Bilinear equations, Bell polynomials and linear superposition principle

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1 Blnear equatons, Bell polynomals and lnear superposton prncple Wen-Xu Ma Department of Mathematcs and Statstcs, Unversty of South Florda, Tampa, FL , USA E-mal: Abstract. A class of blnear dfferental operators s ntroduced through assgnng approprate sgns and used to create blnear dfferental equatons whch generalze Hrota blnear equatons. The resultng blnear dfferental equatons are characterzed by a specal knd of Bell polynomals and the lnear superposton prncple s appled to the constructon of ther lnear subspaces of solutons. Illustratve examples are made by an algorthm usng weghts of dependent varables. 1. Introducton Nonlnear dfferental equatons play a sgnfcant role n explorng physcal phenomena n depth. Hrota presented a drect method to solve a knd of specfc blnear dfferental equatons [1]; and solton solutons are, despte ther dversty, a unversal phenomenon that Hrota blnear equatons descrbe [2]. It s known that under u 2(ln f xx, the KdV equaton can be transformed nto whch reads u t + 6uu x + u xxx 0 (1.1 (D x D t + D 4 xf f 0, (1.2 f xt f f x f t + f xxxx f 4f xxx f x + 3f 2 xx 0. (1.3 Through ths blnear form, general Wronskan solutons, ncludng soltons and complextons, are presented for the KdV equaton [3, 4]. The Hrota D-operators [5] are defned to be D n xf g ( x x n f(xg(x x x n x f(x + x g(x x x 0. (1.4 For example, we have D x f g f x g fg x, D 2 xf g f xx g 2f x g x + fg xx, D 3 xf g f xxx g 3f xx g x + 3f x g xx fg xxx. (1.5 Publshed under lcence by IOP Publshng Ltd 1

2 It s very nterestng that most ntegrable equatons possess the Hrota blnear form. Solton solutons, partcularly three-solton solutons and Wronskan, Gramman and Pfaffan solutons, to Hrota blnear equatons can be generated by the Hrota perturbaton and Pfaffan technques [4, 6, 7, 8]. However, Hrota blnear equatons are specal and there are many other blnear dfferental equatons whch are not wrtten n the Hrota blnear form. Ths report wll ntroduce a knd of generalzed blnear dfferental operators and ther correspondng blnear equatons, whch stll possess nce mathematcal propertes. More mportantly, we wll talk about lnks of the presented blnear equatons wth multvarate Bell exponental polynomals and ther lnear subspaces of solutons by the lnear superposton prncple. 2. Blnear dfferental operators and blnear equatons 2.1. Blnear D p -operators Let p be a gven natural number. We ntroduce blnear dfferental operators as follows: (Dp,xf n g(x ( x + α x n f(xg(x ( n x x α ( x n f(x( xg(x, n 1, (2.1 where the powers of α are determned by 0 α ( 1 r(, where r( mod p wth 0 r( < p, 0. (2.2 Obvously, the case of p 1 gves the normal dervatves, and the cases of p 2k, k N, reduce to Hrota blnear operators. We can observe that the powers α read and thus p 3 : +,, +, +,, +, for 0, 1, 2, ; (2.3 p 5 : +,, +,, +, +,, +,, +, for 0, 1, 2, ; (2.4 p 7 : +,, +,, +,, +, +,, +,, +,, +, for 0, 1, 2, ; (2.5 D 3 3,xf g f 3x g 3f 2x g x + 3f x g 2x + fg 3x, D 5 5,xf g f 5x g 5f 4x g x + 10f 3x g 2x 10f 2x g 3x + 5f x g 4x + fg 5x, D 7 7,xf g f 7x g 7f 6x g x + 21f 5x g 2x 35f 4x g 3x + 35f 3x g 4x 21f 2x g 5x + 7f x g 6x + fg 7x, whch are dfferent from the Hrota blnear dfferental expressons [1]. A common feature that the D p -operators share s the Taylor expanson f we defne f(x + δg(x + αδ g(x + αδ 0 0 1! (D p,xf gδ, (2.6 ( xg(x α δ. (2.7! The case of blnear operators wth more than one dependent varables can be smlarly defned as follows: (D n 1 p,x 1 D n l p, f g(x 1,, ( x1 + α x 1 n1 ( xl + α x l n l f(x 1,, g(x 1,, x l x x, n 1,, n l 1. (2.8 2

