Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations

Transcription

1 International Journal of Modern Physics B Vol ) pages) c World Scientific Publishing Company DOI: /S X Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations Wen-Xiu Ma Yuan Zhou and Rachael Dougherty College of Mathematics and Physics Shanghai University of Electric Power Shanghai P. R. China Department of Mathematics and Statistics University of South Florida Tampa FL USA International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical Sciences North-West University Mafikeng Campus Private Bag X2046 Mmabatho 2735 South Africa mawx@cas.usf.edu zhouy@mail.usf.edu rdougherty@mail.usf.edu Accepted 25 April 2016 Published 5 August 2016 Lump-type solutions rationally localized in many directions in the space are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on f are computed and thus lump-type solutions are presented to the corresponding nonlinear differential equations on u generated from taking a transformation of dependent variables u = 2ln f) x. Keywords: Symbolic computation; generalized bilinear form; positive polynomial solutions; lump and lump-type solutions. PACS numbers: Ik Xm Yv 1. Introduction The Hirota direct method 1 is one of the most powerful approaches for constructing multi-soliton solutions to integrable nonlinear equations. Its successful idea is to use a transformation of dependent variables to convert nonlinear differential equations into bilinear forms defined in terms of bilinear derivatives called the Hirota bilinear derivatives. Such bilinear forms have been generalized by adopting new rules of taking signs. 2 New broader classes of new nonlinear differential equations derived from generalized bilinear equations have been presented and their resonant soliton solutions have been constructed by adopting the linear superposition principle

2 W.-X. Ma Y. Zhou & R. Dougherty In recent years there has been growing interest in rationally localized solutions in the space 4 7 particularly lump solutions localized in all directions in the space see e.g. Refs for typical examples). The KPI equation u t + 6uu x + u xxx ) x u yy = 0 1) has the following lump solution 13 : u = 4 [x + ay + a2 b 2 )t] 2 + b 2 y + 2at) 2 + 3/b 2 {[x + ay + a 2 b 2 )t] 2 + b 2 y + 2at) 2 + 3/b 2 } 2 2) where a and b 0 are two free real constants. More generally the KPI equation 1) admits the following lump solution 14 : u = 2ln f) xx = 4a a 2 5 )f 8a 1 g + a 5 h) 2 f 2 f = g 2 + h 2 + 3a a 2 5 ) 3 a 1 a 6 a 2 a 5 ) 3) where g = a 1 x + a 2 y + a 1a 2 2 a 1 a a 2 a 5 a 6 a 12 + a 5 2 t + a 4 h = a 5 x + a 6 y + 2a 1a 2 a 6 a 2 2 a 5 + a 5 a 6 2 a 12 + a 5 2 t + a 8 the six parameters a 1 a 2 a 4 a 5 a 6 and a 8 being free real constants satisfying a 1 a 6 a 2 a 5 0. Rogue wave solutions which draw big attention from research scientists worldwide are a particularly interesting class of lump-type solutions 1516 and such solutions could be used to describe significant nonlinear wave phenomena in both oceanography 17 and nonlinear optics. 18 There are various discussions on general rational function solutions to integrable equations such as the KdV KP Boussinesq and Toda equations It has become a very interesting topic to search for lump solutions or lump-type solutions rationally localized solutions in many directions in the space to nonlinear differential equations based on Hirota bilinear forms and generalized bilinear forms. Lump solutions to nonlinear differential equations possessing Hirota bilinear forms are analyzed in a recent paper. 24 The basis of success is a set of Hirota bilinear forms and the primary object is a class of multi-variate positive quadratic functions. Necessary and sufficient conditions for the existence of positive quadratic function solutions are presented for general Hirota bilinear equations. Such polynomial solutions yield lump solutions to nonlinear differential equations derived from Hirota bilinear equations under a transformation of either u = 2ln f) x or u = 2ln f) xx. In this paper we would like to consider generalized Hirota bilinear equations and focus on the two classes of generalized bilinear differential equations involving the prime number p = 3 presented in Ref. 2. We will search for their positive quadratic and quartic function solutions by symbolic computation with Maple

