Computers and Mathematics with Applications. Linear superposition principle applying to Hirota bilinear equations

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1 Computers and Mathematcs wth Applcatons 61 (2011) Contents lsts avalable at ScenceDrect Computers and Mathematcs wth Applcatons journal homepage: Lnear superposton prncple applyng to Hrota blnear equatons Wen-Xu Ma a,, Engu Fan b,1 a Department of Mathematcs and Statstcs, Unversty of South Florda, Tampa, FL , USA b School of Mathematcal Scences and Key Laboratory of Mathematcs for Nonlnear Scence, Fudan Unversty, Shangha , PR Chna a r t c l e n f o a b s t r a c t Artcle hstory: Receved 6 October 2010 Accepted 21 December 2010 Keywords: Hrota s blnear form Solton equatons N-wave soluton A lnear superposton prncple of exponental travelng waves s analyzed for Hrota blnear equatons, wth an am to construct a specfc sub-class of N-solton solutons formed by lnear combnatons of exponental travelng waves. Applcatons are made for the dmensonal KP, Jmbo Mwa and BKP equatons, thereby presentng ther partcular N-wave solutons. An opposte queston s also rased and dscussed about generatng Hrota blnear equatons possessng the ndcated N-wave solutons, and a few llustratve examples are presented, together wth an algorthm usng weghts Elsever Ltd. All rghts reserved. 1. Introducton It s sgnfcantly mportant n mathematcal physcs to search for exact solutons to nonlnear dfferental equatons. Exact solutons play a vtal role n understandng varous qualtatve and quanttatve features of nonlnear phenomena. There are dverse classes of nterestng exact solutons, such as travelng wave solutons and solton solutons, but t often needs specfc mathematcal technques to construct exact solutons due to the nonlnearty present n dynamcs (see, e.g., [1,2]). Among the exstng theores, Hrota s blnear technque [3] provdes a drect powerful approach to nonlnear ntegrable equatons [4], and t s wdely used n constructng N-solton solutons [5] and even for ntegrable couplngs by perturbaton [6]. The exstence of N-solton solutons often mples the ntegrablty [7] of the consdered dfferental equatons. Interactons between soltons are elastc and nonlnear, but unfortunately, the lnear superposton prncple does not hold for soltons any more. However, blnear equatons are the nearest neghbors to lnear equatons, and expected to have some features smlar to those of lnear equatons. In ths paper, we would lke to explore a key feature of the lnear superposton prncple that lnear equatons possess, for Hrota blnear equatons, whle amng to construct a specfc sub-class of N-solton solutons formed by lnear combnatons of exponental travelng waves. More specfcally, we wll prove that a lnear superposton prncple can apply to exponental travelng waves of Hrota blnear equatons. Applcatons wll be made to show that the presented lnear superposton prncple s helpful n generatng N-wave solutons to solton equatons, partcularly those n hgher dmensons. Illustratve examples nclude the dmensonal KP, Jmbo Mwa and BKP equatons [8 12]. An opposte procedure s also proposed for generatng Hrota blnear equatons possessng N-wave solutons of lnear combnatons of exponental waves, along wth an algorthm usng weghts. A few new and general such Hrota blnear equatons are therefore computed. Correspondng author. Tel.: ; fax: E-mal addresses: mawx@cas.usf.edu (W.X. Ma), faneg@fudan.edu.cn (E.G. Fan). 1 Tel.: ; fax: /$ see front matter 2010 Elsever Ltd. All rghts reserved. do: /j.camwa