3 2.2. Blnear equatons A blnear dfferental equaton assocated wth a multvarate polynomal F F(x 1,, s defned by F(D p,x1,, D p,xl f f 0, (2.9 whch reduces a Hrota blnear equaton f p 2k, k N. When p 5, we partcularly have the generalzed blnear KdV equaton (D 5,x D 5,t + D 4 5,xf f 2 f xt f 2 f x f t + 2f 4x f 8f 3x f x + 6 f 2 2x 0, (2.10 the generalzed blnear Boussnesq equaton (D 2 5,t + D 4 5,xf f 2 f 2t f 2 f 2 t + 2f 4x f 8f 3x f x + 6 f 2 2x 0, (2.11 and the generalzed blnear KP equaton (D 5,t D 5,x +D 4 5,x +D 2 5,yf f 2 f xt f 2 f x f t +2f 4x f 8f 3x f x +6 f 2 2x +2 f 2y f 2 f 2 y 0. (2.12 Such generalzed blnear equatons have two common characterstcs: Blnear: The nearest neghbors to lnear equatons. Usng the D p -operators: Nce mathematcal operators. Two basc questons n the mathematcal theory of blnear equatons are: How can one characterze blnear equatons defned by (2.9? What knd of exact solutons are there to blnear equatons defned by (2.9? In ths report, we would lke to dscuss those two questons, and provde solutons to both questons through the Bell exponental polynomals and the lnear superposton prncple, respectvely. 3. Relatons wth Bell exponental polynomals 3.1. Bell polynomals To begn wth, let y be a C functon of x and ntroduce The Bell polynomals are defned by y r r xy, r 1. (3.1 Y nx (y Y n (y 1,, y n e y n x e y, n 1, (3.2 n combnatoral mathematcs [9]. A drect computaton tells Y 1 y 1, Y 2 y y 2, Y 3 y y 1y 2 + y 3, Y 4 y y2 1 y 2 + 4y 1 y 3 + 3y y 4, Y 5 y y3 1 y y 2 1 y y 1 y y 1y y 2 y 3 + y 5. (3.3 A specal case of the Faà d Bruno formula (see, e.g., [10] presents the Bell polynomals precsely: Y nx (y n! m 1! m n!(1! m 1 (n! m n y m 1 1 yn mn, (3.4 3

4 where the sum s over all n-tuples of nonnegatve ntegers (m 1,, m n satsfyng the constrant m 1 + 2m nm n n. The Bell polynomals can also be computed from y r exp( r! tr 1 + r1 n1 Y n (y 1,, y n t n. (3.5 n! The general formula (3.4 mmedately tells the homogeneous property Y n (αy 1, α 2 y 2,, α n y n α n Y n (y 1,, y n, (3.6 whose left-hand sde s evaluated through frst substtutng all αy 1, α 2 y 2,, α n y n nto Y n and then collectng powers of α and computng them by the rule (2.2. On the other hand, the general Lebnz rule ( n (fg 1 x n (fg (f 1 x n f(g 1 xg (3.7 0 shows the addton formula for the Bell polynomals: Y nx (y + y 0 ( n Y (n x (yy x (y. (3.8 Those two propertes wll be used to lnk blnear equatons to a specal knd of Bell polynomals Bnary Bell polynomals We frst explore a relaton of the Bell polynomals to the D p -operators. computatonal convenence, we assume that For the sake of Then usng (3.6 and (3.8, we have (fg 1 D n p,xf g f e ξ(x, g e η(x. (3.9 ( n α (f 1 x n f(g 1 xg ( n α Y (n x (ξy x (η 0 0 Y n (y 1,, y n yrξ rx+α r η rx, (3.10 where ξ rx r xξ and η rx r xη, r 1. Smlarly to the case of the Hrota D-operators [11], we ntroduce bnary Bell polynomals Y p;nx (v, w Y n (y 1,, y n yr 1 2 (wrx+vrx+ 1 2 αr (w rx v rx, n 1, (3.11 where v rx r xv and w rx r xw, r 1. For example, we have Y 3;x (v, w v x, Y 3;2x (v, w v 2 x + w 2x, Y 3;3x (v, w v 3 x + 3v x w 2x + w 3x, Y 3;4x (v, w v 4 x + 6v 2 xw 2x + 3w 2 2x + 4v xw 3x + v 4x, Y 3;5x (v, w v x v 3 xw 2x + 15v x w 2 2x + 10v2 xw 3x + 10w 2x w 3x + 5v x v 4x + w 5x. (3.12 Ths way, upon settng that w ξ + η, v ξ η, (3.13 4