3 Lump-type solutions to nonlinear differential equations Further we will present lump-type solutions to the corresponding nonlinear differential equations generated by u = 2ln f) x. It is hoped that the study will help us recognize characteristics of nonlinearity more concretely. A few concluding remarks will be given at the end of the paper. 2. Generalized Bilinear Equations 2.1. Hirota bilinear derivatives and bilinear equations Let M be a natural number and x = x 1 x 2... x M ) T be a column vector of R M. For f g C R M ) the Hirota bilinear derivatives are defined as follows: D n1 1 Dn2 2 Dn M M f g := M xi x i ) ni fx)gx ) x =x 4) i=1 where x = x 1 x 2... x M )T and n 1... n M are arbitrary nonnegative integers. For example we can compute D i f g = f xi g fg xi D i D j f g = f xix j g + fg xix j f xi g xj f xj g xi. Assume that D = D 1 D 2... D M ) where each D i is the first-order Hirota bilinear derivative with respect to x i. One important property of the Hirota bilinear derivatives is that D i1 D i2 D ik f g = 1) k D i1 D i2 D ik g f where 1 i 1 i 2... i k M need not be distinct. It then follows that if k is odd we have D i1 D i2 D ik f f = 0. We are interested in the following general Hirota bilinear equation: P D)f f = P D 1 D 2... D M )f f = 0 5) where P is a polynomial in M variables. This equation is bilinear indeed. Since the terms of odd powers are all zeros we can assume that P is an even polynomial i.e. P x) = P x) while discussing Hirota bilinear equations. For convenience s sake we adopt the notation Therefore f i1i 2 i k = k f x i1 x i2 x ik 1 i 1 i 2... i k M. 6) D i D j f f = 2f ij f f i f j ) 1 i j M 7) and D i D j D k D l f f = 2[f ijkl f f ijk f l f ijl f k f ikl f j f jkl f i + f ij f kl + f ik f jl + f il f jk ] 1 i j k l M. 8)

4 W.-X. Ma Y. Zhou & R. Dougherty Motivated by Bell polynomial theories on soliton equations we take one of the transformations: u = 2ln f) x1 u = 2ln f) x1x 1 9) to formulate nonlinear differential equations from Hirota bilinear equations. All integrable nonlinear equations are such examples. 128 For example for the KdV equation and the KPI and KPII equations u t + 6uu x + u xxx = 0 u t + 6uu x + u xxx ) x + σu yy = 0 σ = 1 the transformation u = 2ln f) xx provides a link to the 1+1)-dimensional Hirota bilinear form and the 2+1)-dimensional Hirota bilinear form D x D t + D 4 x)f f = 0 10) D x D t + D 4 x + σd 2 y)f f = 0 11) respectively. If a polynomial solution f to a bilinear equation is positive then the solution u defined by any of the transformations in 9) is analytical and most likely rationally localized in all directions in the space and thus it often presents a lump solution to the corresponding nonlinear differential equation Generalized bilinear derivatives Let p N be given. For f g C R M ) the so-called generalized bilinear derivatives are defined as follows 2 : M D n1 p1 Dn2 p2 Dn M pm f g)x) := xi + α x i ) ni fx)gx ) x =x = i=1 M n i i=1 j i=0 α j l ni j i ) x ni ji i fx) x ji i gx) 12) where x = x 1 x 2... x M )T n 1... n M are arbitrary nonnegative integers and for any integer m the mth power of α is defined by α m = 1) rm) with rm) being the remainder of m divided by p: m = rm) mod p with 0 rm) < p and thus m rm) = kp for some integer k. It is easy to see that where 1 i M and p 2. D n pi = D n i 1 n p

5 Lump-type solutions to nonlinear differential equations When p = 2k k N) the generalized bilinear derivatives reduce to the Hirota bilinear derivatives since m rm) is even and we have α m = 1) rm) = 1) m. Therefore D 2ki = D i for 1 i M. Now we consider p = 3 and then we have It is direct to get α = 1 α 2 = α 3 = 1 α 4 = 1 α 5 = α 6 = D 3i f g = D i f g =f i g fg i D 3i D 3j f g = D i D j f g =f ij g f j g i f i g j +fg ij D 3i D 3j D 3k f g = f ijk g f ij g k f ik g j +f i g jk f jk g i +f j g ik +f k g ij +fg ijk D 3i D 3j D 3k D 3l f g = f ijkl g f ijk g l f ijl g k +f ij g kl f ikl g j +f ik g jl + f il g jk +f i g jkl f jkl g i +f jk g il +f jl g ik +f j g ikl + f kl g ij +f k g ijl +f l g ijk fg ijkl. Therefore when g = f we obtain D 3i f f = f i f ff i = 0 D 3i D 3j f f = 2f ij f f i f j ) In particular D 3i D 3j D 3k f f = 2f ijk f D 3i D 3j D 3k D 3l f f = 2f ij f kl + f ik f jl + f il f jk ). 13) D 2 3if f = 2f ii f f 2 i ) D 3 3if f = 2f iii f D 4 3if f = 6f 2 ii. 14) 2.3. Generalized bilinear equations and polynomial solutions Let P x) be a polynomial in x R M with degree d P and P 0) = 0 P may not be even). Suppose that p 2 is an integer. Formulate a generalized bilinear equation as follows: P D p )f f = 0 15) where D p = D p1 D p2... D pm ). We consider polynomial solutions fx) with the independent variable x R M. For a monomial P k x) = x n1 1 xn M M noting that i) degf) = 0 if and only if f = const. 0 and ii) deg0) = we have degp k D p )f f) 2 degf) degp k ) since P k D p )f f = J j1 α J x j1 1 j M x j M M fx) n1 j1 x n1 j1 1 n M j M x n M j M M fx)

