On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields
aa r X i v : . [ n li n . S I] J un On the relationship between nonlinear equations integrable by themethod of characteristics and equations associated with commutingvector fields.
A. I. Zenchuk , § Institute of Problems of Chemecal Physics, Russian Academy of Sciences, Chernogolovka,Moscow reg., 142432
November 3, 2018
Abstract
It was shown recently that Frobenius reduction of the matrix fields reveals interestingrelations among the nonlinear Partial Differential Equations (PDEs) integrable by theInverse Spectral Transform Method ( S -integrable PDEs), linearizable by the Hopf-Colesubstitution ( C -integrable PDEs) and integrable by the method of characteristics ( Ch -integrable PDEs). However, only two classes of S -integrable PDEs have been involved:soliton equations like Korteweg-de Vries, Nonlinear Shr¨odinger, Kadomtsev-Petviashviliand Davey-Stewartson equations, and GL ( N, C ) Self-dual type PDEs, like Yang-Millsequation. In this paper we consider the simple five-dimensional nonlinear PDE from an-other class of S -integrable PDEs, namely, scalar nonlinear PDE which is commutativitycondition of the pair of vector fields. We show its origin from the (1+1)-dimensionalhierarchy of Ch -integrable PDEs after certain composition of Frobenius type and differ-ential reductions imposed on the matrix fields. Matrix generalization of the above scalarnonlinear PDE will be derived as well. We continue study of the relationship among nonlinear integrable Partial Differential Equa-tions (PDEs) with different integrability algorithms in the spirit of refs.[1, 2], where the blockFrobenious Reduction (FR) of the matrix fields has been introduced for this purpose. First ofall, we recall that there are several classes of completely integrable nonlinear PDEs dependingon their integration algorithm. Most famous classes are following.1. Nonlinear PDEs linearizable by some direct substitution (or C -integrable PDEs) [3, 4,5, 6, 7, 8]. Mostly remarkable nonlinear PDEs of this type are PDEs linearizable by theHopf-Cole substitution [9] and by its multidimensional generalization [10].2. Scalar equations integrable by the method of characteristics [11] (or Ch -integrable PDEs)and their matrix generalizations [12, 13].3. Nonlinear PDEs integrable by the inverse spectral transform method [14, 15, 16] andby the dressing method [17, 18, 19, 20, 21] ( S -integrable PDEs). We underline threesubclasses of these equations:(a) Soliton equations in (1+1)-dimensions, such as Korteweg-de Vries (KdV) [14, 22] andNonlinear Shr¨odinger (NLS) [23] equations , and soliton (2+1)-dimensional equa-tions, such as Kadomtsev-Petwiashvili (KP) [24] and Davey-Stewartson (DS) [25]equations. Hereafter we call this subclass S -integrable PDEs.1b) Self-dual type PDEs having instanton solutions, like Self-dual Yang-Mills equation(SDYM) [26, 27] and its multidimensional generalizations. This subclass will becalled S -integrable PDEs.(c) PDEs assotiated with commutativity of vector fields [28, 29, 30, 31, 32, 33, 34].These equations may be either scalar or vector ( S -integrable PDEs).It was shown in [2] that C and Ch -integrable matrix PDEs supplemented by the Frobeniusreduction of the matrix fields lead to the proper class of S -integrable PDEs. Thus, matrix PDEslinearizable by the Hopf-Cole substitution generate (2+1)-dimensional S -integrable PDEs;matrix equations integrable by the method of characteristics generate S -integrable PDEs or,in the particular case of constant characteristics, (1+1)-dimensional S -integrable PDEs.However, the technique developed in [2] does not allow one to involve S -integrable PDEsinto consideration. In this paper we consider the particular example of S -integrable PDEswhich may be incorporated into the above mentioned algorithm with minor modifications.Namely, we consider the following representative of S -integrable PDEs [34]: u z t − u z t + u z u z x − u z u z x = 0 , (1)whose Lax pair reads: ψ t i ( λ ; ~x ) + λψ z i ( λ ; ~x ) + u z i ( ~x ) ψ x ( λ ; ~x ) = 0 , i = 1 , , (2)where ψ is a scalar spectral function, λ is a spectral parameter and ~x = ( x, z , z , t , t ) is a setof all independent variables of nonlinear PDE. We show (for the restricted manifold of solutionsto the eq.(1)) that eq.(1) originates from the S -integrable GL ( N S , C ) SDYM ( N S is somepositive integer) with independent variables ( z , z , t , t ) after two reductions imposed on thematrix field:1. the Differential Reduction introducing one more variable x (DR( x ))2. the Frobenius Type Reduction (FTR) (or, in particular, just Frobenius reduction).Matrix generalization of eq.(1) will be introduced as well. Remember that S -integrable GL ( N S , C ) SDYM originates from the (1+1)-dimensional Ch -integrable hierarchy of nonlinearPDEs through the Frobenius reduction [2]. All in all, the diagram in Fig.(1) illustrates the chainof transformations leading from the (1+1)-dimensional matrix N × N (where N = 2 N n M ) Ch -integrable hierarchy to the scalar S -integrable PDE (1).2ig.1 The chain of transformations from the (1+1)-dimensional Ch -integrable hierarchy to the five-dimensional S integrable PDE (1). Here N = 2 N n M , N S = 2 n M , I N S and I M are N S - and M -dimensional identity matrices respectively, W is N × N matrix, W ( l ) are 2 n M × n M matrices v ( i ) and w ( i ) are n M × n M matrices, v ( i ; l ) and w ( i ; l ) are M × M matrices, p ( i ) and q ( i ) are definedby eqs.(29) Note that the reduction z = x, t = t, t = − z = y (3)reduces eq.(1) into the following one u xt + u yy + u x u xy − u y u xx = 0 , (4)which has been studied in [35, 36, 37, 38].The structure of this paper is following. In Sec.2 we recall one of the results of [2]. Namely,we describe a version of the dressing method relating the (1+1)-dimensional Ch -integrablehierarchy of nonlinear PDEs with the four-dimensional S -integrable GL ( N S , C ) SDYM. Thelater admits differential reduction of the special type introduced in Sec.3. After that, using FTRwe derive assotiated five-dimensional matrix system of two matrix nonlinear PDEs in Sec.4.Scalar case of this equation yelds the S -integrable eq.(1). Solution spaces to the nonlinearPDEs derived in Secs.3 and 4 will be considered in Sec.5. Conclusions are given in Sec.6. GL ( N S , C ) SDYM from the hierarchy ofmatrix
C h -integrable nonlinear PDE.
