = − x, z ( x, t ) = ∂ − x q T ( x, t ) r ( x, t ) + z , ( z = lim | x |→∞ z ( x, t )) , (82)10onsider the change of z resulting from nonlocal reduction (50) with a general real symmetricmatrix W . By analysis one can find z ( − x, − t ) = 12 ( Z − x −∞ − Z + ∞− x ) q T ( x, − t ) r ( x, − t )d x = − 12 ( Z x + ∞ − Z −∞ x ) q T ( − x ′ , − t ) r ( − x ′ , − t )d x ′ = − 12 ( Z x −∞ − Z + ∞ x ) q T ( − x ′ , − t ) W q ( x ′ , t )d x ′ = − z ( x, t ) , (83a)and y ( − x, − t ) = − y ( x, t ) , (83b)where we have taken z = 0. It is remarkable that ∂ − x = R − x −∞ · d x or R + ∞− x · d x cannot lead torelation (83a). Further we have W u ( − y, − z ) = W u ( − y ( x, t ) , − z ( x, t )) = W u ( y ( − x, − t ) , z ( − x, − t ))= W q ( − x, − t ) = r ( x, t )= v ( y, z ) , (84)which is nothing but the reverse-( y, z ) reduction (75).Eq.(83) records the change of ( y, z ) resulting from a reverse change of ( x, t ) in hodographlink. Based on such a relation it is found nonlocal reductions on ( u , v ) and ( q , r ) are covariant,as shown in (84). Employing similar analysis we examine all the nonlocal reductions of vectorSP systems. Below we list out independent variables changes in nonlocal reductions. AKNS( − 1) ( y, z ) correspondence in (66) SP(49) with (50) y ( − x, − t ) = − y ( x, t ) , z ( − x, − t ) = − z ( x, t ) (76) with (75)(51) with (52) y ( − x, t ) = − y ( x, t ) , z ( − x, t ) = − z ( x, t ) (76) with (75)(53) with (54) y ( x, − t ) = y ( x, t ) , z ( x, − t ) = z ( x, t ) (72) with (71)(55) with (56) y ( − x, − t ) = − y ( x, t ) , z ( − x, − t ) = − z ( x, t ) (78) with (77)(57) with (58) y ( − x, t ) = − y ( x, t ) , z ( − x, t ) = − z ( x, t ) (78) with (77)(59) with (60) y ( x, − t ) = y ( x, t ) , z ( x, − t ) = z ( x, t ) (74) with (73)Table 1: ( y, z ) correspondence in nonlocal hodograph transformation (66) Let us sort out links between the integrable models involved in the paper. Pedlosky’s ABsystem and Konno et al ’s CD systems are essentially the same as the AKNS( − 1) system (upto reductions). The AKNS( − 1) system can be considered as a nonpotential form of the sGequation. Direct links of AB and CD system to the sG equation were independently presentedin [15] and [30, 31]. The SP equation(s) is related to the WKI spectral problem (4) but canbe derived from the AKNS( − 1) system via hodograph transformations. Thus, we can havea clear map for the relations of the AB, CD, sG, SP and AKNS( − 1) system. In this papersuch relations are extended to multi-component and nonlocal cases. Among these systems theAKNS( − 1) is in a central position and the fruitful results of the AKNS spectral problem canbe used to investigate the others. Usually we take z = 0 but it is not necessary to be zero for the case when z ( − x, t ) = z ( x, t ) and z ( x, − t ) = z ( x, t ). z with respect to reverse-( x, t ), e.g.(83a), which further leads to covariant relation of ( u , v ) and ( q , r ) in nonlocal reductions. Withnonlocal hodograph links one can study solutions of nonlocal SP models from those of nonlocalAKNS( − 1) obtained in [22]. Finally, as for solutions to nonlocal systems, we would like tomention two direct approaches. One is to make use of independent coordinate transformation[29], the other is to use solutions of unreduced systems [22, 32]. Acknowledgments This work was supported by the NSF of China [grant numbers 11371241, 11631007].