5 from (3.10, we have a combnatoral formula for the D p -operators: (fg 1 Dp,xf n g Y p;nx (v ln f, w ln fg. (3.14 g To characterze blnear equatons, we further ntroduce P-polynomals: P p;nx (q Y p;nx (0, q, (3.15 the frst few of whch n the case of p 3 read { P3;x (q 0, P 3;2x (q q 2x, P 3;3x (q q 3x, P 3;4x (q 3q 2 2x, P 3;5x q 5x + 10q 2x q 3x, P 3;6x q 6x + 15q 2 2x + 10q2 3x. (3.16 In terms of q w v 2 ln g, v ln f g, (3.17 the combnatoral formula (3.14 becomes (fg 1 D n p,xf g Y p;nx (v, v + q. (3.18 Lettng f g, ths tells a relaton between blnear expressons and the P-polynomals: Therefore, a blnear equaton f 2 D n p,xf f P p;nx (q 2 ln f. (3.19 F(D p,x f f 0 wth F(x c x (3.20 s equvalent to an equaton on a lnear combnaton of P-polynomals n q 2 ln f: 0 c P p;x (q 2 ln f 0. ( Ths s a characterzaton for our generalzed blnear equatons n one dmensonal case Multvarate bnary Bell polynomals For a C functon y y(x 1,,, defne the varables [12]: y r1,,r l y r1 x 1,,r l r 1 x 1 r l y(x 1,,, r 1,, r l 0, and the multvarate Bell polynomals l r 1, ( Y n1 x 1,,n l (y Y n1,,n l (y r1,,r l e y n 1 x 1 n l e y, n 1,, n l 0, whch can be computed through exp( r r l 1 r 1,, r l 0 y r1,,r l r 1! r l! tr 1 1 tr l l 1 + n n l 1 n 1,, n l 0 l n 1, ( Y n1,,n l n 1! n l! tn 1 1 tn l l. (3.24 5

6 Three examples n dfferental polynomal form are { Yx,t y xt + y x y t, Y 2x,t y 2x,t + y 2x y t + 2y xt y x + y 2 xy t, Y 3x,t y 3x,t + y 3x y t + 3y 2x,t y x + 3y 2x y xt + 3y 2x y x y t + 3y 2 xy xt + y 3 xy t. (3.25 Based on (3.24, we can show the homogeneous property: and the general Lebntz rule (fg 1 n 1 x 1 n l fg Y n1,,n l (α r 1+ r l y r1,,r l α n 1+ +n l Y n1,,n l (y r1,,r l, (3.26 n n l l l 0 j1 ( nj mples the addton formula for the multvarate Bell polynomals: Y n1 x 1,,n l (y + y n n l l l 0 j1 j ( nj j (f 1 n 1 1 x 1 n l l f(g 1 1 x1 l g (3.27 Y (n1 1 x 1,,(n l l (yy 1 x 1,, l (y. (3.28 Smlarly for the sake of computatonal convenence, we assume that Then by (3.26 and (3.28, we can compute that where (fg 1 D n 1 p,x 1 D n l n n n p, f g n l l ( α nj j l 0 j1 j n l l ( α nj j l 0 j1 j n l l ( nj l 0 j1 j f e ξ(x 1,,, g e η(x 1,,. (3.29 (e ξ n 1 1 x 1 n l l e ξ (e η 1 x1 l e η Y (n1 1 x 1,,(n l l (ξy 1 x 1,, l (η Y (n1 1 x 1,,(n l l (ξ r1,,r l Y 1 x 1,, l (α r 1+ +r l η r1,,r l Y n1,,n l (y r1,,r l ξ r1,,r l + α r 1+ +r l η r1,,r l. (3.30 Let us now ntroduce bnary multvarate Bell polynomals n dfferental polynomal form: Y p;n1 x 1,,n l (v, w Y n1,,n l ( yr1,,r l ξ r1 x 1,,r l + α r 1+ +r l η r1 x 1,,r l, (3.31 Then from (3.30, a combnatoral formula follows w ξ + η, v ξ η. (3.32 (fg 1 D n 1 p,x 1 D n l p, f g Y p;n1 x 1,,n l (v ln f, w ln fg. (3.33 g Further settng the followng multvarate P-polynomals: P p;n1 x 1,,n l (q Y p;n1 x 1,,n l (v 0, w q. (3.34 6