In this section we describe briefly the algorithm relating the simplest (1+1)-dimensional hierar-chy of the matrix N × N Ch -integrable nonlinear PDEs from one side and GL ( N S , C ) SDYMfrom another side. Here N = N N S and N is an arbitrary positive integer. We start with thefollowing linear equation [1, 2] χ Λ =
W χ (5)3ith χ given as a solution to the following linear PDEs: χ t n + χ z n Λ = 0 , n = 1 , , . . . . (6)Here χ and W are 2 N n M × N n M matrix functions and Λ is a diagonal 2 N n M × N n M matrix function. Parameters M , n and N are arbitrary positive integers. Meaning of theseparameters is clarified in Fig.1. Eq.(5) must be viewed as algebraic equation for W with given χ and Λ: W = χ Λ χ − . (7)Note that the diagonal matrix Λ is not constant in general. In fact, compatibility condition ofthe eqs.(5) and (6) reads χ (Λ t n + Λ z n Λ) = ( W t n + W z n W ) χ, n = 1 , , . . . , (8)which suggests us the following nonlinear PDEs for ΛΛ t n + Λ z n Λ = 0 , n = 1 , , . . . (9)and for W W t n + W z n W = 0 , n = 1 , , . . . . (10)Eq. (9) is integrable by the method of characteristics [12, 1, 2]. Now both χ and Λ are fixed,so that W (solution to eq.(10)) may be found using eq.(7).Similar to [2], in order to derive SDYM we take W in the Frobenius form: W = W (1) W (2) · · · W ( N − W ( N ) I Mn Mn · · · Mn Mn Mn I Mn · · · Mn Mn · · · · · · · · · · · · · · · Mn Mn · · · I Mn Mn , (11)where I J and 0 J are J × J identity and zero matrices respectively, W ( i ) are 2 n M × n M matrix functions. Let matrix parameter Λ be given in the following block-diagonal formΛ = diag(Λ (1) , . . . , Λ ( N ) ) , (12)where Λ ( j ) are 2 n M × n M diagonal matrices. Substituting (11) and (12) into eq.(5) weobtain that χ must have the following block structure: χ = χ (1) · · · χ ( N ) χ (1) (Λ (1) ) − · · · χ ( N ) (Λ ( N ) ) − · · · · · · · · · χ (1) (Λ (1) ) − N +1 · · · χ ( N ) (Λ ( N ) ) − N +1 , (13)where χ ( i ) are 2 n M × n M matrix functions. In turn, 2 N n M × N n M matrix equation(5) reduces to the following set of 2 n M × n M matrix equations: χ ( j ) Λ ( j ) = N X i =1 W ( i ) χ ( j ) (Λ ( j ) ) − i +1 , j = 1 , . . . , N , (14)4hile eqs.(6) and (9) yield χ ( j ) t n + χ ( j ) z n Λ ( j ) = 0 , (15)Λ ( j ) t n + Λ ( j ) z n Λ ( j ) = 0 , j = 1 , . . . , N , n = 1 , , . . . . (16)Then compatibility condition of eqs. (14) and (15) yields the following chain [1, 2]: W ( i ) t n + W (1) z n W ( i ) + W ( i +1) = 0 , i = 1 , . . . , N , W ( N +1) = 0 , n = 1 , , . . . , (17)which may be obtained directly after substitution eq.(11) into eq.(10). Putting i = 1 andeliminating W (2) using two equations (17) with n = 1 , GL ( N S , C ) SDYM, N S =2 n M : W (1) z n t m − W (1) z m t n + W (1) z m W (1) z n − W (1) z n W (1) z m = 0 . (18) χ and assotiated system ofnonlinear PDEs Introduce one more reduction. Namely, let matrices χ ( j ) and Λ ( j ) have the following blockstructures: χ ( j ) = (cid:18) Ψ (2 j − Ψ (2 j ) Ψ (2 j − x Ψ (2 j ) x (cid:19) , Λ ( j ) = diag( ˜Λ (2 j − , ˜Λ (2 j ) ) , j = 1 , . . . , N , (19)where Ψ ( m ) are n M × n M matrix functions and ˜Λ ( m ) are n M × n M diagonal matrix functions.Eqs.(15) and (16) yield:Ψ ( m ) t n + Ψ ( m ) z n ˜Λ ( m ) = 0 , (20)˜Λ ( m ) t n + ˜Λ ( m ) z n ˜Λ ( m ) = 0 , m = 1 , . . . , N , n = 1 , , . . . . (21) x -dependence of Ψ ( m ) is introduced by the following second order PDE: E (0) := Ψ ( m ) xx = a Ψ ( m ) ˜Λ ( m ) + ν Ψ ( m ) x + µ Ψ ( m ) , m = 1 , . . . , N , (22)where a , ν and µ are n M × n M diagonal matrix parameters. These parameters must beconstant and ˜Λ ( m ) must be independent on x . In fact, compatibility condition of (20) and (22)yields, along with (21), the following conditions:˜Λ ( m ) x = 0 , m = 1 , . . . , N , a, ν, µ are constant diagonal matrices , m = 1 , . . . , N . (23)The block structures of χ ( j ) and Λ ( j ) (19) suggest us the relevant block structures of W ( j ) : W ( j ) = (cid:18) w ( j ) v ( j ) p ( j ) q ( j ) (cid:19) , (24)where w ( j ) , v ( j ) , q ( j ) and p ( j ) are n M × n M matrix functions. Now set of 2 n M × n M eqs.(14) may be written as two n M × N n M equations: E (1) := ΨΛ = N X i =1 (cid:16) v ( i ) Ψ x + w ( i ) Ψ (cid:17) Λ − i +1 , (25) E (2) := Ψ x Λ = N X i =1 (cid:16) q ( i ) Ψ x + p ( i ) Ψ (cid:17) Λ − i +1 , (26)5here Ψ = (Ψ (1) , . . . , Ψ (2 N ) ) . (27)Compatibility condition of eqs.(25) and (26), E (1) x = E (2) , (28)yields the expressions for p ( j ) and q ( j ) in terms of v ( j ) and w ( j ) : p ( j ) = w ( j ) x + v ( j ) µ + v ( j +1) a + v (1) aw ( j ) , q ( j ) = v ( j ) x + v ( j ) ν + w ( j ) + v (1) av ( j ) . (29)Thus only two blocks of W ( j ) are independent, i.e. w ( j ) and v ( j ) .Matrix equation (25) may be considered as the uniquely solvable system of scalar linearalgebraic equations for the functions v ( i ) and w ( i ) , i = 1 , . . . , N , while eq.(26) is the consequenceof eq.(25). In order to derive the nonlinear PDEs for v ( i ) and w ( i ) , we turn to the compatibilitycondition of the eqs.