2 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) Lnear superposton prncple We begn wth a Hrota blnear equaton P(D x1, D x2,..., D xm )f f = 0, (2.1) where P s a polynomal n the ndcated varables satsfyng P(0, 0,..., 0) = 0, M and D x, 1 M, are Hrota s dfferental operators defned by D p y f (y) g(y) = ( y y ) p f (y)g(y ) y = p =y y f (y + y )g(y y ) y =0, p 1. (2.2) Varous nonlnear equatons of mathematcal physcs are wrtten as Hrota forms through dependent varable transformatons [3,5]. Let us ntroduce N wave varables η = k 1, x 1 + k 2, x k M, x M, 1 N, (2.3) and N exponental wave functons f = e η = e k 1,x 1 +k 2, x 2 + +k M, x M, 1 N, (2.4) where the k,j s are constants. Observng that we have a blnear dentty [3]: P(D x1, D x2,..., D xm ) e η e η j = P(k 1, k 1,j, k 2, k 2,j,..., k M, k M,j ) e η +η j, (2.5) t follows mmedately from (2.2) that all exponental wave functons f, 1 N, solve the Hrota blnear equaton (2.1). Now consder an N-wave testng functon f = ε 1 f 1 + ε 2 f ε N f N = ε 1 e η 1 + ε 2 e η ε N e η N, (2.6) where ε, 1 N, are arbtrary constants. Ths s a general lnear combnaton of N exponental travelng wave solutons. Naturally, we would lke to ask f t wll stll present a soluton to the Hrota blnear equaton (2.1) as each f does. The answer s postve. We wll show that a lnear superposton prncple of those exponental waves wll apply to Hrota blnear equatons, under some addtonal condton on the exponental waves and possbly on the polynomal P as well. Followng (2.5), we can compute that P(D x1, D x2,..., D xm )f f = = ε ε j P(D x1, D x2,..., D xm )e η e η j,j=1 ε ε j P(k 1, k 1,j, k 2, k 2,j,..., k M, k M,j )e η +η j. (2.7),j=1 Ths blnear property wll play a promnent role n establshng the lnear superposton prncple for the exponental waves e η, 1 N, and t s also a key tool n constructng quas-perodc wave solutons (see, e.g., [13 15]). It now follows drectly from (2.7) that any lnear combnaton of the N exponental wave solutons e η, 1 N, solves the Hrota blnear equaton (2.1) f the followng condton P(k 1, k 1,j, k 2, k 2,j,..., k M, k M,j ) = 0, 1 j N, (2.8) s satsfed. In ths condton (2.8), we excluded the case of = j, snce that case s just a consequence of (2.2). The condton (2.8) gves us a system of nonlnear algebrac equatons on the wave related numbers k,j s, when the polynomal P s fxed. We wll see that there s a better possblty of exstence of solutons for the varables k,j s n hgher dmensonal cases, because there are more varables to solve for n the resultng system of algebrac equatons. We conclude the above analyss n the followng theorem. Theorem 2.1 (Lnear Superposton Prncple). Let P(x 1, x 2,..., x M ) be a multvarate polynomal satsfyng (2.2) and the wave varables η, 1 N, be defned by (2.3). Then any lnear combnaton of the exponental waves e η, 1 N, solves the Hrota blnear equaton (2.1) f the condton (2.8) s satsfed.

3 952 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) Ths shows a lnear superposton prncple of exponental wave solutons that apples to Hrota blnear equatons, and paves a way of constructng N-wave solutons from lnear combnatons of exponental waves wthn the Hrota blnear formulaton. The system (2.8) s a key condton we need to handle. Once we get a soluton of the wave related numbers k,j s by solvng the system (2.8), we can present an N-wave soluton, formed by (2.6), to the consdered nonlnear equaton. Takng one of the wave varables η, 1 N, to be a constant, for example, takng η 0 = ε 0,.e., k,0 = 0, 1 M, where 1 0 N s fxed, the N-wave soluton condton (2.8) subsequently requres all other wave related numbers to satsfy the dsperson relaton P(k 1,, k 2,,..., k M, ) = 0, 1 N, 0. (2.10) Ths corresponds to a specfc case of N-solton solutons by the Hrota perturbaton technque [3], truncated at the secondorder perturbaton term. But generally, we want to emphasze that t s not necessary to satsfy the dsperson relaton. 