Paper Info
Title
Travelling wave solutions of nonlinear partial equations by using the first integral method
Abstract
In this paper, the first integral method is employed for constructing the new exact travelling wave solutions of nonlinear partial differential equations. The power of this manageable method is confirmed by applying it for two selected nonlinear partial equations. This approach can also be applied to other systems of nonlinear differential equations.
Year
DOI
Venue
2010
10.1016/j.amc.2010.02.028
Applied Mathematics and Computation
Keywords
Field
DocType
and using (4) and (5),0 ) on both sides of eq. (51),k 3 = 2 r 1 - λ 2 l k 2 l,suppose that g ( x ) = a 0 + a 1 x,then we obtain a system of nonlinear algebraic equations and by solving it,g 3 ⩽ 0 lead to solitary wave solutions. if δ = 0,where a i ( x ) ( i = 0,v 1 ( x,u ix 2 t,we have (26a) a 2 ′ ( x ) = a 2 ( x ) h ( x ),due to the division theorem,we obtain (60) y ( ξ ) = - a 0 - 2 a λ x - abx 2 2 ± 1 4 a 0 2 - d . combining (60) with (49a),if q ( x,we can get (49a) x ′ ( ξ ) = y ( ξ ),the nonlinear evolution equation (1) is reduced to a nonlinear ordinary differential systems (3) h i = u i ( ξ ),(26d) a 0 ( x ) g ( x ) = a 1 ( x ) ( k 1 x 3 - k 2 x 2 - k 3 x + k 4 ) . since a i ( x ) ( i = 0,x n ] not containing 1 admits at least one zero in l n . (2) let x = ( x 1,where a 0,y ] such that (18) d q d ξ = ∂ q ∂ x d x d ξ + ∂ q ∂ y d y d ξ = ( g ( x ) + h ( x ) y ) ∑ i = 0 m a i ( x ) y i . in this example,r 2 = - 1 2 k λ lc 1,w ( ξ ) = - s λ bl - s a 2 λ 2 + 2 aba 0 abl tanh 1 2 a 2 λ 2 + 2 aba 0 ( k ( x + ly + sz - λ t ) + ξ 0 ) . where ξ 0 is arbitrary constant. case 2: suppose that m = 2,then we find a 1 ( x ) and a 0 ( x ) as follows (27a) a 1 ( x ) = c 1 + a 0 x + 1 2 a 1 x 2,then we find a 0 ( x ) (20) a 0 ( x ) = c 1 + a 0 x + 1 2 a 1 x 2,we obtain (29) y ( ξ ) = - 1 4 a 1 x 2 - 1 2 a 0 x - 1 2 c 1 . combining (29) with (16a),c 1 and c 2 are arbitrary integration constants. substituting a 0 ( x ),(2+1)-dimensional broer-kaup-kupershmidt system,u ix 1 x 1,where e 1 = | g 3 3 | in eq. (41) and e 1 = 1 2 | g 3 | 3 in eq. (42) . from above discussion,eq. (45) will be reduced into the (2,r 1 = - 2 lc 1 a 1,we obtain (59) a 1 = 0 . using the conditions (59) in eq. (50),then we find (53) a 0 ( x ) = a 0 + ( - a λ + a 1 ) x - ab 2 x 2,g 3 > 0 lead to periodic solutions,v ( x,eq. (35) can be simplified to (37) x ( ξ ) = f 0 + r ′ ( f 0 ) 4 ℘ ( ξ,a 1,v t + v xx + 2 ( uv ) x = 0 . the bkk system has been widely applied in many branches of physics like plasma physics,1915. [28] e.t. whittaker g.n. watson a course of modern analysis 1927 cambridge univ. press cambridge [29] k. chandrasekharan elliptic functions 1985 springer berlin [30] h.w. schürmann travelling wave solutions for cubic–quintic nonlinear schrodinger equation phys. rev. e 54 1996 4312 4320 [31] m. abramovitz i.a. stegun handbook of mathematical functions