7 For example, we have It now follows that Thus, a blnear equaton P 3;x,t (q 0, P 3;2x,t 1 4 q2 xq t, P 3;3x,t 3 4 q2 xq xt q3 xq t q xq 2x q t. (3.35 f 2 D n 1 p,x 1 D n l p, f f P p;n1 x 1,,n l (q 2 ln f. (3.36 F(D p,x1, D p,xl f f 0 wth F(x 1,, 1,, l 0 c 1,, l x 1 1 x l l (3.37 s equvalent to an equaton on a lnear combnaton of multvarate P-polynomals n q 2 ln f: 1,, l 0 c 1,, l P p;1 x 1,, l (q 2 ln f 0, (3.38 where the coeffcents c 1,, l s are constants. Ths s a characterzaton for generalzed blnear equatons defned through the D p -operators. 4. Lnear superposton prncple 4.1. Subspaces of solutons Let F(x 1,, be a multvarate polynomal. Consder a blnear equaton Defne a set of N wave varables F(D p,x1,, D p,xl f f 0. (4.1 θ k 1, x k l,, 1 N, (4.2 where the k j, s are constants, and form a lnear combnaton of N exponental waves f N ε e θ 1 N ε e k 1,x 1 + +k l,, (4.3 1 where all ε s are arbtrary constants. Note that we have the blnear denttes: F(D p,x1,, D p,xl e θ e θ j F(k 1, + αk 1,j,, k l, + αk l,j e θ +θ j, 1, j N, (4.4 where the powers of α obey the rule (2.2. Therefore, we can have the followng crteron for obtanng the lnear subspaces of solutons defned by (4.3 (see also [13, 14]. Theorem: Let N 1. An arbtrary lnear combnaton of N exponental waves defned by (4.3 solves the generalzed blnear equaton (4.1 ff the constants k j, s satsfy F(k 1, + αk 1,j,, k l, + αk l,j + F(k 1,j + αk 1,,, k l,j + αk l, 0, 1 j N. (4.5 7

8 Let F be a multvarate polynomal defned as n (3.37. Obvously, the formula (3.33 yelds D 1 p,x1 D l p, e θ e θ j Y p;1 x 1,, l (θ θ j, θ + θ j e θ +θ j, 1, j N. (4.6 Thus, we obtan an equvalent theorem on the lnear subspaces of exponental N-wave solutons. Theorem : Let F be defned by (3.37 and N 1. An arbtrary lnear combnaton of N exponental waves defned by (4.3 presents a soluton to the generalzed blnear equaton (4.1 ff the wave varables θ s satsfy 1,, l 0 c 1,, l [Y p;1 x 1,, l (θ θ j, θ + θ j + Y p;1 x 1,, l (θ j θ, θ j + θ ] 0, 1 j N. Ths theorem has an advantage that the wave varables θ s can be nonlnear functons of dependent varables, but the frst theorem only works for lnear wave varables θ s. Gven a multvarate polynomal F, below s one way of solvng the system (4.5 or (4.7 for k j, and c 1,, l, n order to obtan blnear equatons and ther lnear subspaces of solutons (see, e.g., [13, 15]. We adopt a knd of parameterzaton for wave numbers and frequences and lst the sequental soluton procedure as follows: Introduce weghts for the ndependent varables: (4.7 (w(x 1,, w( (w 1,, w l, (4.8 where the weghts w s can be both postve and negatve. Form a homogeneous polynomal F(x 1,,, defned by (3.37, n some weght. Parameterze k 1,,, k l, usng a parameter k : k j, b j k w j, 1 j l, (4.9 and then determne the proportonal constants b j s and the coeffcents c 1,, l s by solvng the system (4.5 or ( Illustratve examples To present llustratve examples, we consder the (3 + 1-dmensonal case wth and θ k x + l y + m z ω t, l b 1 k wy (w(x, w(y, w(z, w(t (w x, w y, w z, w t, (4.10, m b 2 k wz, ω b 3 k wt, 1 N. (4.11 Then, upon formng a homogeneous multvarate polynomal n some weght F 1, 2, 3, 4 1 c 1, 2, 3, 4 x 1 y 2 z 3 t 4, (4.12 we solve the system (4.5 or (4.7 for the proportonal constants b 1, b 2, b 3 and the coeffcents c 1, 2, 3, 4 s so that we can determne the correspondng blnear equatons and ther assocated lnear subspaces of solutons consstng of lnear combnatons of exponental waves. We are gong to present two concrete llustratve examples by applyng ths general dea below. 8