3. Applcatons to solton equatons Let us shed lght on the lnear superposton prncple n Theorem 2.1 by three applcaton examples of constructng N-wave solutons dmensonal KP equaton The frst example s the dmensonal KP equaton [8,9]: (u t 6uu x + u xxx ) x + 3u yy + 3u zz = 0. Through the dependent varable transformaton u = 2(ln f ) xx, the dmensonal KP equaton (3.1) s wrtten as (D 4 + x D td x + 3D 2 + y 3D2 z )f f = 0, whch s equvalent to f xxxx f 4f xxx f x + 3f 2 xx + f txf f t f x + 3(f yy f f 2 y + f zzf f 2 z ) = 0. Assume that the N wave varables (2.3) are determned by η = k x + l y + m z + ω t, 1 N, (3.3) and then the N-wave soluton condton (2.8) becomes k 4 4k 3 k j + 6k 2 k2 j 4k k 3 j + k 4 j + k ω k ω j k j ω + k j ω j + 3l 2 6l l j + 3l 2 j + 3m 2 6m m j + 3m 2 j = 0, 1 j N. (3.4) By nspecton, a soluton to ths equaton s l = ak 2, m = bk 2, ω = 4k 3, 1 N, (3.5) where a 2 + b 2 = 1. Therefore by the lnear superposton prncple n Theorem 2.1, the dmensonal KP equaton (3.1) has the followng N-wave soluton u = 2(ln f ) xx, f = ε f = (2.9) (3.1) (3.2) ε e k x+ak 2 y+bk2 z 4k3 t, (3.6) where a 2 + b 2 = 1, and the k s and ε s are arbtrary constants. In ths soluton f, each exponental wave f satsfes the correspondng nonlnear dsperson relaton dmensonal Jmbo Mwa equaton The second example s the dmensonal Jmbo Mwa equaton [10]: u xxxy + 3(u x u y ) x + 2u yt 3u zz = 0. Through the dependent varable transformaton u = 2(ln f ) x, the 3+1 dmensonal Jmbo Mwa equaton (3.7) s wrtten as (D 3 x D y + 2D t D y 3D 2 z )f f = 0, whch equvalently reads (f xxxy + 2f ty 3f zz )f 3f xxy f x + 3f xy f xx f y f xxx 2f t f y + 3f 2 z = 0. (3.7) (3.8)

4 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) Assume that the N wave varables (2.3) are determned by (3.3), and then the N-wave soluton condton (2.8) becomes k 3 l k 3 l j 3k 2 k jl + 3k 2 k jl j + 3k k 2 j l 3k k 2 j l j k 3 j l + k 3 j l j + 2ω l 2ω l j 2ω j l + 2ω j l j 3m 2 + 6m m j 3m 2 j = 0, 1 j N. (3.9) Smlarly by nspecton, a soluton to ths equaton s l = a 2 k, m = ak 2, ω = 2k 3, 1 N, (3.10) where a s an arbtrary constant. Therefore by the lnear superposton prncple n Theorem 2.1, the dmensonal Jmbo Mwa equaton (3.7) has the followng N-wave soluton u = 2(ln f ) x, f = ε f = ε e k x a 2 k y+ak 2 z 2k3 t, (3.11) where a, and the k s and ε s are arbtrary constants. In ths soluton f, each exponental wave f satsfes the correspondng nonlnear dsperson relaton dmensonal BKP equaton As the thrd example, let us form and consder a dmensonal generalzaton of the BKP equaton: u zt u xxxy 3(u x u y ) x + 3u xx = 0. (3.12) If z = y, ths dmensonal BKP equaton reduces to the BKP equaton [11,12]: u yt u xxxy 3(u x u y ) x + 3u xx = 0. (3.13) Through the dependent varable transformaton u = 2(ln f ) x, the dmensonal BKP equaton (3.12) s wrtten as (D t D z D 3 x D y + 3D 2 x )f f = 0, (3.14) whch s equvalent to (f tz f xxxy + 3f xx )f f t f z + f xxx f y + 3f xxy f x 3f xx f xy 3f 2 x = 0. Assume that the N wave varables (2.3) are determned by (3.3), and then the N-wave soluton condton (2.8) becomes ω m ω m j ω j m + ω j m j k 3 l + k 3 l j + 3k 2 k jl 3k 2 k jl j 3k k 2 j l + 3k k 2 j l j + k 3 j l k 3 j l j + 3k 2 6k k j + 3k 2 j = 0, 1 j N. (3.15) Smlarly by nspecton, a soluton to ths equaton s l = k 1, m = ak 1, ω = 1 a k3, 1 N, (3.16) where a s an arbtrary non-zero constant. It then follows from the lnear superposton prncple n Theorem 2.1 that the dmensonal BKP equaton (3.12) has the followng N-wave soluton u = 2(ln f ) x, f = ε f = ε e k x+k 1 y+ak 1 z+(1/a)k 3 t, (3.17) where a and the k s are arbtrary non-zero constants and the ε s are arbtrary constants. However, none of the N exponental waves f, 1 N, n the soluton f satsfes the correspondng nonlnear dsperson relaton. The case a = 1 produces z = y, and so t gves an N-wave soluton to the BKP equaton (3.13): u = 2(ln f ) x, f = ε e k x+k 1 y+k 3 t. (3.18) 4. An opposte queston We would lke to propose an opposte procedure for conversely constructng Hrota blnear equatons that possess N-wave solutons formed by lnear combnatons of exponental waves. Ths s an opposte queston on applyng the lnear superposton prncple n Theorem 2.1.