ninth ed. 1972 dover new york [32] y.j. ping l.s. yue multilinear variable separation approach in (3 + 1)-dimensions: the burgers equation chin. phys. lett. 20 2003 1448 1451 [33] s.s. peng p.z. liang z. jun new exact solution to (3 + 1)-dimensional burgers equation commun. theor. phys. 42 2004 49 50 [34] q. wang y. chen h. zhang a new riccati equation rational expansion method and its application to (2 + 1)-dimensional burgers equation chaos solitons fract. 25 2005 1019 1028 [35] x.y. tang s. lou variable separation solutions for the (2 + 1)-dimensional burgers equation chin. phys. lett. 20 2003 335 337 [36] a.m. wazwaz multiple soliton solutions and multiple singular soliton solutions for the (3 + 1)-dimensional burgers equations appl. math. comput. 200 2008 942 948,where a = - 1 / ( k ( 1 + l 2 + s 2 ) ) and b = ( α l + β / l + γ s / l ),r 1 = klc 1,(27b) a 0 ( x ) = - 4 k 1 a 1 2 8 x 4 + 4 k 2 + 3 a 0 a 1 6 x 3 + 2 k 3 + a 0 2 + a 1 2 x 2 + ( - 2 k 4 + a 0 c 1 ) x + c 2,9],su ′ = lw ′ . integrating the last two equations gives u = lv = l s w where the constants of integration are considered zeros. the first equation becomes: (47) λ u ′ + ( α l + β / l + γ s / l ) uu ′ + k ( 1 + l 2 + s 2 ) u ″ = 0 . rewrite this second-order ordinary differential equation as follows (48) u ″ ( ξ ) - a λ u ′ ( ξ ) - abu ( ξ ) u ′ ( ξ ) = 0,by equating the coefficients of y i ( i = 2,y ) = ∑ i = 0 m a i ( x ) y i = 0 is an irreducible polynomial in the complex domain c such that (50) q [ x ( ξ ),where k,then we find a 1 ( x ) and a 0 ( x ) (58a) a 1 ( x ) = a 0 + ( - 2 a λ + a 1 ) x - abx 2,y ) = ∑ i = 0 m a i ( x ) y i = 0 is an irreducible polynomial in the complex domain c such that (17) q [ x ( ξ ),(49b) y ′ ( ξ ) = a λ y ( ξ ) + abx ( ξ ) y ( ξ ) . according to the first integral method,(57c) a 0 ′ ( x ) = a 1 ( x ) g ( x ) - a λ a 1 ( x ) - aba 1 ( x ) x + a 0 ( x ) h ( x ),we can obtain one first integral to eq. (5) which can reduces eq. (3) to a first-order integrable ordinary differential equation. an exact travelling solution to eq. (1) is then obtained by solving this equation directly. division theorem suppose that p ( x,+ 1)-dimensional burgers equations [35] . firstly,y ) = p ( x,we obtain (28) c 2 = c 1 2 4,g 3 ) - g 2 ℘ ( ξ,suppose that g ( x ) = a 0,we change eq. (7) into a system of odes given by (9a) - λ lu ″ - klu ‴ + 2 l ( uu ′ ) ′ + 2 v ″ = 0,y,such as inverse scattering method [1],we obtain the exact solution to (48) and then the exact solutions to (3 + 1)-dimensional integrable burgers equations can be written as: (61) u ( ξ ) = - λ b + c ab tan 1 2 c ( k ( x + ly + sz - λ t ) + ξ 0 ),t ) = f 0 + r ( f 0 ) d ℘ ( ξ,(26c) a 0 ′ ( x ) = a 1 ( x ) g ( x ) + 2 a 2 ( x ) ( k 1 x 3 - k 2 x 2 - k 3 x + k 4 ) + a 0 ( x ) h ( x ),etc,(52c) a 0 ( x ) g ( x ) = 0 . since a i ( x ) ( i = 0,v ( ξ ) = - λ bl + c abl tan 1 2 c ( k ( x + ly + sz - λ t ) + ξ 0 ),x n ],k 4 = 2 r 2 k 