6 W.-X. Ma Y. Zhou & R. Dougherty for some real constants α J with J = j 1... j M ). As a corollary for a linear or quadratic function f i.e. degf) 2) we have P k D p )f f = 0 16) when degp k ) 5. In general if f is a polynomial solution to the generalized bilinear equation 15) then the coefficients of f satisfy a group of nonlinear algebraic equations. For example if fx t) = ax 2 + 2bxt + ct 2 + dx + et + g is a solution of the bilinear KdV equation 10) then we have ab = ac = ae = bc = cd = 12a 2 + 2bg de = 0 17) which leads to the following three classes of solutions: i) fx t) = 2bxt + dx + et + de/2b) ii) fx t) = ct 2 + et + g iii) fx t) = dx + g. For the bilinear KdV equation with P = P t x) = xt + x 4 we have when p = 2 and P D)f f = 2f xt f f x f t + f xxxx f 4f xxx f x + 3f 2 xx) = 0 P D 3 )f f = 2f xt f f x f t + 3f 2 xx) = 0 when p = 3. Therefore the corresponding bilinear differential equations depend on the value of p. However if f is quadratic then we can easily find that D n pif f = D n i f f n 1 18) where 1 i M and p 2 and thus the same quadratic function f can solve all generalized bilinear differential equations with different values of p 2. We list this result as follows. Theorem 1. The generalized bilinear equations 15) with a given same polynomial P but different integers p 2 possess the same set of quadratic function solutions. Let us denote a general quadratic function by fx) = x T Ax 2b T x + c 19) where A = A T R M M b R M and c R and write the polynomial P x) defining the generalized bilinear equation 15) as follows: P x) = N M k=1 i 1...i k =1 p i1 i k x i1 x ik 20) where N = degp ) 1 is an integer and p i1 i k 1 i 1... i k M 1 k N are real constants

7 Lump-type solutions to nonlinear differential equations It has been proved 24 that a quadratic function f defined by 19) is positive everywhere in R M i.e. fx) > 0 x R M if and only if A is positive semidefinite b rangea) and c b T A + b > 0 where A + being the Moore Penrose pseudoinverse of A. Noting the properties in 16) and 18) we can see that only the coefficients p ij and p ijkl take effect in computing quadratic function solutions. Necessary and sufficient conditions on quadratic function solutions to Hirota bilinear equations have been presented in the previous paper 24 indeed. Based on Theorem 1 we can have the same criterion on quadratic function solutions to generalized bilinear equations stated below. Theorem 2. Let A = a ij ) M M R M M be symmetric and positive semidefinite b R M and c R. The quadratic function f defined by 19) solves the generalized bilinear equation 15) with P x) defined by 20) if and only if and 2 M ijkl=1 p ijkl a ij a kl + a ik a jl + a il a jk ) + d M p ij a ij = 0 21) ij=1 M p ij a ij A A i A T j A j A T i ) = 0 22) ij=1 where A i denotes the ith column vector of the symmetric matrix A for 1 i M and d = c b T A + b. We point out that for distinct p the generalized bilinear equations by 15) may have different polynomial solutions of higher order than two. For example any C 3 -differentiable function is a solution to the equation D 3 xf f = 0. But if f = fx t) = x 4 + t 2 based on 14) we can have D 3 3xf f = 48xx 4 + t 2 ) 0 which means that this quartic function f does not solve D 3 3xf f = Lump-Type Solutions to Two Classes of Nonlinear Differential Equations In general it is difficult to find rational function solutions to nonlinear differential equations. But using Mathematical software such as Maple we can find polynomial solutions to generalized bilinear differential equations. In this section we will try to search for positive quadratic or quartic polynomial solutions to generalized bilinear equations. From those polynomial solutions f we will be able to construct lump-type solutions to nonlinear differential equations via the transformation of dependent variables u = 2 ln f) x