5 954 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) We frst fnd a multvarate polynomal P(x 1, x 2,..., x M ) wth no constant term such that P(k 1,1 k 2,1, k 1,2 k 2,2,..., k 1,M k 2,M ) = 0, (4.1) for two sets of parameters k,1, k,2,..., k,m, = 1, 2, each of whch would better contan at least one free parameter. Then formulate a Hrota blnear equaton through (2.1) usng the polynomal P. Theorem 2.1 tells that the resultng Hrota blnear equaton possesses multple wave solutons of lnear combnatons of exponental travelng waves. Such a multvarate polynomal P can be normally found by balancng the nvolved free parameters n (4.1), and often upon assumng that two sets of parameters satsfy the dsperson relaton P(k,1, k,2,..., k,m ) = 0, = 1, 2. (4.2) Ths constructon procedure also brngs us an nterestng problem: how do we construct multvarate polynomals whch satsfy the property (4.1) guaranteeng the lnear superposton prncple for exponental wave solutons, when (4.2) holds? 4.1. Examples of equatons expressed n u One example s the followng polynomal P(x, y, z, t) = x 3 y + tx + ty z 2, whose correspondng condton (4.1) s gven by (4.3) P(k 1 k 2, l 1 l 2, m 1 m 2, ω 1 ω 2 ) = k 3 1 l 1 k 3 1 l 2 3k 2 1 k 2l 1 + 3k 2 1 k 2l 2 + 3k 1 k 2 2 l 1 3k 1 k 2 2 l 2 k 3 2 l 1 + k 3 2 l 2 + ω 1 l 1 ω 1 l 2 ω 2 l 1 + ω 2 l 2 + ω 1 k 1 ω 1 k 2 ω 2 k 1 + ω 2 k 2 m m 1m 2 m 2 2 = 0. Obvously, the correspondng Hrota blnear equaton reads that s, (D 3 x D y + D t D x + D t D y D 2 z )f f = 0, (f xxxy + f tx + f ty f zz )f 3f xxy f x + 3f xy f xx f y f xxx f t f x f t f y + f 2 z = 0. Under the dependent varable transformaton u = 2(ln f ) x, ths equaton s transformed nto u xxxy + 3(u x u y ) x + u tx + u ty u zz = 0. Followng the lnear superposton prncple of exponental waves, we have ts N-wave soluton u = 2(ln f ) x, f = ε f = ε e k x (1/3)a 2 k y+ak 2 z+4[a2 /(3 a 2 )]k 3 t, (4.6) where a ± 3, and the ε s and k s are arbtrary parameters. Each exponental wave f n the soluton f satsfes the correspondng nonlnear dsperson relaton. The other example s the followng polynomal P(x, y, z, t) = ty x 3 y + 3xz, whose correspondng condton (4.1) s gven by P(k 1 k 2, l 1 l 2, m 1 m 2, ω 1 ω 2 ) = ω 1 l 1 ω 1 l 2 ω 2 l 1 + ω 2 l 2 k 3 1 l 1 + k 3 1 l 2 + 3k 2 1 k 2l 1 3k 2 1 k 2l 2 Obvously, the correspondng Hrota blnear equaton reads that s, (D t D y D 3 x D y + 3D x D z )f f = 0, (f ty f xxxy + 3f xz )f f t f y + f xxx f y + 3f xxy f x 3f xx f xy 3f x f z = 0. 3k 1 k 2 2 l 1 + 3k 1 k 2 2 l 2 + k 3 2 l 1 k 3 2 l 2 + 3k 1 m 1 3k 1 m 2 3k 2 m 1 + 3k 2 m 2 = 0. Under the dependent varable transformaton u = 2(ln f ) x, ths equaton s transformed nto u yt u xxxy 3(u x u y ) x + 3u xz = 0. If z = x, the equaton (4.9) reduces to the BKP equaton (3.13). Followng the lnear superposton prncple of exponental waves, we have an N-wave soluton for (4.9): u = 2(ln f ) x, f = ε f = (4.4) (4.5) (4.7) (4.8) (4.9) ε e k x+(1/a)k 1 y+ak z+k 3 t, (4.10) where a 0, and the ε s and k s are arbtrary parameters. However, each exponental wave f n the soluton f does not satsfy the correspondng nonlnear dsperson relaton. The case of a = 1 produces z = x and gves the same N-wave soluton to the BKP equaton (3.13) as n (3.18).