(20) and (25): E (1) t n + E (1) z n Λ ⇒ N X i =1 E (1 i ) Ψ x Λ − i +1 + N X i =1 E (0 i ) ΨΛ − i +1 = 0 , n = 1 , , . . . , (30)which generates the following chains of nonlinear PDEs for v ( i ) and w ( i ) , i = 1 , . . . , N : E (1 i ) n := v ( i ) t n + v (1) z n ( w ( i ) + v ( i ) x + v ( i ) ν + v (1) av ( i ) ) + w (1) z n v ( i ) + v ( i +1) z n = 0 , (31) v ( N +1) = 0 .E (0 i ) n := w ( i ) t n + v (1) z n ( w ( i ) x + v ( i ) µ + v (1) aw ( i ) + v ( i +1) a ) + w (1) z n w ( i ) + w ( i +1) z n = 0 , (32) w ( N +1) = 0 , n = 1 , , . . . . In addition, we must take into account the compatibility condition of the eqs.(22) and (25) E (1) xx = E (0) Λ ⇒ N X i =0 ( ˜ E (1 i ) Ψ x + ˜ E (0 i ) Ψ)Λ − i +1 = 0 , (33)which gives us the non-evolutionary part of the system of nonlinear chains, i = 1 , . . . , N :˜ E (1 i ) := v ( i ) xx + [ w ( i ) , ν ] + 2 w ( i ) x + 2 v ( i ) x ν − νv ( i ) x + [ v ( i ) , µ ] + [ v ( i ) , ν ] ν + [ v (1) , ν ] av ( i ) + (34)2 v (1) x av ( i ) + [ w (1) , a ] v ( i ) + [ v (1) , a ]( w ( i ) + v ( i ) x + v ( i ) ν + v (1) av ( i ) ) + [ v ( i +1) , a ] = 0 , ˜ E (0 i ) := w ( i ) xx + [ w ( i ) , µ ] + [ w (1) , a ] w ( i ) + 2 v ( i ) x µ + [ v ( i ) , ν ] µ + [ v (1) , ν ] aw ( i ) + (35)2 v (1) x aw ( i ) − νw ( i ) x + [ v (1) , a ]( w ( i ) x + v ( i ) µ + v (1) aw ( i ) + v ( i +1) a ) + [ w ( i +1) , a ] +2 v ( i +1) x a + [ v ( i +1) , ν ] a = 0Of course, the system (31,32,34,35) may be derived directly from the system (17) using reduction(24). However, the derivation of this system as the compatibility condition of the linear system(20,22,25) is more illustrative.The complete system of nonlinear PDEs for v ( i ) and w ( i ) , i = 1 ,
2, is represented by theeqs.(31,32) with fixed n (say n = 1) and i = 1 and by the eqs.(34,35) with i = 1: E (11)1 := v (1) t + v (1) z ( w (1) + v (1) x + v (1) ν + v (1) av (1) ) + w (1) z v (1) + v (2) z = 0 , (36) E (01)1 := w (1) t + v (1) z ( w (1) x + v (1) µ + v (1) aw (1) + v (2) a ) + w (1) z w (1) + w (2) z = 0 , (37)6 E (11) := v (1) xx + [ w (1) , ν ] + 2 w (1) x + 2 v (1) x ν − νv (1) x + [ v (1) , µ ] + (38)[ v (1) , ν ]( ν + av (1) ) + 2 v (1) x av (1) +[ w (1) , a ] v (1) + [ v (1) , a ]( w (1) + v (1) x + v (1) ν + v (1) av (1) ) + [ v (2) , a ] = 0 , ˜ E (01) := w (1) xx + [ w (1) , µ ] + [ w (1) , a ] w (1) + 2 v (1) x µ + [ v (1) , ν ] µ + [ v (1) , ν ] aw (1) + (39)2 v (1) x aw (1) − νw (1) x + [ v (1) , a ]( w (1) x + v (1) µ + v (1) aw (1) + v (2) a ) + [ w (2) , a ] +2 v (2) x a + [ v (2) , ν ] a = 0The scalar version ( n M = 1) of the system (36-39) reads: E = v (1) t + v (1) z ( w (1) + v (1) x + νv (1) + av (1) v (1) ) + w (1) z v (1) + v (2) z = 0 , (40) w (1) t + v (1) z ( w (1) x + v (1) µ + av (1) w (1) + av (2) ) + w (1) z w (1) + w (2) z = 0 , (41) v (1) xx + 2 w (1) x + v (1) x ν + 2 av (1) x v (1) = 0 , (42) w (1) xx + 2 v (1) x µ + 2 av (1) x w (1) − νw (1) x + 2 av (2) x = 0 (43) It is remarkable that the chains of nonlinear PDEs (31,32,34,35) admit the following FTR: v ( i ) = { v ( i ; kl ) , k, l = 1 , . . . , n } , w ( i ) = { w ( i ; kl ) , k, l = 1 , . . . , n } , (44) v ( i ; kl ) = δ k v ( i ; l ) + δ k ( l + n ( i )) I M ,w ( i ; kl ) = δ k w ( i ; l ) + δ k ( l + n ( i )) I M , where n is a positive integer parameter, n ( i ) and n ( i ) are arbitrary positive integer functionsof positive integer argument, v ( i ; l ) and w ( i ; l ) are M × M matrix fields. In particular, if n ( i ) = n ( i ) = 1, then this reduction becomes Frobenius one [2], which is shown in Fig.1. Eq.(44)requires the following diagonal block-structures for a , ν and µ : a = diag(˜ a, . . . , ˜ a | {z } n ) , ν = diag(˜ ν, . . . , ˜ ν | {z } n ) , µ = diag(˜ µ, . . . , ˜ µ | {z } n ) , (45)(˜ ν , ˜ µ and ˜ a are M × M diagonal constant matrices) and the appropriate block-structures forthe functions Ψ ( l ) and ˜Λ ( l ) :Ψ ( l ) = { Ψ ( l ; nm ) , n, m = 1 , . . . , n } , (46)˜Λ ( l ) = diag( ˜Λ ( l ;1) , . . . , ˜Λ ( l ; n ) ) , l = 1 , . . . , N . Here Ψ ( l ; nm ) are M × M matrix functions and ˜Λ ( l ; n ) are M × M diagonal matrix functions.Eq.(25) reduces to the following one:Ψ ( l ; nm ) ˜Λ ( l ; m ) = N X i =1 n X j =1 h(cid:16) δ n v ( i ; j ) + δ n ( j + n ( i )) (cid:17) Ψ ( l ; jm ) x + (cid:16) δ n w ( i ; j ) + (47) δ n ( j + n ( i )) (cid:17) Ψ ( l ; jm ) i ( ˜Λ ( l ; m ) ) − i +1 ,l = 1 , . . . , N n, m = 1 , . . . , n , ( l ; jm ) t n + Ψ ( l ; jm ) z n ˜Λ ( l ; m ) = 0 , (48)Ψ ( l ; jm ) xx = ˜ a Ψ ( l ; jm ) ˜Λ ( m ) + ˜ ν Ψ ( l ; jm ) x + ˜ µ Ψ ( l ; jm ) , (49)˜Λ ( l ; m ) t n + ˜Λ ( l ; m ) z n ˜Λ ( l ; m ) = 0 , ˜Λ ( l ; m ) x = 0 , (50) n = 1 , , . . . , l = 1 , . . . , N , j, m = 1 , . . . , n . Then the chains of nonlinear PDEs (31,32,34,35) get the following block structures: E ( mi ) n = { E ( mi ; l ) n δ k , k, l, = 1 , . . . , n } = 0 , (51)˜ E ( mi ) = { ˜ E ( mi ; l ) δ k , k, l, = 1 , . . . , n } = 0 ,m = 0 , , i = 1 , . . . , N . where E (1 i ; l ) n := v ( i ; l ) t n + v (1;1) z n ( w ( i ; l ) + v ( i ; l ) x + v ( i ; l ) ˜ ν ) + ( v (1;1) z n v (1;1) + v (1;1+ n (1)) z n )˜ av ( i ; l ) + (52) v (1;1) z n v (1; l + n ( i )) ˜ a + w (1;1) z n v ( i ; l ) + ( Q ( i ; l )1 ) z n = 0 ,E (0 i ; l ) n := w ( i ; l ) t n + v (1;1) z n ( w ( i ; l ) x + v ( i ; l ) ˜ µ + v ( i +1; l ) ˜ a ) + ( v (1;1) z n v (1;1) + v (1;1+ n (1)) z n )˜ aw ( i ; l ) + (53) v (1;1) z n v (1; l + n ( i )) ˜ a + w (1;1) z n w ( i ; l ) + ( Q ( i ; l )2 ) z n = 0 , ˜ E (1 i ; l ) := v ( i ; l ) xx + [ w ( i ; l ) , ˜ ν ] + 2 w ( i ; l ) x + 2 v ( i ; l ) x ˜ ν − νv ( i ; l ) x + [ v ( i ; l ) , ˜ µ ] + [ v ( i ; l ) , ˜ ν ]˜ ν + (54)[ v (1;1) , ˜ ν ]˜ av ( i ; l ) + [ v (1; l + n ( i )) , ˜ ν ]˜ a + 2 v (1;1) x ˜ av ( i ; l ) + 2 v (1; l + n ( i )) x ˜ a + [ w (1;1) , ˜ a ] v ( i ; l ) +[ v (1;1) , ˜ a ]( w ( i ; l ) + v ( i ; l ) x + v ( i ; l ) ˜ ν ) + ([ v (1;1) , ˜ a ] v (1;1) + [ v (1;1+ n (1)) , ˜ a ])˜ av ( i ; l ) +[ v (1;1) , ˜ a ] v (1; l + n ( i )) ˜ a + [ Q ( i ; l )1 , ˜ a ] = 0 , ˜ E (0 i ; l ) := w ( i ; l ) xx + [ w ( i ; l ) , µ ] + [ w (1;1) , a ] w ( i ; l ) + 2 v ( i ; l ) x µ + [ v ( i ; l ) , ˜ ν ]˜ µ + [ v (1;1) , ˜ ν ]˜ aw ( i ; l ) + (55)[ v (1; l + n ( i )) , ˜ ν ]˜ a + 2 v (1;1) x ˜ aw ( i ; l ) + 2 v (1; l + n ( i )) x ˜ a − ˜ νw ( i ; l ) x +[ v (1;1) , ˜ a ]( w ( i ; l ) x + v ( i ; l ) ˜ µ + v ( i +1; l ) ˜ a ) + ([ v (1;1) , ˜ a ] v (1;1) + [ v (1;1+ n (1)) , ˜ a ])˜ aw ( i ; l ) +[ v (1;1) , ˜ a ] v (1; l + n ( i )) ˜ a + 2 v ( i +1; l ) x ˜ a + [ v ( i +1; l ) , ˜ ν ]˜ a + [ Q ( i ; l )2 , ˜ a ] = 0 ,Q ( i ; l )1 = v (1; l + n ( i )+ n (1)) ˜ a + v (1; l + n ( i )) + v (1; l + n ( i )) ˜ ν + w (1; l + n ( i )) + v ( i +1; l ) , (56) Q ( i ; l )2 = v (1; l + n ( i )+ n (1)) ˜ a + v (1; l + n ( i )) ˜ µ + v (1; l + n ( i +1)) ˜ a + w (1; l + n ( i )) + w ( i +1; l ) . We see that after the reduction (44) the chains of nonlinear PDEs (31,32,34,35) acquire onemore discrete variable.To write the complete system of nonlinear PDEs we, first of all, put i = l = 1 in the chainsof PDEs (52-55) and take two values of n ( n = 1 ,
2) in eqs.(52,53) (remember, that derivingthe complete system (36-39) in Sec.3 we fixed n = 1 in eqs.(31,32)): E (11;1) n := v (1;1) t n + v (1;1) z n ( w (1;1) + v (1;1) x + v (1;1) ˜ ν ) + ( v (1;1) z n v (1;1) + (57) v (1;1+ n (1)) z n )˜ av (1;1) + v (1;1) z n v (1;1+ n (1)) ˜ a + w (1;1) z n v (1;1) + ( Q (1;1)1 ) z n = 0 ,E (01;1) n := w (1;1) t n + v (1;1) z n ( w (1;1) x + v (1;1) ˜ µ + v (2;1) ˜ a ) + ( v (1;1) z n v (1;1) + (58) v (1;1+ n (1)) z n )˜ aw (1;1) + v (1;1) z n v (1;1+ n (1)) ˜ a + w (1;1) z n w (1;1) + ( Q (1;1)2 ) z n = 0 , E (11;1) := v (1;1) xx + [ w (1;1) , ˜ ν ] + 2 w (1;1) x + 2 v (1;1) x ˜ ν − ˜ νv (1;1) x + [ v (1;1) , ˜ µ ] + [ v (1;1) , ˜ ν ]˜ ν + (59)[ v (1;1) , ˜ ν ]˜ av (1;1) + [ v (1;1+ n (1)) , ˜ ν ]˜ a + 2 v (1;1) x ˜ av (1;1) + 2 v (1;1+ n (1)) x ˜ a +[ w (1;1) , ˜ a ] v (1;1) + [ v (1;1) , ˜ a ]( w (1;1) + v (1;1) x + v (1;1) ˜ ν ) +([ v (1;1) , ˜ a ] v (1;1) + [ v (1;1+ n (1)) , ˜ a ])˜ av (1;1) + [ v (1;1) , ˜ a ] v (1;1+ n (1)) ˜ a + [ Q (1;1)1 , ˜ a ] = 0 , ˜ E (01;1) := w (1;1) xx + [ w (1;1) , ˜ µ ] + [ w (1;1) , ˜ a ] w (1;1) + 2 v (1;1) x ˜ µ + (60)[ v (1;1) , ˜ ν ]˜ µ + [ v (1;1) , ˜ ν ]˜ aw (1;1) + [ v (1;1+ n (1)) , ˜ ν ]˜ a +2 v (1;1) x ˜ aw (1;1) + 2 v (1;1+ n (1)) x ˜ a − ˜ νw (1;1) x + [ v (1;1) , ˜ a ]( w (1;1) x + v (1;1) ˜ µ + v (2;1) ˜ a ) +([ v (1;1) , ˜ a ] v (1;1) + [ v (1;1+ n (1)) , ˜ a ])˜ aw (1;1) + [ v (1;1) , ˜ a ] v (1;1+ n (1)) ˜ a + 2 v (2;1) x ˜ a +[ v (2;1) , ˜ ν ]˜ a + [ Q (1;1)2 , ˜ a ] = 0 , where Q (1;1)1 = v (1;1+2 n (1)) ˜ a + v (1;1+ n (1)) + v (1;1+ n (1)) ˜ ν + w (1;1+ n (1)) + v (2;1) , (61) Q (1;1)2 = v (1;1+ n (1)+ n (1)) ˜ a + v (1;1+ n (1)) ˜ µ + v (1;1+ n (2)) ˜ a + w (1;1+ n (1)) + w (2;1) . One can eliminate Q (1;1)1 from the system (57,59) resulting in the following complete system oftwo PDEs for the matrix fields u = v (1;1) and q = w (1;1) + v (1;1+ n (1)) :( E (11;1)1 ) z − ( E (11;1)2 ) z = 0 , (62)[ E (11;1)1 , ˜ a ] − ( ˜ E (11;1) ) z = 0 . (63)Three equations (58,60) are not important because they introduce three more fields ( w (1;1) , Q (1;1)2 and p = v (2;1) + v (1;1+ n (1)) ) which do not appear in the system (62,63). In particular, if˜ a is a scalar, eq.(63) must be replaced by the following one:˜ E (11;1) = 0 . (64)Consider the scalar case ( M = 1). Then only field u = v (1;1) remains in the eq. (62)which becomes eq.(1). Eq. (63) is not important in this case. Thus the system (62,63) may beconsidered as a matrix generalization of eq.(1). Its integrability must be studied more carefully.All in all, we have the following chain of transformations relating four systems of nonlinearPDEs having different integrability properties (compare with Fig.1):eq.(10) eq.( ) −→ eq.(18) eq.( ) −→ eqs.(36 − eqs.( ) −→ eqs.(62 , scalar case −→ eq.