9 Example 1 - Example wth postve weghts Let us set the weghts of ndependent varables: (w(x, w(y, w(z, w(t (1, 2, 4, 5, (4.13 and consder a polynomal beng homogeneous n weght 6: F c 1 x 6 + c 2 x 4 y + c 3 x 2 z + c 4 xt + c 5 yz. (4.14 Followng the parameterzaton of wave numbers and frequences n (4.11, we set the wave varables θ k x + b 1 k 2 y + b 2 k 4 z + b 3 k 5 t, 1 N, (4.15 where k, 1 N, are arbtrary constants but the proportonal constants b 1, b 2 and b 3 are to be determned by (4.5. Now, a drect computaton shows that the correspondng blnear equaton reads F(D 3,x, D 3,y, D 3,z, D 3,t f f 2c 1 f 6x f + 20c 1 f 2 3x + 2c 2 f 4x,y f 2c 2 f 4x f y 8c 2 f 3x,y f x +8c 2 f 3x f xy + 12c 2 f 2x,y f 2x + 2c 3 f 2x,z f + 2c 4 f xt f 2c 4 f x f t + 2c 5 f yz f 2c 5 f y f z 0. (4.16 The correspondng lnear subspace of N-wave solutons s gven by f N ε e θ 1 N ε e k x+b 1 k 2y+b 2k 3z+b 3k 4t, ( where ε, 1 N, are arbtrary constants but the proportonal constants b 1, b 2 and b 3 are defned by b 1 25c 3, b 2 5c 2, b 3 10c 1, (4.18 8c 5 c 5 c 4 when the coeffcents of the polynomal F satsfy Example 2 - Example wth postve and negatve weghts Let us set the weghts of ndependent varables: 4c 1 c 5 5c 2 c 3. (4.19 (w(x, w(y, w(z, w(t (1, 1, 2, 3, (4.20 and consder a polynomal beng homogeneous n weght 3: F c 1 x 3 + c 2 x 5 z + c 3 xyt. (4.21 Followng the parameterzaton of wave numbers and frequences n (4.11, we set the wave varables θ k x + b 1 k 1 y + b 2 k 2 z + b 3 k 3 t, 1 N, (4.22 where k, 1 N, are arbtrary constants but the proportonal constants b 1, b 2 and b 3 are to be determned by (4.5. Smlarly, a smlar drect computaton shows that the correspondng blnear equaton reads F(D 3,x, D 3,y, D 3,z, D 3,t f f 2c 1 f 3x f + 2c 2 f 5x,z f + 20c 2 f 3x f 2x,z + 2c 3 f xyt f 0, (4.23 and t possesses the lnear subspace of N-wave solutons determned by f N ε e θ 1 where the ε s and k s are arbtrary but b 1, b 2 and b 3 need to satsfy N 1 ε e k x+b 1 k 1 y+b 2 k 2 z+b 3 k 3t, (4.24 c c 2 b 2 + c 3 b 1 b 3 0. (4.25 9