8 W.-X. Ma Y. Zhou & R. Dougherty 3.1. First class of nonlinear equations Let us begin with the following polynomial [see formula 18) in Ref. 2]: P = c 1 x 5 + c 2 x 3 y + c 3 x 2 z + c 4 xt + c 5 yz where the coefficients c i 1 i 5 are free real constants. The associated generalized bilinear differential equation with p = 3 reads [see 19) in Ref. 2]: P D 3x D 3y D 3z D 3t )f f = 2c 1 f xxxxx f 5f xxxx f x + 10f xxx f xx ) + 6c 2 f xx f xy + 2c 3 f xxz f + 2c 4 f xt f f x f t ) + 2c 5 f yz f f y f z ) = 0. 23) Taking u = 2ln f) x generates the corresponding nonlinear differential equation: P D 3x D 3y D 3z D 3t )f f x f 2 = c u3 x u3 u xx u4 u x + 10u x u xxx u2 u 2 x + 15uu x u xx + 10u 2 xx + u xxxxx ) + c 2 [ 3 8 u3 u y u xu xy u2 u xy u xxu y uu xu y + 3 ] 8 3u2 u x + 2uu xx + 2u 2 x)v + c 3 [uu xz + u xxz u xu z u2 u z + 1 ] 2 u xx + uu x )w + c 4 u xt + c 5 u yz = 0 where u y = v x and u z = w x. Therefore if f solves the generalized bilinear equation 23) then u = 2ln f) x solves the nonlinear differential equation 24). 24) Quadratic function solutions Let us first consider quadratic function solutions to the generalized bilinear equation 23) which involve a sum of two squares. Based on the discussion in Sec. 2 we know that such solutions have nothing to do with c 1 and c 3. Therefore the coefficients c 1 and c 3 will be arbitrary real constants. Three cases of such solutions by symbolic computation with Maple are displayed as follows. 1) When c 4 0 but c 2 and c 5 are arbitrary we have a4 a 7 a 8 c 5 f = t + a 2 x a ) 2 7a 8 y + a 4 z + a 5 a 22 c 4 a 2 + a 4a 8 c 5 t + a 7 x + a 8 y + a ) 2 4a 7 z + a 10 + a 11 a 2 c 4 a

9 Lump-type solutions to nonlinear differential equations where a 2 a 4 a 5 a 7 a 8 a 10 and a 11 are arbitrary real constants satisfying a 2 0 and a 11 > 0. 2) When c 2 c 4 0 but c 5 is arbitrary we have [ a3 a 2 4 a 11 c 5 3a 3 ] 2 7 a 9 c 2 )c 5 f = t + a 3 y + a 4 z + a 5 3a 74 c 2 c 4 [ ) ] 2 a3 a 3 4 3a7 c 2 + a 9 a 11 c 5 c5 + t + a 7 x a 3a 4 a 11 c 5 y + a 9 z + a 10 + a 11 3a 74 c 2 c 4 3a 73 c 2 where a 3 a 4 a 5 a 7 a 9 a 10 and a 11 are arbitrary real constants satisfying a 7 0 and a 11 > 0. 3) When c 4 c 5 0 but c 2 is arbitrary we have f = [ a 2 a 3 a 4 a 2 a 8 a 9 + a 3 a 7 a 9 + a 4 a 7 a 8 )d 1 t + a 2 x + a 3 y + a 4 z + a 5 ] 2 with + [ a 2 a 3 a 9 + a 2 a 4 a 8 a 3 a 4 a 7 + a 7 a 8 a 9 )d 1 t + a 7 x + a 8 y + a 9 z + a 10 ] 2 + d 2 c 5 d 1 = d 2 = 3a a 2 7 ) 2 a 2 a 3 + a 7 a 8 )c 2 a 22 + a 72 )c 4 a 2 a 9 a 4 a 7 )a 2 a 8 a 3 a 7 )c 5 where a i i = are arbitrary real constants satisfying a 2 a 9 a 4 a 7 0 and a 2 a 8 a 3 a Quartic function solutions Let us now consider quartic function solutions to the generalized bilinear equation 23). A direct symbolic computation with Maple tells us seven classes of positive quartic function solutions. 1) Solutions independent of y: f = a 2 x + a 4 z + a 5 ) 2 + a 7 x + a 9 z + a 10 ) 2 + a 14 z + a 15 ) 4 + a 16 where a 2 a 4 a 5 a 7 a 9 a 10 a 14 a 15 and a 16 > 0 are arbitrary real constants. 2) Solutions independent of z: f = a ) 2 7a 8 x + a 3 y + a 5 + a 7 x + a 8 y + a 10) 2 + a 13y + a 15) 4 + a 16 a 3 where a 3 a 5 a 7 a 8 a 10 a 13 a 15 and a 16 > 0 are arbitrary real constants. 3) Solutions independent of x and y: f = a 1 t + a 4 z + a 5 ) 2 + a 6 t + a 9 z + a 10 ) 2 + a 11 t + a 14 z + a 15 ) 4 + a 16 where a 1 a 4 a 5 a 6 a 9 a 10 a 11 a 14 a 15 and a 16 > 0 are arbitrary real constants. 4) Solutions independent of x and z: f = a 1 t + a 3 y + a 5 ) 2 + a 6 t + a 8 y + a 10 ) 2 + a 11 t + a 13 y + a 15 ) 4 + a 16 where a 1 a 3 a 5 a 6 a 8 a 10 a 11 a 13 a 15 and a 16 > 0 are arbitrary real constants