2 l . using (4) and (5),t ) = λ 2 - 4 kc 1 + λ 2 2 tanh 4 kc 1 + λ 2 2 k ( k ( x + ly - λ t ) + ξ 0 ),y ] such that (6) q ( x,then there exists a polynomial h ( x,darboux transformation method [4,k 2 = 3 λ k 2,for the set of polynomials of k [ x 1,t ) = v ( ξ ),where a 0 is arbitrary integration constant. substituting a 0 ( x ) and g ( x ) into (52c) and setting all the coefficients of powers x to be zero,whereas the conditions (41) δ = 0,u ix 3,5],vol. v,(19b) a 0 ′ ( x ) = a 1 ( x ) g ( x ) + a 0 ( x ) h ( x ),the tanh-method [11,(1) f i ( u i,y ) are polynomials in c [ x,y = ( y 1,which can be derived from the parameter dependent symmetry constraint of the kadomtsev–petviashvili (kp) equation [22–26] . by considering the wave transformations (8) u ( x,l i and λ are constants to be determined later,take a 1 ( x ) = 1 . balancing the degrees of g ( x ) and a 0 ( x ),we have different behavior of ℘ ( ξ ) . the conditions [30] (40) δ ≠ 0 or δ = 0,(58b) a 0 ( x ) = a 2 b 2 4 x 4 + r 1 x 3 + r 2 x 2 + r 3 x + d,λ = - 2 a 0 a 1,negative or zero,g 3 ) - g 3,we obtain (22) y ( ξ ) = - c 1 + λ k x ( ξ ) - 1 k x ( ξ ) 2 . combining (22) with (16a),we conclude that deg ( g ( x ) ) = 0 or deg ( g ( x ) ) = 1 only. if deg ( g ( x ) ) = 1,and so on. the first integral method was first proposed by feng [17] in solving burgers–kdv equation which is based on the ring theory of commutative algebra. recently,12],y n ) be two elements of l n,and solitary wave solutions by (43) x ( ξ ) = f 0 + r ′ ( f 0 ) 4 e 1 + 3 e 1 csch 2 ( 3 e 1 ξ ) - 1 24 r ″ ( f 0 ),a 0 = 0 . using the conditions (32) in eq. (16),a 1 = 2 k,(2 + 1)-dimensional broer–kaup–kupershmidt system (3 + 1)-dimensional burgers equations 1 introduction nonlinear phenomena appear in a wide variety of scientific applications such as plasma physics,we obtain (24) y ( ξ ) = - c 1 - λ k x ( ξ ) + 1 k x ( ξ ) 2,(31b) a 0 ( x ) = - 1 2 k 1 x 4 + 2 3 k 2 x 3 + 1 2 2 k 3 + a 0 2 x 2 + ( - 2 k 4 + a 0 c 1 ) x + c 2,y ) . the division theorem follows immediately from the hilbert–nullstellensatz theorem: hilbert–nullsellensatz theorem let k be a field and l an algebraic closure of k. (1) every ideal γ of k [ x 1,m ) are polynomials of x and a m ( x ) ≠ 0 . eq. (17) is called the first integral to (16),y ) and q ( x,x n ] to be maximal,g 3 > 0,0 ) on both sides of eq. (18),assuming that m = 1 and m = 2 in eq. (17) . c ase 1: suppose that m = 1,u i ′ ( ξ ),(19c) a 0 ( x ) g ( x ) = ( k 1 x ( ξ ) 3 - k 2 x ( ξ ) 2 - k 3 x ( ξ ) + k 4 ) a 1 ( x ) . since a i ( x ) ( i = 0,where c = - aba 0 ± ab a 0 2 - 4 d - a 2 λ 2 and ξ 0 is an arbitrary constant. comparing our results with wazwaz’s results [36],k = - 4 a 1,where r 1 is second integration constant and the first one is taken to zero. rewrite this equation as follows (11) v ( ξ ) = r 1 2 + λ l 2 u - l 2 u 2 + kl 2 u ′ . inserting eq. (11) into (9b) yields (12) k 2 l 2 u ‴ - 3 lu ′ u 2 + 3 λ luu ′ + r 1 - 1 2 λ 2 l u ′ = 0 . integrating eq. (12) once leads to (13) k 2 l 2 u ″ - lu 3 + 3 2 λ lu 2 + r 1 - 1 2 λ 2 l u = r 2,g 2 > 0,…,(52b) a 0 ′ ( x ) = a 1 ( x ) g ( x ) - a 1 ( x ) ( a λ + abx ) + a 0 ( x ) h ( x ),where ξ 0 is an arbitrary constant. case 2: suppose that m = 2,y ],ξ = k ( x 1 + l 1 x 2 + ⋯ + l m - 1 x m + λ t ),then ℘ ( ξ ) degenerates into trigonometric or hyperbolic functions [31] . thus,and if u is independent of z,we suppose that x ( ξ ) and y ( ξ ) are nontrivial solutions of eq. (16),we obtain (21a) a 0 = - λ k,w ( ξ ) = - s λ bl + sc abl tan 1 2 c ( k ( x + ly + sz - λ t ) + ξ 0 ),g 2 ⩾ 0,where (15) k 1 = 2 k 2,u z = w y,there exists a polynomial g ( x ) + h ( x ) y in the complex domain c [ x,it is necessary and sufficient that there exists a k -automorphism s of l such that y i = s ( x i ) for 1 ⩽ i ⩽ n . (3) for an ideal α of k [ x 1,we introduce the wave variable ξ = k ( x + ly + sz - λ t ) into eq. (45) to carry out the system equation (45) into a system of odes given by (46) - λ u ′ = α luu ′ + β vu ′ + γ wu ′ + k ( 1 + l 2 + s 2 ) u ″,we obtain the exact solution to (46) and then the exact solutions to (3 + 1)-dimensional integrable burgers equations can be written as: (56) u ( ξ ) = - λ b - a 2 λ 2 + 2 aba 0 ab tanh 1 2 a 2 λ 2 + 2 aba 0 ( k ( x + ly + sz - λ t ) + ξ 0 ),from eq. (17),we obtain (55) y ( ξ ) = - a 0 + a λ x ( ξ ) + ab 2 x ( ξ ) 2 . combining (55) with (49a),where the primes denote differentiation with respect to x,mathematische werke,where r 2 is an integration constant. rewrite this second-order ordinary differential equation as follows (14) u ″ - k 1 u 3 + k 2 u 2 + k 3 u - k 4 = 0,h ( x ) into (26d) and setting all the coefficients of powers x to be zero,it is necessary and sufficient that there exist an integer m > 0 such that q m ∈ γ . 3 applications example 1 let us first consider the celebrated (2 + 1)-dimensional broer–kaup–kupershmidt system (7) u ty - u xxy + 2 ( uu x ) y + 2 v xx = 0,we can get (16a) x ′ ( ξ ) = y ( ξ ),(57d) a 0 ( x ) g ( x ) = 0 . since a i ( x ) ( i = 0,then from (19a) we deduce that a 1 ( x ) is constant and h ( x ) = 0 . for simplicity,we obtain (32) c 1 = 0,u it,t ) = λ 2 - 4 kc 1 - λ 2 2 tan 4 kc 1 - λ 2 2 k ( k ( x + ly - λ t ) + ξ 0 ),a 1 = - 2 k,x n ] zero at x to be identical with the set of polynomials of k [ x 1,(5b) y i ′ ( ξ ) = g ( x i ( ξ ),y ] such that (51) d q d ξ = ∂ q ∂ x d x d ξ + ∂ q ∂ y d y d ξ = ( g ( x ) + h ( x ) y ) ∑ i = 0 m a i ( x ) y i . in this example,and ℘ ( ξ,u x = v y,we conclude that deg ( g ( x ) ) = 1 only. suppose that g ( x ) = a 0 + a 1 x,then from (52a) we deduce that a 1 ( x ) is constant and h ( x ) = 0 . for simplicity,a 1 ( x ) and g ( x ),t ) = l ( 2 a 1 c 1 - a 0 2 ) a 1 2 1 + tan 2 2 a 1 c 1 - a 0 2 4 ( ξ + ξ 0 ),g 3 ) - 1 24 r ″ ( f 0 ),u ′ = lv ′,1,t ) = klc 1 2 + λ 2 l 8 + - 1 2 klc 1 + 1 8 l λ 2 tanh 2 4 kc 1 + λ 2 2 k ( k ( x + ly - λ t ) + ξ 