6 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) Examples of equatons expressed n f An algorthm can be gven to use the concept of weghts, to compute examples of Hrota blnear equatons that possess the lnear superposton prncple of exponental waves. Let us frst defne the weghts of ndependent varables: (w(x 1 ), w(x 2 ),..., (w(x M ))), where each weght w(x ) s an nteger, and then form a homogeneous polynomal P(x 1, x 2,..., x M ) of some weght to check f t wll satsfy the condton (4.1). A nce dea to start our checkng s to assume that the wave varables η s nvolve arbtrary constants. Ths way, we can compare powers of those arbtrary constants n (4.1) to obtan algebrac equatons on other constants and/or coeffcents to solve. The followng are a few examples whch apply ths algorthm usng weghts Examples wth N-waves satsfyng the dsperson relaton Example 1. Weghts (w(x), w(y), w(z), w(t)) = (1, 2, 3, 3): Let us frst ntroduce the weghts of ndependent varables: (w(x), w(y), w(z), w(t)) = (1, 2, 3, 3). (4.11) Then, a general homogeneous polynomal of weght 4 reads P = c 1 x 4 + c 2 x 2 y + c 3 xz + c 4 xt + c 5 y 2. (4.12) Assume that the wave varables are η = k x + b 1 k 2 y + b 2k 3 z + b 3k 3 t, 1 N, (4.13) where k, 1 N, are arbtrary constants, but b 1, b 2 and b 3 are constants to be determned. Ths way, a drect computaton tells that we must have c 2 = 0 to keep the non-trvalty b 1 b 2 b 3 0, and b 1, b 2 and b 3 need to satsfy c 5 b 2 1 3c 1 = 0, c 3 b 2 + c 4 b 3 + 4c 1 = 0. (4.14) It follows now that the correspondng Hrota blnear equaton reads (c 1 D 4 + x c 3D x D z + c 4 D x D t + c 5 D 2 y )f f = 0, (4.15) and t possesses an N-wave soluton f = ε f = ε e k x+b 1 k 2 y+b 2k 3 z+b 3k 3 t, (4.16) where the ε s and k s are arbtrary, but b 1, b 2 and b 3 satsfy (4.14). The frst equaton n (4.14) requres c 1 c 5 > 0, to guarantee a non-zero real soluton for b 1. The soluton of the second equaton of b 2 and b 3 n (4.14) contans an arbtrary constant f c 3 c 4 0. We pont out that when both those condtons, c 1 c 5 > 0 and c 3 c 4 0, are satsfed, the resultng equaton (4.15) actually can be transformed nto the KP equaton. Example 2. Weghts (w(x), w(y), w(z), w(t)) = (1, 2, 2, 3): Let us second ntroduce the weghts of ndependent varables: (w(x), w(y), w(z), w(t)) = (1, 2, 2, 3). (4.17) Then, a general homogeneous polynomal of weght 4 reads P = c 1 x 4 + c 2 x 2 y + c 3 x 2 z + c 4 xt + c 5 y 2 + c 6 yz + c 7 z 2. (4.18) Assume that the wave varables are η = k x + b 1 k 2 y + b 2k 2 z + b 3k 3 t, 1 N, (4.19) where k, 1 N, are arbtrary constants, but b 1, b 2 and b 3 are constants to be determned. Ths way, a drect computaton tells that the correspondng Hrota blnear equaton P(D x, D y, D z, D t )f f = 0 possesses an N-wave soluton f = ε f = ε e k x+b 1 k 2 y+b 2k 2 z+b 3k 3 t, (4.20)

7 956 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) where the ε s and k s are arbtrary, but b 1, b 2 and b 3 satsfy c 4 b 3 + 4c 1 = 0, c 2 b 1 + c 3 b 2 = 0, c 5 b c 7b c 6b 1 b 2 = 3c 1. (4.21) The frst equaton above determnes b 3 unquely f c 4 0 or does not present any condton on b 3 f c 1 = c 4 = 0. There are two cases to determne b 1 and b 2, whch are depcted as follows. (a) The case of c 2 c 3 0: Then we need c 1 (c 2 2 c 7 c 2 c 3 c 6 + c 2 3 c 5) > 0, to have non-zero b 1 and b 2. Under ths condton, the solutons for b 1 and b 2 are gven by c3 3c1 d (b 1, b 2 ) =, c 2 3c1 d, (b 1, b 2 ) = c 3 3c1 d, c 2 3c1 d, (4.23) d d d d where d = c 2 2 c 7 c 2 c 3 c 6 + c 2 3 c 5. (b) The case of c 2 = c 3 = 0: Then f c 5 0, b 2 needs to satsfy (c 2 6 