Generalized bilinear differential equations

Generalized bilinear differential equations Wen-Xiu Ma Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Abstract We introduce a kind of bilinear differential equations by generalizing Hirota bilinear operators, and explore when the linear superposition principle can apply to the resulting generalized bilinear differential equations. Together with an algorithm using weights, two examples of generalized bilinear differential equations are computed to shed light on the presented general scheme for constructing bilinear differential equations which possess linear subspaces of solutions. Key words: Bilinear differential equation, Linear superposition principle, Subspace of solutions PACS: 02.30.Ik, 02.30.Xx, 05.45.Yv 1 Introduction It is known that the Hirota bilinear form provides an efficient tool to solve nonlinear differential equations of mathematical physics [1]. Particularly, based on the Hirota bilinear form, soliton solutions can be generated by the Hirota perturbation technique [1] or the multiple exp-function algorithm [2]. It is interesting to note that the linear superposition principle can also apply to Hirota bilinear equations and present their resonant soliton solutions [3]. Such solutions would be dense in the solution set of a function space, appropriately equipped with an inner product, and play an important role in formulating basic approximate solutions to initial value problems. Many important equations of mathematical physics are rewritten in the Hirota bilinear form through dependent variable transformations [1, 4]. For instance, the KdV equation u t + 6uu x + u xxx = 0, the Boussinesq equation u tt + (u 2 ) xx + u xxxx = 0, Email: mawx@cas.usf.edu, Tel: 1-813-9749563, Fax: 1-813-9742700 1

and the KP equation which can be expressed as under the transformation u = 2(ln f) xx, under the transformation u = 6(ln f) xx, and (u t + 6uu x + u xxx ) x + u yy = 0, (D x D t + D 4 x)f f = 0 (D 2 t + D 4 x)f f = 0 (D t D x + D 4 x + D 2 y)f f = 0 under the transformation u = 2(ln f) xx, respectively. In the above equations, D x, D y and D t are Hirota bilinear operators, and generally, we have Dx m Dy n Dt k f g = ( x ) m ( x y ) n ( y t ) kf(x, y, t)g(x, y, t ) t, x =x,y =y,t =t for nonnegative integers m, n and k. Wronskian solutions, including solitons, positons and complexitons [5]-[8], can be presented precisely, on the basis of the Hirota bilinear form. In this letter, we would like to introduce a kind of generalized bilinear differential operators and explore when the linear superposition principle applies to the corresponding bilinear differential equations. The resulting theory paves a way to construct a new kind of bilinear differential equations which possess linear subspaces of solutions. The considered solutions are linear combinations of exponential traveling wave solutions, and the involved exponential wave solutions may and may not satisfy the corresponding dispersion relations. All the obtained results will exhibit that there are bilinear differential equations different from Hirota bilinear equations, which share some common features with linear differential equations. The letter is structured as follows. In Section 2, we will generalize Hirota bilinear operators and introduce a kind of generalized bilinear differential equations. In Section 3, we will analyze the linear superposition principle for exponential traveling waves and establish a criterion for guaranteeing the existence of linear subspaces of exponential traveling wave solutions to generalized bilinear differential equations. In Section 4, we will present two examples of newly introduced bilinear differential equations, together with an algorithm using weights to compute. In the final section, we will make a few concluding remarks. 2 Bilinear differential operators and equations Let M, p N be given. We introduce a kind of bilinear differential operators: = D n 1 p,x 1 D n M p,xm f g M ( x i + α x i ) ni f(x 1,, x M )g(x 1,, x M) x, (2.1) 1 =x 1,,x M =x M 2