10 5. Concluson and further questons We created a knd of generalzed blnear dfferental operators D p,x, dscussed ther lnks wth the Bell polynomals, and appled the lnear superposton prncple to the correspondng blnear equatons. Two llustratve examples n the case of p 3 were made to shed lght on the general framework. There are, however, many other questons whch are worth further nvestgaton. Below s just a few of them. Queston 1 - Mxed blnear equatons We can mx D p -operators wth dfferent natural numbers p to formulate a more general blnear equaton, for example, m p 1,,p l 1 1,, l 0 c p 1,,p l 1,, l D 1 p1,x 1 D l p l, f f 0, where the coeffcents c p 1,,p l 1,, l s are constants. Ths knd of mxed combnatons wll brng dversty n establshng lnks wth bnary Bell polynomals and formulatons for lnear subspaces of solutons. Queston 2 - Geometres related to multvarate polynomals What knd of geometres of a multvarate polynomal F does the equaton F(k 1 + αk 1,, k l + αk l + F(k 1 + αk 1,, k l + αk l 0 defne? It determnes an affne geometry of F when p 2k. Queston 3 - Parameterzatons acheved by multple parameters Parameterze k 1,,, k l, usng multple parameters, for example, two parameters k and l : w k j, b j,r k r l w r, 1 j l. r0 What knd of spaces can exst for the proportonal constants b j,r whch wll solve the system F(k 1, + αk 1,j,, k l, + αk l,j + F(k 1,j + αk 1,,, k l,j + αk l, 0, 1 j N? Queston 4 - Blnear Bäcklund transformatons In the case of the Hrota D-operators, bnary Bell polynomals are used to buld blnear Bäcklund transformatons for solton equatons [16]. Is there any theory n the general case of D p -operators? Ths case should be a lttle bt more dffcult than the Hrota case, notcng a fact that the sgn functon ( 1 r( n the defnton (2.2 does not satsfy ( 1 r(+j ( 1 r(+r(j,, j 0, when p > 1 s odd, whle t s true when p s one or even. It s obvous that the above property holds when p s one or even; but t does not hold because we have ( 1 r(p ( 1 r(p 1+r(1, due to ( 1 r(p 1, ( 1 r(p 1 1, ( 1 r(1 1, when p > 1 s odd. The property s also crucal n dervng Lax pars from blnear Bäcklund transformatons (see, e.g., [11, 12]. Queston 5 - Crteron for multvarate polynomals wth one zero Whle we used multvarate polynomals, whch have one zero and only one zero, to determne blnear equatons wth gven lnear subspaces of solutons, we came up wth an nterestng 10

11 queston [15]: How can one determne f a multvarate polynomal F(x 1,, wth real coeffcents has one and only one zero n R l? Two examples of such multvarate polynomals are as follows: x 2 + y 2 2y + 1, zero (x, y (0, 1; 5x 2 + 4xy + y 2 2x 2y + 2, zero (x, y ( 1, 3. Ths problem s more general than Hlbert s 17th problem, snce all such multvarate polynomals satsfy all the condtons n Hlbert s 17th problem. It s hoped that there would be a defntve answer to the queston. Acknowledgments The work was supported n part by the State Admnstraton of Foregn Experts Affars of Chna, the Natonal Natural Scence Foundaton of Chna (Nos and , Chunhu Plan of the Mnstry of Educaton of Chna, the Natural Scence Foundaton of Shangha (No. 09ZR and the Shangha Leadng Academc Dscplne Project (No. J References [1] Hrota R 2004 The Drect Method n Solton Theory (Cambrdge: Cambrdge Unversty Press [2] Hetarnta J 2005 Phys. AUC 15 (part 1 31 [3] Ma W X 2002 Phys. Lett. A [4] Ma W X and You Y 2005 Trans. Amer. Math. Soc [5] Hrota R 1974 Prog. Theor. Phys [6] Hrota R 1989 J. Phys. Soc. Jpn [7] Hu X B and Wang H Y 2006 Inverse Problems [8] Ma W X, Abdeljabbar A and Asaad M G 2011 Appl. Math. Comput [9] Bell E T 1934 Ann. Math [10] Crak A D D 2005 Amer. Math. Monthly [11] Lambert F, Sprngael J and Wllox R 1994 J. Phys. A: Math. Gen [12] Glson C, Lambert F, Nmmo J and Wllox R 1996 Proc. R. Soc. Lond. A [13] Ma W X and Fan E G 2011 Comput. Math. Appl [14] Ma W X 2012 World Appl. Sc. J. to appear [15] Ma W X, Zhang Y, Tang Y N and Tu J Y 2012 Appl. Math. Comput [16] Lambert F and Sprngael J 1997 J. Phys. Soc. Jpn