4c 5c 7 )b c 1c 5 0, to have a real b 1, and b 1 s determned n terms of b 2 by the thrd equaton n (4.21). The value of b 2 may have lots of chooses. For example, f c 1 c 5 0, c 2 6 4c 5c 7 0, then b 2 s arbtrary; f c 1 c 5 0, c 2 6 4c 5c 7 < 0, then b 2 must be 2 3c 1 c 5 (4c 5 c 7 c 2 6 ) c 2 6 4c 5c 7 b 2 2 Symmetrcally, f c 7 0, b 2 needs to satsfy (c 2 6 4c 5c 7 )b c 1c 7 0, 3c 1 c 5 (4c 5 c 7 c 2) 6 c c 5c 7 to have a real b 2, and b 2 s determned n terms of b 1 by the thrd equaton n (4.21). The value of b 1 may have lots of chooses. For example, f c 1 c 7 0, c 2 6 4c 5c 7 0, then b 1 s arbtrary; f c 1 c 7 0, c 2 6 4c 5c 7 > 0, then b 1 must be b 1 2 3c 1 c 7 (4c 5 c 7 c 2 6 ) c 2 6 4c 5c 7 or b 1 2 3c 1 c 7 (4c 5 c 7 c 2) 6 c c 5c 7 We pont out that n ths case, f c 1 c 5 c 6 c 7 0, then there wll defntely be non-zero b 1 and b 2. Ths s because we have c 7 b 2 2 = 3c 1, (c 2 6 4c 5c 7 )b c 1c 5 = 0, whch leads to c 6 = 0, f, for example, b 1 has to be zero. Example 3. Weghts (w(x), w(y), w(z), w(t)) = (1, 1, 2, 3): Let us thrd ntroduce the weghts of ndependent varables: (4.22) (4.24) (4.25) (w(x), w(y), w(z), w(t)) = (1, 1, 2, 3). (4.26) Then, a general homogeneous polynomal of weght 4 reads P = c 1 x 4 + c 2 y 4 + c 3 x 3 y + c 4 xy 3 + c 5 x 2 y 2 + c 6 x 2 z + c 7 y 2 z + c 8 xt + c 9 yt + c 10 z 2. (4.27)

8 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) Assume that the wave varables are η = k x + b 1 k y + b 2 k 2 z + b 3k 3 t, 1 N, (4.28) where k, 1 N, are arbtrary constants, but b 1, b 2 and b 3 are constants to be determned. In ths example, the correspondng Hrota blnear equaton P(D x, D y, D z, D t )f f = 0 has an assocated N-wave solutons f = ε f = ε e k x+b 1 k y+b 2 k 2 z+b 3k 3 t. (4.29) There are four cases to determne the nvolved constants b 1, b 2 and b 3 as follows. (a) The case of c 7 0 and c 10 0: In ths case, a drect computaton yelds and where b 2 1 = c 6 c 7, (4.30) b 3 = 4(c 2b c 4b c 5b c 3b 1 + c 1 ) c 9 b 1 + c 8, b 2 2 = 3b 3(c 9 b 1 + c 8 ) 4c 10, (4.31) c 9 b 1 + c 8 0, c 2 b c 4b c 5b c 3b 1 + c 1 0, (4.32) whch actually gve two condtons on the Hrota blnear equaton consdered. (b) The case of c 7 = 0 but c 10 0: In ths case, we automatcally have c 6 = 0, to keep the non-trvalty b 1 b 2 b 3 0. A smlar drect computaton shows that b 1 needs to satsfy (4.32), and that b 2 and b 3 are defned by (4.31). (c) The case of c 7 = c 10 = 0 but c 9 0: In ths case, we automatcally have c 6 = 0 and c 8 0, to keep the non-trvalty b 1 b 2 b 3 0. A smlar drect computaton tells that b 1 = c 8 c 9 (4.33) and b 2 s arbtrary, and that c 2 c 4 8 c 4c 3 8 c 9 + c 5 c 2 8 c2 9 c 3c 8 c c 1c 4 9 = 0. (4.34) Ths condton on the Hrota blnear equaton consdered s equvalent to c 2 b c 4b c 5b c 3b 1 + c 1 = 0. (4.35) (d) The case of c 7 = c 9 = c 10 = 0: In ths case, we automatcally have c 6 = c 8 = 0, to keep the non-trvalty b 1 b 2 b 3 0. Ths tme, b 1 must satsfy (4.35), but b 2 can be arbtrary Examples wth N-waves not satsfyng the dsperson relaton Example 4. Weghts (w(x), w(y), w(z), w(t)) = (1, 1, 1, 3): Let us now ntroduce the weghts of ndependent varables: (w(x), w(y), w(z), w(t)) = (1, 1, 1, 3). (4.36) Then, a homogeneous polynomal of weght 2 reads P = c 1 ty + c 2 tz + c 3 x 2 + c 4 x 3 y + c 5 x 3 z + c 6 x 4 y 2 + c 7 x 4 z 2. (4.37) Assume that the wave varables are η = k x + b 1 k 1 y + b 2 k 1 z + b 3 k 3 t, 1 N, (4.38) where k, 1 N, are arbtrary constants, but b 1, b 2 and b 3 are constants to be determned. Now, a smlar drect computaton tells that the correspondng Hrota blnear equaton (c 1 D t D y + c 2 D t D z + c 3 D 2 + x c 4D 3 x D y + c 5 D 3 x D z + c 6 D 4 x D2 + y c 7D 4 x D2 z )f f = 0 (4.39)