where n 1,, n M are arbitrary nonnegative integers and for an integer m, the mth power of α is defined by α m = ( 1) r(m), if m r(m) mod p with 0 r(m) < p. (2.2) For example, if p = 2k (k N), all above bilinear differential operators are Hirota bilinear operators, since D 2k,x = D x [1, 9]. If p = 3, we have which tells the pattern of symbols and if p = 5, we have which tells the pattern of symbols α = 1, α 2 = 1, α 3 = 1, α 4 = 1, α 5 = 1, α 6 = 1,,, +, +,, +, +, (p = 3); α = 1, α 2 = 1, α 3 = 1, α 4 = 1, α 5 = 1, α 6 = 1, α 7 = 1, α 8 = 1, α 9 = 1, α 10 = 1,,, +,, +, +,, +,, +, +, (p = 5). Note that the pattern of symbols for defining Hirota bilinear operators is, +,, +,, +, (p = 2), and similarly when p = 7, the pattern of symbols is, +,, +,, +, +,, +,, +,, +, +, (p = 7). Following those patterns of symbols, some new bilinear differential operators can be computed: and D 3,x f g = f x g fg x, D 3,x D 3,t f g = f xt g f x g t f t g x + fg xt, D 3 3,xf g = f xxx g 3f xx g x + 3f x g xx + fg xxx, D 2 3,xD 3,t f g = f xxt g f xx g t 2f xt g x + 2f x g xt + f t g xx + fg xxt, D 3 3,xD 3,t f g = f xxxt g f xxx g t 3f xxt g x + 3f xt g xx + 3f xx g xt + 3f x g xxt + f t g xxx fg xxxt, D 4 3,xf g = f xxxx g 4f xxx g x + 6f xx g xx + 4f x g xxx fg xxxx, D 5 3,xf g = f xxxxx g 5f xxxx g x + 10f xxx g xx + 10f xx g xxx 5f x g xxxx + fg xxxxx ; D 5,x f g = f x g fg x, D 5,x D 5,t f g = f xt g f x g t f t g x + fg xt, D 3 5,xf g = f xxx g 3f xx g x + 3f x g xx fg xxx, D 2 5,xD 5,t f g = f xxt g f xx g t 2f xt g x + 2f x g xt + f t g xx fg xxt, D 3 5,xD 5,t f g = f xxxt g f xxx g t 3f xxt g x + 3f xt g xx + 3f xx g xt 3f x g xxt f t g xxx + fg xxxt, D 4 5,xf g = f xxxx g 4f xxx g x + 6f xx g xx 4f x g xxx + fg xxxx, D 5 5,xf g = f xxxxx g 5f xxxx g x + 10f xxx g xx 10f xx g xxx + 5f x g xxxx + fg xxxxx. 3

Immediately from those formulas, we see that except D 3 5,xf f = 0, D 3 3,xf f, D 5 3,xf f and D 5 5,xf f do not equal to zero, which is different from the Hirota case: D 3 xf f = D 5 xf f = 0. Now let P be a polynomial in M variables and introduce a generalized bilinear differential equation: P (D p,x1,, D p,xm )f f = 0. (2.3) Particularly, when p = 3, we have the generalized bilinear KdV equation (D 3,x D 3,t + D 4 3,x)f f = 2 f xt f 2 f x f t + 6 f xx 2 = 0, (2.4) the generalized bilinear Boussinesq equation and the generalized bilinear KP equation (D 2 3,t + D 4 3,x)f f = 2 f tt f 2 f t 2 + 6 f xx 2 = 0, (2.5) (D 3,t D 3,x + D 4 3,x + D 2 3,y)f f = 2 f xt f 2 f x f t + 6 f xx 2 + 2 f yy f 2 f y 2 = 0. (2.6) We would like to discuss linear subspaces of solutions to the generalized bilinear differential equations defined by (2.3). More exactly, as in the Hirota case [3], we want to explore when the linear superposition principle will apply to the generalized bilinear differential equations (2.3). 3 Linear superposition principle Let us now fix N N and introduce N wave variables and N exponential wave functions η i = k 1,i x 1 + + k M,i x M, 1 i N, (3.1) f i = e η i = e k 1,ix 1 + +k M,i x M, 1 i N, (3.2) where the k j,i s are constants. Note that we have a bilinear identity: P (D p,x1,, D p,xm ) e ηi e η j = P (k 1,i + αk 1,j,, k M,i + αk M,j ) e η i+η j, (3.3) where the powers of α obey the rule (2.2). Then we consider a linear combination solution to the bilinear differential equation (2.3): f = ε 1 f 1 + + ε N f N = ε 1 e η 1 + + ε N e η N, (3.4) 4