(1) . (65) Solutions to the system of matrix nonlinear PDEs (36-39) may be written in terms of thefunctions Ψ ( m ) and ˜Λ ( m ) , m = 1 , . . . , N , taken as solutions to the linear systems (20,22) and(21,23) respectively with n = 1. They read:Ψ ( l ) αβ ( ~x ) = X i =1 ψ ( l ; i ) αβ ( z − ˜Λ ( l ) β t ) e k ( l ; i ) αβ x , (66)˜Λ ( l ) β = E ( l ) β ( z − ˜Λ ( l ) β t ) , α, β = 1 , . . . , n M, l = 1 , . . . , N . (67)9here ψ ( l ) αβ ( y ) and E ( l ) β ( y ) are arbitrary scalar functions of single scalar variable, ~x = ( x, z , t )is the list of all independent variables of the nonlinear PDEs, k ( l ; i ) αβ are the roots of the charac-teristics equation assotiated with eq.(22): k − a α Λ ( l ) β − ν α k − µ α = 0 ⇒ (68) k ( l ;1) αβ = 12 (cid:18) ν α + q ( ν α ) + 4 a ˜Λ ( l ) β + 4 µ α (cid:19) , k ( l ;2) αβ = 12 (cid:18) ν α − q ( ν α ) + 4 a ˜Λ ( l ) β + 4 µ α (cid:19) . Now, we can use eq.(25) in order to find v ( i ) and w ( i ) . The simplest nontrivial case corre-sponds to N = 2. Then eq.(25) reduces to the following four matrix equations:Ψ ( m ) ˜Λ ( m ) = X i =1 (cid:16) v ( i ) Ψ ( m ) x + w ( i ) Ψ ( m ) (cid:17)(cid:16) ˜Λ ( m ) (cid:17) − i +1 , m = 1 , , , . (69)These equations, in general, are uniquely solvable for the matrix fields v ( i ) and w ( i ) , i = 1 , To define restrictions, generated by our algorithm, on the solution space to eqs.(36-39) we, firstof all, write equation (25) in the following compact form:Ψ ( m ) ˜Λ ( m ) = N X n =1 ~V ( n ) ˆΠ ( nm ) , m = 1 , . . . , N , (70)where ~V ( n ) = ( v ( n ) w ( n ) ) and ˆΠ ( nm ) is the following 2 N n M × N n M invertible operator:ˆΠ ( nm ) = (cid:18) Π ( nm ) x Π ( nm ) (cid:19) , Π ( nm ) = Ψ ( m ) Λ ( − n +1) , n = 1 , . . . , N , m = 1 , . . . , N . (71)Introduce operator ˜Π ( mj ) by the formula N X m =1 ˆΠ ( nm ) ˜Π ( mj ) = δ nj I n M , ˜Π ( mn ) = ( ˜Π ( mn )1 ˜Π ( mn )2 ) , n, j = 1 , . . . , N . (72)Then eq.(70) yields v ( n ) = N X m =1 Ψ ( m ) ˜Λ ( m ) ˜Π ( mn )1 , w ( n ) = N X m =1 Ψ ( m ) ˜Λ ( m ) ˜Π ( mn )2 , n = 1 , . . . , N . (73)Now let us turn to the eq.(66). We represent arbitrary functions ψ ( l ; i ) αβ ( y ) in the followingintegral form: ψ ( l ; i ) αβ ( y ) = Z dq ˆ ψ ( l ; i ) αβ ( q ) e qy , i = 1 , , l = 1 , . . . , N , α, β = 1 , . . . , n M (74)10here ˆ ψ ( l ; i ) αβ ( q ) are arbitrary functions of argument and one integrates over the whole space ofthe parameter q which is complex in general. Then eq.(66) may be written in the followingform: Ψ ( l ) αβ ( ~x ) = X i =1 Z dq ˆ ψ ( l ; i ) αβ ( q ) e qz − ˜Λ ( l ) β qt + k ( l ; i ) αβ x , l = 1 , . . . , N , αβ = 1 , . . . , n M. (75)Substituting expression (75) for Ψ ( m ) into eqs.(73) we write them as follows: v ( j ) αβ ( ~x ) = n M X γ =1 2 N X m =1 2 X i =1 Z dq ˆ ψ ( m ; i ) αγ ( q ) ˜Λ ( m ) e qz − ˜Λ ( m ) γ qt + k ( m ; i ) αγ x ( ˜Π ( mj )1 ( ~x )) γβ , (76) w ( j ) αβ ( ~x ) = n M X γ =1 2 N X m =1 2 X i =1 Z dq ˆ ψ ( m ; i ) αγ ( q ) ˜Λ ( m ) e qz − ˜Λ ( m ) γ qt + k ( m ; i ) αγ x ( ˜Π ( mj )2 ( ~x )) γβ ,j = 1 , . . . , N , α, β = 1 , . . . , n M. These formulae might be considered as integral representations of v ( j ) and w ( j ) if ˜Π ( mj ) n ( n = 1 , ~x . However, ˜Π ( mj ) do depend on ~x , so that eq.(76) has more complicatedsense. Nevertheless, in order to estimate the dimensionality of the solution space we considereqs.(76) as integral representations of v ( j ) and w ( j ) with the kernel R ( i ) αγ ( x, z , t ; q, Λ ( m ) γ ) = e qz − ˜Λ ( m ) γ qt + k ( m ; i ) αγ x , i = 1 , . (77)Formulae (76) transform functions v ( j ) αβ and w ( j ) αβ of three continues variables x , z and t intofunctions ˆ ψ ( m ; i ) αβ ( q ) ( i = 1 ,
2) depending on continues variable q and discrete variable m . Discretevariable m is assotiated with ˜Λ ( m ) in the kernel of the integral transformation, where m =1 , . . . , N and N is an arbitrary positive integer. Thus, we may state that the solutionspace to the eqs.(36-39) has freedom of two arbitrary functions, ψ ( m ;1) αβ ( q ) and ψ ( m ;2) αβ ( q ), of onecontinues and one discrete variable, i.e. one has two-dimensional solution space. Using thesefunctions we may approximate (at least formally) two initial conditions, for instance,( v (1) αβ | t =0 , w (1) αβ | t =0 ) → ( ψ ( m ;1) αβ ( q ) , ψ ( m ;2) αβ ( q )) , (78)and, as a consequence, we are able to solve the initial value problem (IVP) for the system (36-39) (which describes evolution of v (1) and w (1) ) with ”approximate” initial conditions. We say”approximate initial condition” since functions ψ ( m ; i ) αβ ( q ) ( i = 1 ,
2) have one continues and onediscrete variable, while both variables must be continues in order to represent arbitrary initialconditions precisely. We conclude that one provides the same variety of solutions to eqs.(36-39)as Sato approach does to the classical S -integrable PDEs [39]. Eqs.(47) have been derived as an algebraic system for the fields v ( i ; j ) and w ( i ; j ) in this case. Wesplit the system (47) into two subsystems. First one corresponds to n = 1:Ψ ( l ;1 m ) ˜Λ ( l ; m ) = N X i =1 n X j =1 (cid:2) v ( i ; j ) Ψ ( l ; jm ) x + w ( i ; j ) Ψ ( l ; jm ) (cid:3) ( ˜Λ ( l ; m ) ) − i +1 , (79) l = 1 , . . . , N , m = 1 , . . . , n . N n linear algebraic M × M matrix equations for the same numberof matrix fields v ( i ; l ) and w ( i ; l ) , i = 1 , . . . , N , j = 1 , . . . , n . Namely eqs.(79) yield functions u = v (1;1) and q = w (1;1) + v (1;1+ n (1)) as solution to the system (62,63). Second subsystemcorresponds to n > ( l ; nm ) ˜Λ ( l ; m ) = N X i =0 (cid:2) Ψ ( l ;( n − n ( i )) m ) x + Ψ ( l ;( n − n ( i )) m ) (cid:3) ( ˜Λ ( l ; m ) ) − i +1 , (80)Ψ ( l ; ij ) = 0 , if , i ≤ ,l = 1 , . . . , N , n, m = 1 , . . . , n , This equation expresses recursively the functions Ψ ( l ; nm ) , n >