9 958 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) possesses an N-wave soluton f = ε f = ε e k x+b 1 k 1 y+b 2 k 1 where the ε s and k s are arbtrary, but b 1, b 2 and b 3 satsfy c 3 + 3c 4 b 1 + 3c 5 b 2 = 0, c 3 3c 1 b 1 b 3 3c 2 b 2 b 3 = 0, c 6 b c 7b 2 = 0. 2 z+b 3 k 3 t, (4.40) The BKP equaton (4.8) s just one specal example wth c 2 = 1, c 3 = 3, c 4 = 1, and the other coeffcents beng zero. An analyss on the exstence of non-zero real b 1 and b 2 can be gven smlarly. Example 5. Weghts (w(x), w(y), w(z), w(t)) = (1, 1, 2, 3): Let us fnally ntroduce the weghts of ndependent varables: (4.41) (w(x), w(y), w(z), w(t)) = (1, 1, 2, 3). (4.42) Then, a homogeneous polynomal of weght 2 reads P = c 1 ty + c 2 x 3 y + c 3 x 4 y 2 + c 5 x 4 z + c 6 x 2. (4.43) Assume that the wave varables are η = k x + b 1 k 1 y + b 2 k 2 z + b 3 k 3 t, 1 N, (4.44) where k, 1 N, are arbtrary constants, but b 1, b 2 and b 3 are constants to be determned. Now, a smlar drect computaton tells that the correspondng Hrota blnear equaton (c 1 D t D y + c 2 D 3 x D y + c 3 D 4 x D2 + y c 5D 4 x D z + c 6 D 2 x )f f = 0 (4.45) possesses an N-wave soluton f = ε f = ε e k x+b 1 k 1 y+b 2 k 2 where the ε s and k s are arbtrary, but b 1, b 2 and b 3 satsfy 3c 2 b 1 + c 6 = 0, c 1 b 1 b 3 + c 2 b 1 = 0, c 3 b 2 = 1 c 5b 2 = 0. Then, takng c 3 = c 5 = 0 tells that the Hrota blnear equaton z+b 3 k 3 t, (4.46) (4.47) (c 1 D t D y + c 2 D 3 x D y + c 6 D 2 x )f f = 0, (4.48) has the N-wave soluton defned by (4.46) wth b 1 = c 6 3c 2 and b 3 = c 2 c Concludng remarks To summarze, we analyzed a specfc sub-class of N-solton solutons, formed by lnear combnatons of exponental travelng waves, for Hrota blnear equatons. The startng pont s to solve a system of nonlnear algebrac equatons for the wave related numbers, called the N-wave soluton condton. The resultng system tells what Hrota blnear equatons the lnear superposton prncple of exponental waves wll apply to. Hgher dmensonal Hrota blnear equatons have a better opportunty to satsfy the N-wave soluton condton snce there are more parameters to choose from. Applcatons were made for the 3+1 dmensonal KP, Jmbo Mwa and BKP equatons, thereby presentng ther partcular exact multple wave solutons. Moreover, an opposte procedure was proposed for conversely generatng Hrota blnear equatons possessng multple wave solutons formed by lnear combnatons of exponental waves, and a few such examples were computed. Generally speakng, the lnear superposton prncple does not apply to nonlnear dfferental equatons. But, t can hold for some specal knd of wave solutons to Hrota blnear equatons, for example, for exponental waves as we explored. Ths also tells, to some extent, why Hrota blnear forms are so effectve n presentng determnant solutons and pfaffan solutons [16 18], and of course, determnant and pfaffan solutons are more general than the consdered N-wave solutons. Among further nterestng questons s to establsh the lnear superposton prncple that apples to constraned solton