where ε i, 1 i N, are arbitrary constants. Thanks to (3.3), we can compute that = = = P (D p,x1,, D p,xm )f f ε i ε j P (D p,x1,, D p,xm ) e ηi e η j i,j=1 ε i ε j P (k 1,i + αk 1,j,, k M,i + αk M,j ) e η i+η j i,j=1 ε [ 2 i P (k 1,i + αk 1,i,, k M,i + αk M,i ) e 2 η i + 1 i<j N ε i ε j [P (k 1,i + αk 1,j,, k M,i + αk M,j ) +P (k 1,j + αk 1,i,, k M,j + αk M,i )] e η i+η j. (3.5) It thus follows that a linear combination function f defined by (3.4) solves the generalized bilinear differential equation (2.3) if and only if P (k 1,i +αk 1,j,, k M,i +αk M,j )+P (k 1,j +αk 1,i,, k M,j +αk M,i ) = 0, 1 i j N, (3.6) are satisfied, where the powers of α obey the rule (2.2). The conditions in (3.6) present a system of nonlinear algebraic equations on the wave related numbers k i,j s and the coefficients of the polynomial P. Generally, it is not easy to solve (3.6). But in many cases, such systems have various sets of solutions. In the next section, we will present an idea to solve. We now summarize our result as follows. Theorem 3.1 (Criterion for linear superposition principle) Let P (x 1,, x M ) be a polynomial in the indicated variables and the N wave variables η i, 1 i N, be defined by η i = k 1,i x 1 + + k M,i x M, 1 i N, where the k i,j s are all constants. Then any linear combination of the exponential waves e η i, 1 i N, solves the generalized bilinear differential equation P (D p,x1,, D p,xm )f f = 0 if and only if the conditions in (3.6) are satisfied. This theorem tells us that the linear superposition principle can apply to generalized bilinear differential equations defined by (2.3). It also paves a way of constructing N-wave solutions to generalized bilinear differential equations. The system (3.6) is a resonance condition we need to deal with (see [10] for resonance of 2-solitons for Hirota bilinear equations). As soon as we find a solution of the wave related numbers k i,j s to the system (3.6), we can present an N-wave solution, formed by (3.4), to the considered generalized bilinear differential equation (2.3). 5

4 Applications Let us now consider how to compute examples of generalized bilinear differential equations, defined by (2.3), with linear subspaces of solutions, by applying Theorem 3.1. The problem is how to construct a multivariate polynomial P (x 1, x M ) such that P (k 1,1 + αk 1,2,, k M,1 + αk M,2 ) + P (k 1,2 + αk 1,1,, k M,2 + αk M,1 ) = 0, (4.1) holds for two sets of constants k 1,i,, k M,i, i = 1, 2, where the powers of α obey the rule (2.2). Our basic idea to solve is to introduce weights for the independent variables and then use parameterizations of wave numbers and frequencies. Let us first introduce the weights for the independent variables: (w(x 1 ),, w(x M )) = (n 1,, n M ), (4.2) where each weight w(x i ) = n i is an integer, and then form a polynomial P (x 1,, x M ) being homogeneous in some weight. Second, for i = 1, 2, we parameterize the constants k 1,i,, k M,i, consisting of wave numbers and frequencies, using a free parameter k i as follows: k j,i = b j k i n i, 1 j M, (4.3) where the b j s are constants to be determined. This parameterization balances the degrees of the free parameters in the system (4.1). Then, plugging the parameterized constants (4.3) into (4.1), we collect terms by powers of the parameters k 1 and k 2, and set the coefficient of each power to be zero, to obtain algebraic equations on the constants b j s and the coefficients of the polynomial P. Finally, solve the resulting algebraic equations to determine the polynomial P and the parameterization. Now, the resulting solution obviously yields a generalized bilinear differential equation defined by (2.3) and a linear subspace of its solutions given by f = ε i f i = ε i e b n 1k i 1x1 n + +b M k i M xm, N 1, (4.4) where the ε i s and k i s are arbitrary constants. In what follows, we present two illustrative examples in 3+1 dimensions, which apply the above parameterization achieved by using one free parameter. Example 1. Example with positive weights: Let us introduce the weights of independent variables: (w(x), w(y), w(z), w(t)) = (1, 2, 3, 4). (4.5) Then, a general even polynomial being homogeneous in weight 5 reads P = c 1 x 5 + c 2 x 3 y + c 3 x 2 z + c 4 xt + c 5 yz. (4.6) 6