1, in terms of the functions Ψ ( l ;1 m ) and their x -derivatives. The simplest case corresponds to n j ( i ) = 1, ∀ i, j (Frobenius reduction).Functions Ψ ( l ; nm ) and ˜Λ ( l ; m ) are solutions to the system (48-50) with n = 1 , ( l ;1 m ) αβ ( ~x ) = X i =1 ψ ( lm ; i ) αβ ( z − ˜Λ ( l ; m ) β t , z − ˜Λ ( l ; m ) β t ) e k ( lm ; i ) αβ x , (81)˜Λ ( l ; m ) β = E ( lm ) β ( z − ˜Λ ( l ; m ) β t , z − ˜Λ ( l ; m ) β t ) , (82) k ( lm ;1) αβ = 12 (cid:18) ˜ ν α + q (˜ ν α ) + 4˜ a ˜Λ ( l ; m ) β + 4˜ µ α (cid:19) ,k ( lm ;2) αβ = 12 (cid:18) ˜ ν α − q (˜ ν α ) + 4˜ a ˜Λ ( l ; m ) β + 4˜ µ α (cid:19) ,m = 1 , . . . , n , l = 1 , . . . , N , α, β = 1 , . . . , M. Here ψ ( lm ; i ) αβ ( y , y ) and E ( lm ) β ( y , y ) are arbitrary scalar functions of two scalar arguments.Since the functions Ψ ( l ;1 m ) depend explicitely on the functions ˜Λ ( l ; m ) , the functions v ( i ; j ) and w ( i ; j ) depend explicitely on ˜Λ ( l ; m ) as well. Functions ˜Λ ( l ; m ) describe the break of the waveprofiles because they are solutions of eqs.(82). Consequently, the functions v ( i ; j ) and w ( i ; j ) (inparticular, solutions to the nonlinear PDEs (62,63)) exhibit the break of the wave profiles aswell, unless ˜Λ ( l ; m ) = const ∀ l, m . One gets explicite solutions in the later case.Consider the scalar case corresponding to eq.(1), i.e. M = 1 and diagonal matrices ˜ a , ˜ ν , ˜ µ and ˜Λ ( l ; m ) become scalars. Eqs.(81,82) readΨ ( l ;1 m ) ( ~x ) = X i =1 ψ ( lm ; i ) ( z − ˜Λ ( l ; m ) t , z − ˜Λ ( l ; m ) t ) e k ( lm ; i ) x (83)˜Λ ( l ; m ) = E ( lm ) ( z − ˜Λ ( l ; m ) t , z − ˜Λ ( l ; m ) t ) , (84) k ( lm ;1) = 12 (cid:18) ˜ ν + q ˜ ν + 4˜ a ˜Λ ( l ; m ) + 4˜ µ (cid:19) , k ( lm ;1) = 12 (cid:18) ˜ ν − q ˜ ν + 4˜ a ˜Λ ( l ; m ) + 4˜ µ (cid:19) ,m = 1 , . . . , n , l = 1 , . . . , N . Reduction (3) corresponding to the eq.(4) reduces the eq.(83) into the following one:Ψ ( l ;1 m ) ( ~x ) = X i =1 ψ ( lm ; i ) e k ( lm ; i ) ( x − ˜Λ ( l ; m ) y − (˜Λ ( l ; m ) ) t ) , (85) m = 1 , . . . , n , l = 1 , . . . , N , ψ ( l ; m ) and ˜Λ ( l ; m ) are arbitrary scalar constant parameters. Emphasise, that ˜Λ ( l ; m ) doesnot depend on variables ~x and there is no arbitrary functions in the available solution manifold. Let us define restrictions, generated by our algorithm, on the solution space to the nonlinearPDEs (62,63). First of all we represent the formal solution to the system (80) as follows:Ψ ( l ; nm ) = Ψ ( l ;1 m ) P ( lmn ) , P ( lm ≡ I M , l = 1 , . . . , N , n, m = 1 , . . . , n , (86)reflecting the fact that eqs.(80) must express Ψ ( l ; nm ) ( n >
1) in terms of Ψ ( l ;1 m ) , ∀ l, m . Nowwe may represent eq.(79) in the following compact form:Ψ ( l ;1 m ) Λ ( l ; m ) = N X i =1 n X j =1 v ( i ; j ) Π ( ij ; lm ) x + w ( i ; j ) Π ( ij ; lm ) = N X i =1 n X j =1 ~V ( i ; j ) ˆΠ ( ij ; lm ) , (87) l = 1 , . . . , N , m = 1 , . . . , n , where ~V ( i ; j ) = ( v ( i ; j ) w ( i ; j ) ) , ˆΠ ( ij ; lm ) = (cid:18) Π ( ij ; lm ) x Π ( ij ; lm ) (cid:19) , Π ( ij ; lm ) = Ψ ( l ;1 m ) P lmj (Λ ( l ; m ) ) − i +1 , (88) i = 1 , . . . , N , j, m = 1 , . . . , n , l = 1 , . . . , N . Introduce operator ˜Π ( ml ; ji ) by the following formula: N X l =1 n X m =1 ˆΠ ( ij ; lm ) ˜Π ( ml ; kn ) = δ in δ jk I M , ˜Π ( ml ; kn ) = (cid:16) ˜Π ( ml ; kn )1 ˜Π ( ml ; kn )2 (cid:17) , (89) i, n = 1 , . . . , N , j, k = 1 , . . . , n . Then eq.(87) yields: v ( n ; j ) = N X m =1 n X l =1 Ψ ( l ;1 m ) ˜Λ ( l ; m ) ˜Π ( ml ; jn )1 , w ( n ; j ) = N X m =1 n X l =1 Ψ ( l ;1 m ) ˜Λ ( l ; m ) ˜Π ( ml ; jn )2 , (90) n = 1 , . . . , N , j = 1 , . . . , n , Now let us turn to eq. (81). We represent arbitrary functions ψ ( lm ; i ) αβ ( y , y ) in the followingintegral form ψ ( lm ; i ) αβ ( y , y ) = Z dq dq ˆ ψ ( lm ; i ) αβ ( q , q ) e q y + q y , (91) l = 1 , . . . , N , m = 1 , . . . , n , α, β = 1 , . . . , M, where ˆ ψ ( lm ; i ) αβ ( q , q ) are arbitrary functions of two arguments and one integrates over the wholetwo dimensional space of variables q and q which are complex in general. Then eq.(81) maybe written as follows:Ψ ( l ;1 m ) αβ ( ~x ) = X i =1 Z dq dq ˆ ψ ( lm ; i ) αβ ( q , q ) e q z + q z − ˜Λ ( l ; m ) β ( q t + q t )+ k ( lm ; i ) αβ x , (92) l = 1 , . . . , N , m = 1 , . . . , n , α, β = 1 , . . . , M. ( l ;1 m ) αγ into eqs.(90) we write them in the following form: v ( n ; j ) αβ ( ~x ) = (93) M X γ =1 2 N X m =1 n X l =1 2 X i =1 Z dq dq ˆ ψ ( lm ; i ) αγ ( q , q ) e q z + q z − ˜Λ ( l ; m ) γ ( q t + q t )+ k ( lm ; i ) αγ x ( ˜Π ( ml ; jn )1 ( ~x )) γβ ,w ( n ; j ) αβ ( ~x ) = M X γ =1 2 N X m =1 n X l =1 2 X i =1 Z dq dq ˆ ψ ( lm ; i ) αγ ( q , q ) e q z + q z − ˜Λ ( l ; m ) γ ( q t + q t )+ k ( lm ; i ) αγ x ( ˜Π ( ml ; jn )2 ( ~x )) γβ ,n = 1 , . . . , N , j = 1 , . . . , n . Now we show that the solution space to the five-dimensional matrix eqs.(62,63) has twoarbitrary M × M matrix functions of two continues and one discrete variable (compare withSec.5.1.1). For this purpose we consider formulae (93) as integral representations of v ( n ; j ) and w ( n ; j ) with the kernel R ( i ) αγ ( x, z , z , t , t ; q , q , Λ ( l ; m ) γ ) = e q z + q z − ˜Λ ( l ; m ) γ ( q t + q t )+ k ( lm ; i ) αγ x , i = 1 , . (94)Formulae (93) transform functions v ( n ; j ) αβ and w ( n ; j ) αβ of five continues variables x , z i and t i ( i = 1 ,