10 W.X. Ma, E.G. Fan / Computers and Mathematcs wth Applcatons 61 (2011) equatons, solton equatons wth self-consstent sources and extended solton equatons (see, e.g., [19], [20] and [21 23], respectvely). We would also lke to repeat a sub-mathematcal problem related to our opposte queston: s t feasble to desgn an algorthm to construct multvarate polynomals P(x 1, x 2,..., x M ) whch satsfy the followng property: when P(k 1,1 k 1,2, k 2,1 k 2,2,..., k M,1 k M,2 ) = 0, P(k 1,, k 2,,..., k M, ) = 0, = 1, 2? Ths s an nterestng mathematcal problem n the study of polynomals whose zeros form a vector space, and t determnes one class of Hrota blnear equatons possessng the lnear superposton prncple of exponental waves. In our dscusson above, we only analyzed some specfc such Hrota blnear equatons. We expect to see more examples and, of curse, a systematcal theory fnally. Acknowledgements The work was supported n part by the Establshed Researcher Grant, the CAS faculty development grant and the CAS Dean research grant of the Unversty of South Florda, Chunhu Plan of the Mnstry of Educaton of Chna, Natonal Key Basc Research Project of Chna, Shangha Shuguang Trackng Project and the State Admnstraton of Foregn Experts Affars of Chna. References [1] W.X. Ma, Dversty of exact solutons to a restrcted Bot Leon Pempnell dspersve long-wave system, Phys. Lett. A 319 (2003) [2] H.C. Hu, B. Tong, S.Y. Lou, Nonsngular poston and complexton solutons for the coupled KdV system, Phys. Lett. A 351 (2006) [3] R. Hrota, The Drect Method n Solton Theory, Cambrdge Unversty Press, [4] A.P. Fordy, Solton Theory: A Survey of Results. Nonlnear Scence: Theory and Applcatons, Manchester Unversty Press, Manchester, [5] J. Hetarnta, Hrota s blnear method and solton solutons, Phys. AUC 15 (part 1) (2005) [6] W.X. Ma, W. Srampp, Blnear forms and Bäcklund transformatons of the perturbaton systems, Phys. Lett. A 341 (2005) [7] W.X. Ma, Integrablty, n: A. Scott (Ed.), Encyclopeda of Nonlnear Scence, Taylor & Francs, New York, 2005, pp [8] M.J. Ablowtz, H. Segur, On the evoluton of packets of water waves, J. Fud Mech. 92 (1979) [9] E. Infeld, G. Rowlands, Three-dmensonal stablty of Korteweg de Vres waves and soltons II, Acta Phys. Polon. A 56 (1979) [10] M. Jmbo, T. Mwa, Soltons and nfnte dmensonal Le algebras, Publ. Res. Inst. Math. Sc. 19 (1983) [11] E. Date, M. Jmbo, M. Kashwara, T. Mwa, Transformaton groups for solton equatons VI KP herarches of orthogonal and symplectc type, J. Phys. Soc. Japan 50 (1981) [12] E. Date, M. Jmbo, M. Kashwara, T. Mwa, Transformaton groups for solton equatons IV a new herarchy of solton equatons of KP-type, Physca D 4 (1981/1982) [13] W.X. Ma, R.G. Zhou, L. Gao, Exact one-perodc and two-perodc wave solutons to Hrota blnear equatons n (2 + 1) dmensons, Modern Phys. Lett. A 21 (2009) [14] E.G. Fan, Quas-perodc waves and an asymptotc property for the asymmetrcal Nzhnk Novkov Veselov equaton, J. Phys. A: Math. Theor. 42 (2009) [15] E.G. Fan, Supersymmetrc KdV Sawada Kotera Raman equaton and ts quas-perodc wave solutons, Phys. Lett. A 374 (2010) [16] R. Hrota, Y. Ohta, J. Satsuma, Wronskan structures of solutons for solton equatons, n: Recent Developments n Solton Theory, Progr. Theoret. Phys. Suppl. 94 (1988) [17] W.X. Ma, Y. You, Solvng the Korteweg de Vres equaton by ts blnear form: Wronskan solutons, Trans. Amer. Math. Soc. 357 (2005) [18] X.B. Hu, J.X. Zhao, Commutatvty of Pfaffanzaton and Bäcklund transformatons: the KP equaton, Inverse Problems 21 (2005) [19] B. Konopelchenko, W. Strampp, The AKNS herarchy as symmetry constrant of the KP herarchy, Inverse Problems 7 (1991) L17 L24. [20] R.L. Ln, Y.B. Zeng, W.X. Ma, Solvng the KdV herarchy wth self-consstent sources by nverse scatterng method, Physca A 291 (2001) [21] X.J. Lu, Y.B. Zeng, R.L. Ln, A new extended KP herarchy, Phys. Lett. A 372 (2008) [22] W.X. Ma, An extended Harry Dym herarchy, J. Phys. A: Math. Theor. 43 (2010) [23] W.X. Ma, Commutatvty of the extended KP flows, Commun. Nonlnear Sc. Numer. Smul. 16 (2011)