10 W.-X. Ma Y. Zhou & R. Dougherty 5) When c 5 0 but c k 1 k 4 are arbitrary we have f = a 7a 8 a 11 t + a 2 x a 7a 8 y a ) 2 2a 11 c 4 z + a 5 a 2 a 13 a 2 a 13 c 5 + a8 a 11 t + a 7 x + a 8 y a ) 2 7a 11 c 4 z + a 10 + a 11t + a 13y + a 15) 4 + a 16 a 13 a 13 c 5 where a 2 a 13 0 a 16 > 0 and all other involved parameters are arbitrary real constants. 6) When c 4 0 but c k 1 k 3 we have f = a ) 2 4a 13 c 5 x + a 4 z + a 5 + a 11t + a 13y + a 15) 4 + a 16 a 11 c 4 where a 11 0 a 16 > 0 and all other involved parameters are arbitrary real constants. 7) When c 4 0 but c k 1 k 3 we have f = a ) 2 3a 9 c 5 t + a 3 y + a 5 + a 7 x + a 9 z + a 10) 2 a 7 c 4 + a ) 4 9a 13 c 5 t + a 13 y + a 15 + a 16 a 7 c 4 where a 7 0 a 16 > 0 and all other involved parameters are arbitrary real constants Discussions Lump solutions are rationally localized in all directions in the space. For the exact solutions we discussed above this characteristic property equivalently requires lim ux y z t) = 0 t R x 2 +y 2 +z 2 where u = 2ln f) x and obviously a sufficient condition for u to be a lump solution is lim fx y z t) = t R. x 2 +y 2 +z 2 We point out that all the solutions presented above do not satisfy this criterion but they are rationally localized in many directions in the space and thus we call them lump-type solutions. We consider a special case of the parameters and coefficients for the class of solutions in Sec ). Choose a 2 = a 11 = a 13 = a 16 = 1 a 5 = a 10 = a 15 = 0 a 7 = 1 a 8 = 2 and c 4 = c 5 = 2. Then we have the following positive quartic function solution to the generalized bilinear Eq. 23): f = 2t + x + 2y z) 2 + 2t x + 2y + z) 2 + t + y)

11 Lump-type solutions to nonlinear differential equations Lump-type solutions to nonlinear equations 11 a) b) c) Fig. 1. Plotsof of 25) at at t = 0 with a) x = 0 0 b) b) y y = = 1 1 and and c) c) z = z 1. = 1. and the corresponding lump-type solution to the nonlinear differential equation 24): where the coefficients c i 1 i 4 are free real constants. The associated generalized u = bilinear 2ln f) differential equation with p = 3 see 34) in Ref. 2 x = ) reads: 8x z) 2t + x + 2y z) 2 + 2t x + 2y + z) 2 + t + y) ) Figure 1 shows three 3d P D plots 3x of Dthis 3y Dlump-type 3t )f f solution at t = 0 with x = 0 y = 1 and z = 1 respectively. = 2c 1 f xx f fx) 2 + 6c 2 f xx f xy + 2c 3 f xxxxyy f +4f xxx f xyy + 6fxxy) 2 + 2c 4 f yt f f y f t ) = 0. 26) 3.2. Second class of nonlinear equations Taking u = 2 ln f) x generates the corresponding nonlinear differential equation Let us now begin with the polynomial [see 29) of Ref. 2]: P D 3x D 3y D P 3t = )f c f 1 x 2 + c 2 x 3 y + c 3 x 4 y 2 + c 4 yt x f 2 where the coefficients [3 c i 1 = c 1 u xx + c 2 8 u3 u y + 3 i 4 2 u xu xy + 3 are free real 4 u2 u xy + 3 constants. 2 u xxu y + 3 The associated generalized bilinear differential equation with p = 3 [see 34) in Ref. 2] reads: 8 3u2 u x + 2uu xx + 2u 2 x)w ] [ 5 +c P D 3 16 u4 u yy + 12uu 2 xy + 5u y u xxxy u2 xu yy + 16u xy u xxy + 5u x u xxyy + 5 3x D 3y D 3t )f f = 2c 1 f xx f fx) 2 + 6c 2 f xx f xy + 2c 3 f xxxxyy f 2 u3 u 2 y +2uu xxxyy + u 3 + 4f xxx f u xyy + 7u xx u xyy + 3 xyy + 6fxxy) 2 2 u2 u xxyy u + 2c 4 f yt f f y f xxxu yy + 7u xx u 2 t ) = 0. 26) y + 14uu xxy u y Taking u = 2 ln f) +9uu x u xyy + 21 x generates the corresponding nonlinear differential equation P D 4 u2 u x u yy + 24u x u y u xy + 12u 2 u xy u y + 8uu xx u yy + 18uu x u 2 3x D 3y D 3t )f f y x f 2 +u xxxxyy + u xxxxy + 8u x u xxy + 2uu xxxy + 3u 2 u xxy + 5 [ 2 u3 u xy + 15uu x u xy 3 = c u xxxu y + 10u xx u xy + 2uu xxx u4 y + 11uu y u xx u2 u x u y u xx +c 2 ) 8 u3 u y u xu xy u2 u xy u xxu y + 3 ] 8 3u2 u x +2uu xx +2u 2 x)w 4 u2 xu y w + 1 [ 2 u 5 xxxx u2 u xx + 2uu xxx u xu xx u3 u x + 9 ) 2 uu2 x wy + 1 +c 4 u 3 xxxx u4 u yy +12uu 2 xy +5u y u xxxy u2 u xx u xu xx + uu xxx u2 xu yy +16u xy u xxy +5u x u xxyy u3 u x + 15 ) 4 uu2 x w 2 ] 2 u3 u 2 y + c 4 u yt = 0 27) +2uu xxxyy +u 3 u xyy +7u xx u xyy u2 u xxyy u xxxu yy +7u xx u 2 y +14uu xxy u y where u y = w x. +9uu In what x u xyy follows + 21 we will try to search for positive quadratic and quartic solutions to the generalized 4 u2 u x u yy +24u x u y u xy +12u 2 u xy u y +8uu xx u yy +18uu x u 2 y bilinear equation 26) by symbolic computations with Maple. The resulting polynomial solutions will yield lump-type and lump solutions to the +u nonlinear xxxxyy + u differential xxxxy +8u equation x u xxy +2uu 27). xxxy +3u 2 u xxy u3 u xy +15uu x u xy