2) into functions ˆ ψ ( lm ; i ) αβ ( q , q ) ( i = 1 ,
2) depending on two continues variables q , q and two discrete variables l, m . However, two discrete variables l and m are assotiatedwith single discrete variable ˜Λ ( l ; m ) in the kernel of the integral transformation, where l =1 , . . . , N , m = 1 , . . . , n and N , n are arbitrary positive integers. Thus, we may state thatthe solution space to the eqs.(62,63) (describing the evolution of v (1;1) ) has freedom of twoarbitrary functions, ˆ ψ ( lm ;1) αβ ( q , q ) and ˆ ψ ( lm ;2) αβ ( q , q ), of two continues and one discrete variable,i.e. one has three-dimensional solution space with two arbitrary functions of three variable.Note that this freedom is not enough in order to solve, for instance, the IVP for the matrixsystem (62,63) even with ”approximate initial conditions” like it was done in Sec.5.1.1. In fact,in order to approximate arbitrary initial condition one needs an (infinitely) big number of sucharbitrary functions of three variables. Thus, further modifications of our algorithm with thepurpose to increase the dimensionality of the solution space to the derived nonlinear PDEsbecomes an important problem to be resolved.Solution space to the scalar equation (1) has the same dimensionality so that formulae(86-94) remains valid with M = 1.Consider reduction (3) and estimate the dimensionality of the available solution space tothe (2+1)-dimensional PDE (4). As we have seen in both this subsection and Sec.5.1.1 thedimensionality of the solution space is completely defined by the kernel of the integral trans-formation, see eqs.(77) and (94). Reduction (3) reduces eq.(94) with M = 1 to the followingone R ( i ) ( x, y, t ; Λ ( l ; m ) ) = e k ( lm ; i ) ( x − ˜Λ ( l ; m ) y − (˜Λ ( l ; m ) ) t ) , i = 1 , . (95)This kernel has one discrete parameter Λ ( l ; m ) and, as a consequence, the available solution spaceis one-dimensional with two arbitrary functions of single discrete variable. Number of arbitraryfunctions is predicted by two different values of the superscript i in eq.(95).14 .2.2 Simplest example of the solution to eq.(1) The simplest nontrivial example of the solution to eq.(1) is assotiated with N = n = 1. Theneq.(83) reads Ψ ( l ;11) ( ~x ) = X i =1 ψ ( l i ) ( z − ˜Λ ( l ;1) t , z − ˜Λ ( l ;1) t ) e k ( l i ) x , l = 1 , , (96)where ˜Λ ( l ;1) ( l = 1 ,
2) are solutions to the nonlinear algebraic equations (84) which reduce tothe following ones: ˜Λ ( l ;1) = E ( l ( z − ˜Λ ( l ;1) t , z − ˜Λ ( l ;1) t ) , l = 1 , . (97)Then eqs.(79) become the system of two following equations:Ψ ( l ;11) ˜Λ ( l ;1) = v (1;1) Ψ ( l ;11) x + w (1;1) Ψ ( l ;11) , l = 1 , . (98)It has the following solution: u ≡ v (1;1) = ∆ ∆ , ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ (1;11) x Ψ (1;11) Ψ (2;11) x Ψ (2;11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (99) X i,j =1 (cid:16) ( − i +1 K − ( − j +1 K (cid:17) e (cid:16) ˜ ν +( − i +1 K +( − j +1 K (cid:17) x ψ (11; i ) ( y , y ) ψ (21; j ) ( y , y ) ,K l = 12 q ˜ ν + 4˜ a ˜Λ ( l ;1) + 4˜ µ, y = z − ˜Λ ( l ;1) t , y = z − ˜Λ ( l ;1) t , ∆ = (cid:12)(cid:12)(cid:12)(cid:12) Ψ (1;11) ˜Λ (1;1) Ψ (1;11) Ψ (2;11) ˜Λ (2;1) Ψ (2;11) (cid:12)(cid:12)(cid:12)(cid:12) =( ˜Λ (1;1) − ˜Λ (2;1) ) X i,j =1 e (cid:16) ˜ ν +( − i +1 K +( − j +1 K (cid:17) x ψ (11; i ) ( y , y ) ψ (21; j ) ( y , y ) ,w (1;1) = ∆ ∆ , ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ (1;11) x Ψ (1;11) ˜Λ (1;1) Ψ (2;11) x Ψ (2;11) ˜Λ (2;1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (100)Formulae (99,100) have six arbitrary functions of two variables ψ ( l i ) ( y , y ), ˜Λ ( l ;1) = E ( l ( y , y ), l, i = 1 ,
2. It is evident that solution u has no singularities if ∆ = 0, i.e., for instance, if K > K > , ψ (11;2) < , ψ (11;1) , ψ (21;1) , ψ (21;2) > , ∀ y , y . (101)Since ˜Λ ( l ;1) is implicitly given by the eq.(97), constructed function u describes the break of thewave profile unless ˜Λ ( l ;1) = const .Write the explicite form of the solution corresponding to K = 0 ⇒ ˜Λ (2;1) = − ˜ ν + 4˜ µ a = const, (102) ψ (11;1) ( y , y ) = ξ ( y , y ) > , ψ (11;2) ( y , y ) = − ξ ( y , y ) < . u = ( ˜Λ (1;1) − ˜Λ (2;1) ) (cid:0) e K x ξ ( y , y ) − e − K x ξ ( y , y ) (cid:1) K ( e K x ξ ( y , y ) + e − K x ξ ( y , y )) , (103)Let us consider the particular case ˜Λ ( l ;1) = const , l = 1 ,