Following the parameterization of wave numbers and frequency in (4.3), the wave variables read η i = k i x + b 1 k i 2 y + b 2 k i 3 z + b 3 k i 4 t, 1 i N, where k i, 1 i N, are arbitrary constants, but b 1, b 2 and b 3 are constants to be determined. In this example, the corresponding generalized bilinear differential equation reads P (D 3,x, D 3,y, D 3,z, D 3,t )f f = 2 c 1 f xxxxx f 10 c 1 f xxxx f x + 20 c 1 f xxx f xx +6 c 2 f xx f xy + 2 c 3 f xxz f + 2 c 4 f xt f 2 c 4 f x f t + 2 c 5 f yz f 2 c 5 f y f z = 0. (4.7) The corresponding linear subspace of N-wave solutions is given by f = ε i f i = ε i e k ix+b 1 k 2 i y+b 2 k 3 i z+b 3 k 4t i, (4.8) where ε i, 1 i N, are arbitrary constants but b 1, b 2, b 3 need to satisfy c 5 b 1 b 2 + c 4 b 3 + c 1 + c 3 b 2 = 0, 5 c 1 c 4 b 3 = 0, 10 c 1 c 5 b 1 b 2 + 3 c 2 b 1 = 0. A special solution to (4.9) is when the coefficients of the polynomial P satisfy (4.9) b 1 = 5 c 3 c 5, b 2 = 3 c 2 c 5, b 3 = 15 c 2c 3 c 4 c 5, (4.10) c 1 = 3 c 2c 3 c 5. (4.11) If c 4 = 0, then c 1 = 0, and further a non-trivial (e.g., b 1 b 2 0) solution of b 1 and b 2 is given by but b 3 is arbitrary. If c 4 0, then b 1 = c 3 c 5, b 2 = 3 c 2 c 5, (4.12) b 3 = 5 c 1 c 4, (4.13) and further two non-trivial (e.g., b 1 b 2 0) solutions of b 1 and b 2 are determined by { c3 b 2 4 c 1 = 0, 3 c 2 b 1 + 10 c 1 = 0, (4.14) when c 5 = 0, and { c3 c 5 b 2 2 + (6 c 1 c 5 3 c 2 c 3 )b 2 + 12 c 1 c 2 = 0, c 5 b 1 b 2 + c 3 b 2 4 c 1 = 0, (4.15) 7