12 W.-X. Ma Y. Zhou & R. Dougherty u xxxu y +10u xx u xy +2uu xxx u4 u y +11uu y u xx u2 u x u y + 45 ) 4 u2 xu y w u xxxx+ 9 4 u2 u xx +2uu xxx u xu xx u3 u x + 9 ) 2 uu2 x w y u xxxx u2 u xx u xu xx +uu xxx u3 u x + 15 ) ] 4 uu2 x w 2 +c 4 u yt = 0 where u y = w x. In what follows we will try to search for positive quadratic and quartic solutions to the generalized bilinear equation 26) by symbolic computations with Maple. The resulting polynomial solutions will yield lump-type and lump solutions to the nonlinear differential equation 27) Quadratic function solutions A direct symbolic computation tells the following four classes of positive quadratic function solutions to the generalized bilinear equation 26). 1) When c 1 c 4 0 but c 1 and c 2 are arbitrary we have [ f = a 1 2 a 2 + 2a 1 a 4 a 5 a 2 a 2 ] 2 4 )c 1 t + a 1 x + a 2 y + a 3 a 22 + a 52 )c 4 + [ 2 a1 a 5 2a 1 a 2 a 4 a 2 ] 2 4 a 5 )c 1 t + a 4 x + a 5 y + a 6 a 22 + a 52 )c 4 3a 1a 2 + a 4 a 5 )a a 4 2 )a a 5 2 )c 2 a 1 a 5 a 2 a 4 ) 2 c 1 where a i 1 i 6 are arbitrary real constants satisfying a 1 a 5 a 2 a 4 0 and a 1 a 2 + a 4 a 5 )c 1 c 2 < 0. 2) When c 1 c 4 0 but c 1 and c 2 are arbitrary we have 27) f = [ a4 2 a 6 2 a 1 2 a 3 2 )c 1 a 2 a 32 + a 62 )c 4 t + a 1 x + a 2 y + a 3 ] 2 [ a4 a 6 a 1 a 3 )a 3 a 4 a 1 a 6 )c 1 + t a 2 a 32 + a 62 )c 4 + a 4 x + a ] 2 2a 3 a 4 + a 1 a 6 ) y + a 6 3a 2a 3 a a 2 6 )a a 2 4 )c 2 a 1 a 3 a 4 a 6 a 2 6 a 1a 3 a 4 a 6 )c 1 where a i 1 i 4 and a 6 are arbitrary real constants satisfying a 2 0 a 6 0 a 1 a 3 a 4 a 6 0 and a 2 a 3 a 1 a 3 a 4 a 6 )c 1 c 2 < 0. 3) When c 1 c 4 0 but c 2 and c 3 are arbitrary we have with f = [a 7 t + a 1 x + a 2 y + a 3 ] 2 + [a 8 t + a 4 x + a 5 y + a 6 ] 2 + a 9 a 7 = a 2a 4 2 2a 1 a 4 a 5 a 1 2 a 2 )c 1 a 22 + a 52 )c