2. Then the solution to eq.(1) isgiven explicitely by the eq.(99) with four arbitrary functions of two arguments ψ ( l i ) ( y , y ), l, i = 1 ,
2. Consider solution given by formula (103) with Im ( K ) = 0 and Im ( ξ i ) = 0, i = 1 , u | z i = const,t i = const is kink. Behaviour of u | x = const as a function of z i and t i depends onthe shapes of the arbitrary functions ξ i ( y , y ), i = 1 ,
2. In particular, u x = const may be soliton. In accordance with eq.(85), there is no arbitrary functions in the constructed solution space.The simplest nontrivial example of the solution to the eq.(1) is assotiated with N = n = 1.As we have seen above, reduction (3) requires ˜Λ ( l ;1) = const , l = 1 ,
2. Eq. (85) readsΨ ( l ;11) ( ~x ) = X i =1 ψ ( l i ) e k ( l i ) ( x − ˜Λ ( l ;1) y − (˜Λ ( l ;1) ) t ) l = 1 , , ψ ( l i ) = const. (104)Eq. (98) remains correct. It yields u ≡ v (1;1) = ∆ ∆ , (105)∆ = X i,j =1 (cid:16) ( − i +1 K − ( − j +1 K (cid:17) e ˜ ν ( q / q / − i +1 K q +( − j +1 K q ψ (11; i ) ψ (21; j ) ,q l ( y, t ) = x − ˜Λ ( l ;1) y − ( ˜Λ ( l ;1) ) t, ∆ = ( ˜Λ (1;1) − ˜Λ (2;1) ) X i,j =1 e ˜ ν ( q / q / − i +1 K q +( − j +1 K q ψ (11; i ) ψ (21; j ) . It is evident that solution u has no singularities in the case (101).Write the explicite form of the solution corresponding to conditions (102) with constant ξ i , i = 1 , u = ( ˜Λ (1;1) − ˜Λ (2;1) ) (cid:0) e K q ξ − e − K q ξ (cid:1) K ( e K q ξ + e − K q ξ ) , (106)If the parameter K is real, then the function u is kink. The remarkable relations among C -, Ch -, S - and S -integrable nonlinear PDEs have beenfound in [2] using a version of the dressing method. We represent a modification of this dressingalgorithm allowing one to involve a simple example of the S -integrable scalar equation (i.e.eq.(1)) in the above list. In addition, a matrix version of this equation is derived. We have foundthat (at least on the restricted manifold of solutions, as it is described in Sec.5) eq.(1) originates16rom the GL ( N S , C ) SDYM (which is S -integrable equation) after the special differentialreduction followed by the Frobenius type reduction imposed on the matrix field. In turn, GL ( N S , C ) SDYM may be derived from the (1+1)-dimensional hierarchy of Ch -integrablePDEs [1, 2]. Thus, the important result is that S -integrable equation (1) has been derived fromthe (1+1)-dimensional hierarchy of nonlinear Ch -integrable PDEs by means of the followingset of tree reductions imposed on the matrix fields:FR −→ DR( x ) −→ FTR . (107)The first and the third reductions in eq.(107) introduce the new discrete variables into thehierarchy of commuting nonlinear PDEs. The second reduction introduces the new independentvariable x . After the third reduction, one must to couple two equations of the commutinghierarchy of chains in order to write the complete system of nonlinear PDEs.Solution spaces to both S -integrable PDEs with differential reduction and S -integrablesystem are considered. It is shown that our algorithm allows one to construct a big manifoldof solutions to eq.(1) and to its matrix generalization (62,63) involving solutions with breakof the wave profile and soliton-like solutions. However, solution space to S -integrable PDEsavailable by our method is not full so that IVP may not be formally solved. The problem ofimproving this algorithm with the purpose to describe the full solution space is an importantproblem for further study.It is clear that the chain (107) may be prolonged:FTR ( l ) −→ DR ( x ) −→ FTR ( l ) −→ DR ( x ) −→ · · · (108) −→ DR K − ( x K − ) −→ FTR K ( l K ) . Each FTR i ( l i ) introduces the new discrete parameter l i , while each DR i ( x i ) introduces thenew independent variable x i into the hierarchy of chains of nonlinear PDEs. After the lastreduction, one has to couple two (or may be more) chains from the hierarchy in order to writethe complete system of nonlinear PDEs. Study of these PDEs and their possible relations withthe classical integrable models is postponed for the future work.Another evident modification of our algorithm is generalization of the differential reduction.Namely, instead of the second order linear PDE (22) one can use K th order linear PDE witharbitrary positive integer K : ∂ Kx Ψ ( m ) = K − X j =0 K − j − X i =0 a ( ij ) ∂ ix Ψ ( m ) ( ˜Λ ( m ) ) j , m = 1 , . . . , N . (109)Assotiated nonlinear PDEs will be considered elsewhere.Author thanks Professor P.M.Santini for useful discussion. The work was supported by theRFBR grants 07-01-00446 and 06-01-92053 and by the grant NS-4887.2008.2. References [1] A.I.Zenchuk, J. Math. Phys. (2008) 063502[2] A.I.Zenchuk and P.M.Santini,J.Phys.A:Math.Theor., (2008) 185209[3] Calogero F in What is Integrability ed V E Zakharov (Berlin: Springer) (1990) 1174] Calogero F and Xiaoda Ji, J. Math. Phys. (1991) 875[5] Calogero F and Xiaoda Ji, J. Math. Phys. (1991) 2703[6] Calogero F, J. Math. Phys. (1992) 1257[7] Calogero F, J. Math. Phys. (1993) 3197[8] Calogero F and Xiaoda Ji, J. Math. Phys. (1993) 5810[9] Hopf E, Commun. Pure Appl. Math.
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