when c 5 0. The formula (4.15) provides a large class of generalized bilinear equations which possess the discussed N-wave solutions. Example 2. Example with positive and negative weights: Let us introduce the weights of independent variables: Then, a polynomial being homogeneous in weight 2 reads (w(x), w(y), w(t)) = (1, 1, 3). (4.16) P = c 1 x 2 + c 2 x 3 y + c 3 x 4 y 2 + c 4 yt. (4.17) Following the parameterization of wave numbers and frequency in (4.3), the wave variables read η i = k i x + b 1 k i 1 y + b 2 k i 3 t, 1 i N, where k i, 1 i N, are arbitrary constants, but b 1 and b 2 are constants to be determined. Now, a similar direct computation tells that the corresponding generalized bilinear differential equation (c 1 D 2 3,x + c 2 D 3 3,xD 3,y + c 3 D 4 3,xD 2 3,y + c 4 D 3,t D 3,y )f f = 0 possesses the linear subspace of N-wave solutions determined by f = ε i f i = ε i e k ix+b 1 k 1 i y+b 2 k 3t i, (4.18) where the ε i s and k i s are arbitrary, but b 1 and b 2 satisfy c 1 + 6 c 3 b 1 2 = 0, c 4 b 1 b 2 + 4 c 3 b 1 2 = 0, c 4 b 1 b 2 + c 3 b 1 2 + c 1 + 3 c 2 b 1 = 0. (4.19) Therefore, the coefficients of the polynomial P need satisfy 121 c 1 c 3 = 54 c 2 2, (4.20) and then the corresponding generalized bilinear differential equation P (D 3,x, D 3,y, D 3,z, D 3,t )f f = 2 c 1 f xx f 2 c 1 f x 2 + 6 c 2 f xx f xy + 2 c 3 f xxxxyy f +8 c 3 f xxx f xyy + 12 c 3 f xxy 2 + 2 c 4 f yt f 2 c 4 f y f t = 0 (4.21) has the N-wave solution defined by (4.18) with b 1 = 11 c 1 18 c 2, b 2 = 12 c 2 11 c 4. (4.22) 8

5 Concluding remarks We introduced a new kind of bilinear differential operators and analyzed when the corresponding generalized bilinear equations possess the linear superposition principle. Particularly, we computed two examples by an algorithm using weights and their linear subspaces of exponential traveling wave solutions. The balance requirement of weights allows us to present a class of parameterizations of wave numbers and frequencies, Theorem 3.1 presents a sufficient and necessary criterion for guaranteeing the applicability of the linear superposition principle to a kind of generalized bilinear differential equations. It also follows from Theorem 3.1 that if we begin with an arbitrary multivariate polynomial P (x 1,, x M ), then a sufficient and necessary condition to guarantee the applicability of the discussed linear superposition principle is P (k 1,i + αk 1,j,, k M,i + αk M,j ) + P (k 1,j + αk 1,i,, k M,j + αk M,i ) = 0, 1 i j N, This also serves as a starting point for us to search for generalized bilinear differential equations which possess linear subspaces of solutions. Our results generalize Hirota bilinear operators and establish a bridge between bilinear differential equations and linear differential equations, which exhibits a kind of integrability of nonlinear differential equations [11]. The existence of linear subspaces of solutions amends diversity of exact solutions generated by various analytical methods (see, for example, [12]- [16]). The generalized bilinear operators (2.1) definitely bring more opportunities to generate non-trivial trilinear differential equations. It is also extremely interesting to explore other mathematical properties that the generalized bilinear differential equations (2.3) possess and how to construct their non-resonant soliton solutions. Acknowledgements: The work was supported in part by the Natural Science Foundation of Shanghai (No. 09ZR1410800) and the Shanghai Leading Academic Discipline Project (No. J50101), Zhejiang Innovation Project (Grant No. T200905), the State Administration of Foreign Experts Affairs of China, the National Natural Science Foundation of China (Nos. 10831003, 61072147 and 11071159), and Chunhui Plan of the Ministry of Education of China. I am also grateful to Prof. Huiqun Zhang for her fruitful discussions and valuable suggestions. References [1] Hirota, R., 2004. The Direct Method in Soliton Theory. Cambridge University Press, Cambridge. [2] Ma, W.X., Huang, T.W. and Zhang, Y., 2010. A multiple exp-function method for nonlinear differential equations and its application. Phys. Scripta, 82:065003. [3] Ma, W.X. and Fan, E.G., 2011. Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl., 61:950-959. 9

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