13 Lump-type solutions to nonlinear differential equations a 8 = a 1 2 a 5 2a 1 a 2 a 4 a 4 2 a 5 )c 1 a 22 + a 52 )c 4 a 9 = 3a 1a 2 + a 4 a 5 )a a 4 2 )a a 6 2 )c 2 a 1 a 5 a 2 a 4 ) 2 c 1 where a i 1 i 6 are arbitrary real constants satisfying a 1 a 5 a 2 a 4 0 and a 9 > 0. 4) When c 1 c 4 0 but c 2 and c 3 are arbitrary we have f = a 7 t + a 1 x + a 2 y + a 3 ) 2 + a 8 t + a 4 x + a 5 y + a 6 ) 2 + a 9 with a 5 = a 2a 1 a 6 + a 3 a 4 ) a 1 a 3 a 4 a 6 a 7 = a 1 2 a 3 2 a 4 2 a 6 2 )c 1 a 2 a 32 + a 62 )c 4 a 8 = a 1a 3 a 4 a 6 )a 1 a 6 a 3 a 4 ) a 2 a 32 + a 62 )c 4 a 9 = 3a a 4 2 )a a 6 2 )a 2 a 3 c 2 a 1 a 3 a 4 a 6 )a 62 c 1 where a i 1 i 4 and a 6 are arbitrary real constants satisfying a 2 0 a 1 a 3 a 4 a 6 0 and a 9 > Quartic function solutions A direct symbolic computation with f involving a sum of three squares leads to the following three classess of positive quartic function solutions to the generalized bilinear equation 26). 1) Case I involving a sum of three squares: [ f = a 1 y + a 2 ) 2 + a 3 y + a 4 ) 2 + a 5 ty + a 1a 2 + a 3 a 4 )a 5 t a 12 + a 2 3 ] 2 a 1a 4 a 2 a 3 ) 2 a 12 + a 2 3 where a i 1 i 5 are arbitrary real constants satisfying a a ) Case II involving a sum of three squares: f = a 1 a ) 2 2a 3 t + a 2 t + a 3 ) 2 + a 4 t + a 5 ty) 2 a 2 1 a 2 3 a 1 where a 1 0 and a i 2 i 5 are arbitrary real constants. 3) Case III involving a sum of three squares: f = a 1 a 2a 3 t+ a ) 2 1a 5 y + a 2 t+ a ) 2 3a 5 y+a 3 +a 4 t+a 5 ty) 2 + a 2 2 a 2 1 +a 2 3 ) 2 a 1 a 4 a 12 a 2 4 a 4 where a i 1 i 5 are arbitrary real constants satisfying a 1 a

14 W.-X. Ma Y. Zhou & R. Dougherty 14 W. X. Ma Y. Zhou and R. Dougherty a) b) c) Fig Plotsof of 28) at a) t = 0 0 b) b) t t = = and and c) c) t = t 30. = Discussions We andcan thegenerate lump solution lump to solutions 27): from the presented quadratic function solutions to the generalized bilinear ux y t) = 4[ equation 2 a 1c 1 c 4 t + 26) ] a 2 in 1 + Sec. 1)x a 1 y + But a 1 a the 3 + quartic a function 6 solutions in Sec are all independent of the a 1 x + y + a 3 ) 2 + 2a spatial 1c 1 variable x. Therefore c 4 t + x + a 6 ) c 2 c 1. when y is fixed and x goes to f will not tend to. This implies that the presented quartic Particularly function taking solutions a 1 = to 1 the a 3 generalized = 0 a 6 = bilinear 1 andequation c 1 = 1 c26) 2 = will 1 3 c not 4 = produce 2 leads to any the lump following solution lumptosolution: the nonlinear differential equation 27). Let us now present a special class 42x of lump y + tsolutions 1) from the first class of quadratic function ux solutions y t) = in x Sec y) 2 + To t + the x end 1) 2 we + 2 specify. a 28) 2 = a 4 = 1 a 5 = 0 a 1 = sgnc 1 c 2 ). Then we have the positive quadratic function solution The figure 2 shows three 3d plots of this solution at t = respectively. The to 26): plots depict that the lump by 28) does not change much while it travels. 4. Concluding fx y t) remarks = a 1 x + y + a 3 ) 2 + 2a ) 2 1c 1 t + x + a c 2 c 4 c 1 We have analyzed quadratic function solutions to generalized bilinear equations and and the carried lump out solution a search to for 27): positive quadratic and quartic function solutions to two classes of generalized bilinear equations generated from the two specific polynomials. The resulting polynomial [ solutions yield lump-type solutions of finite energy to the corresponding nonlinear 4 2a ] 1c 1 differential t + a )x equations + a 1 y + derived a 1 a 3 + from a 6 the considered c generalized bilinear equations. 4 ux y t) = It is expected that such a ) study 2 will help. us recognize the characteristics of generalized bilinear 2a1 a 1 x + y + a 3 ) 2 c 1 + equations t + x + a 6 and+ nonlinear 6 c 2 differential equations. c 4 c 1 High-order polynomial solutions to generalized bilinear equations is an interesting problem. taking It will a 1 = be 1 important a 3 = 0 ato 6 = see 1 if and therec 1 is= any 1 cnonlinear 2 = 1/3 csuperposition 4 = 2 leads Particularly to formula the following for generating lump solution: lump solutions in terms of higher-order polynomials. Is there any combinatorial relation between higher-order polynomial solutions and generalized bilinear equations? The second 42x interesting y + tproblem 1) ux y t) = is how to construct lump solutions to discrete integrable x equations. + y) 2 + Rational t + x function 1) solutions to the 28) Toda and 2-dimensional Toda lattice equations are successfully presented and expressed Figure in term2 of shows Casoratian three 3D determinant plots of this 2329 solution. The third at t problem = 1 15 is 30 torespectively. see if it is possible The plots to classify depictlump that solutions the lump from by 28) a determinant does not change pointmuch of view. while Can it travels. lump solutions be written in Wronskian Casoratian or Pfaffian form?

15 Lump-type solutions to nonlinear differential equations 4. Concluding Remarks We have analyzed quadratic function solutions to generalized bilinear equations and carried out a search for positive quadratic and quartic function solutions to two classes of generalized bilinear equations generated from the two specific polynomials. The resulting polynomial solutions yield lump-type solutions of finite energy to the corresponding nonlinear differential equations derived from the considered generalized bilinear equations. It is expected that such a study will help us recognize the characteristics of generalized bilinear equations and nonlinear differential equations. High-order polynomial solutions to generalized bilinear equations is an interesting problem. It will be important to see if there is any nonlinear superposition formula for generating lump solutions in terms of higher-order polynomials. Is there any combinatorial relation between higher-order polynomial solutions and generalized bilinear equations? The second interesting problem is how to construct lump solutions to discrete integrable equations. Rational function solutions to the Toda and 2-dimensional Toda lattice equations are successfully presented and expressed in terms of Casoratian determinant The third problem is to see if it is possible to classify lump solutions from a determinant point of view. Can lump solutions be written in Wronskian Casoratian or Pfaffian form? Acknowledgments The work was supported in part by the NNSFC under the Grant Nos and the distinguished professorships of Shanghai University of Electric Power and Shanghai Second Polytechnic University Natural Science Foundation of Shanghai Grant No. 11ZR ) Zhejiang Innovation Project of China Grant No. T200905) the First-class Discipline of Universities in Shanghai and the Shanghai Univ. Leading Academic Discipline Project Grant No. A ). The authors would also like to thank X. Gu X. Lü S. Manukure M. Mcanally X. L. Yang Y. Q. Yao J. Yu and Y. J. Zhang for their valuable discussions about lump solutions. References 1. R. Hirota The Direct Method in Soliton Theory Cambridge University Press Cambridge 2004). 2. W. X. Ma Stud. Nonlinear Sci ). 3. Ö. Ünsal and W. X. Ma Comput. Math. Appl ). 4. K. A. Gorshkov D. E. Pelinovsky and Yu. A. Stepanyants J. Exp. Theor. Phys ). 5. A. A. Minzoni and N. F. Smyth Wave Motion ). 6. K. Imai Prog. Theor. Phys ). 7. K. M. Berger and P. A. Milewski SIAM J. Appl. Math ). 8. D. J. Kaup J. Math. Phys ). 9. C. R. Gilson and J. J. C. Nimmo Phys. Lett. A )

16 W.-X. Ma Y. Zhou & R. Dougherty 10. J. Satsuma and M. J. Ablowitz J. Math. Phys ). 11. M. J. Ablowitz and P. A. Clarkson Solitons Nonlinear Evolution Equations and Inverse Scattering Cambridge University Press Cambridge 1991). 12. W. X. Ma Z. Y. Qin and X. Lu Nonlinear Dyn ). 13. S. V. Manakov et al. Phys. Lett. A ). 14. W. X. Ma Phys. Lett. A ). 15. S. W. Xu J. S. He and L. H. Wang J. Phys. A: Math. Theor ). 16. Y. Ohta and J. K. Yang Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci ). 17. P. Müller C. Garrett and A. Osborne Oceanography ). 18. D. R. Solli et al. Nature ). 19. W. X. Ma and Y. You Trans. Am. Math. Soc ). 20. M. Adler and J. Moser Commun. Math. Phys ). 21. M. J. Ablowitz and J. Satsuma J. Math. Phys ). 22. W. X. Ma C. X. Li and J. S. He Nonlinear Anal ). 23. W. X. Ma and Y. You Chaos Solitons Fractals ). 24. W. X. Ma and Y. Zhou preprint 2015) arxiv: C. Gilson et al. Proc. R. Soc. Lond. A ). 26. W. X. Ma J. Phys.: Conf. Ser ). 27. W. X. Ma Rep. Math. Phys ). 28. D. Y. Chen Introduction to Solitons Science Press Beijing 2006). 29. W. X. Ma Modern Phys. Lett. B )