0 ),… ) = 0 . using the travelling wave transformation (2) u i ( x 1,f 0 is any constant,g 3 ) d ξ + 1 2 r ′ ( f 0 ) ℘ ( ξ,homogeneous balance method [8,where a 0 and d are arbitrary integration constants,t ) = - klc 1 + λ 2 l 4 + - klc 1 + 1 4 l λ 2 tan 2 4 kc 1 - λ 2 2 k ( k ( x + ly - λ t ) + ξ 0 ),let us recall the first integral method. by using the division theorem for two variables in the complex domain c which is based on the hilbert–nullstellensatz theorem [21],a 1 ( x ) and g ( x ) into (26d) and setting all the coefficients of powers x to be zero,the invariants g 2,x n ),y ) is irreducible in c [ x,x n ] to be zero on the set of zeros in l n of an ideal γ of k [ x 1,c 1 are arbitrary constants. using the conditions (21a) in eq. (17),we conclude that deg ( g ( x ) ) = 0 or deg ( g ( x ) ) = 1 only. if deg ( g ( x ) ) = 0,t ) = - a 0 a 1 - 2 a 1 c 1 - a 0 2 a 1 tan 2 a 1 c 1 - a 0 2 4 ( ξ + ξ 0 ),y ) h ( x,x m,l,y i ( ξ ) ) . now,β and γ are nonzero constants. for w = 0,1 ) are polynomials,r 1 = - 2 3 aba 1 + a 2 b λ,y ( ξ ) ] = ∑ i = 0 m a i ( x ( ξ ) ) y ( ξ ) i = 0,where c 1 is arbitrary integration constant. substituting a 0 ( x ) and g ( x ) into (19c) and setting all the coefficients of powers x to be zero,(21b) a 0 = λ k,solid state physics,then we have (10) - λ lu - klu ′ + lu 2 + 2 v = r 1,where “ ′ ” = d d ( ξ ) . we assume that eq. (3) has solution in the form (4) u i ( ξ ) = x i ( ξ ),fluid dynamics. in order to better understand these nonlinear phenomena,nonlinear optics,heidelberg [5] c.h. gu darboux transformation in solitons theory and geometry applications 1999 shangai science technology press shanghai [6] ryogo hirota exact solution of the korteweg–de vries equation for multiple collisions of solitons phys. rev. lett. 27 1971 192 1194 [7] ryogo hirota the direct method in soliton theory 2004 cambridge university press [8] m.l. wang solitary wave solutions for variant boussinesq equations phys. lett. a. 199 1995 169 172 [9] e.g. fan h.q. zhang a note on the homogeneous balance method phys. lett. a 246 1998 403 406 [10] y. cheng q. wang a new general algebraic method with symbolic computation to construct new travelling wave solution for the (1 + 1)-dimensional dispersive long wave equation appl. math. comput. 168 2005 1189 1204 [11] e.g. fan extended tanh-function method and its applications to nonlinear equations phys. lett. a. 277 2000 212 218 [12] z.y. yan new explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations phys. lett. a. 292 2001 100 106 [13] e.g. fan travelling wave solutions in terms of special functions for nonlinear coupled evolution systems phys. lett. a. 300 2002 43 249 [14] e.g. fan a new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed kdv equation chaos solitons fract. 15 2003 67 574 [15] m.a. abdou an improved generalized f -expansion method and its applications j. comput. appl. math. 214 2008 202 208 [16] emmanuel yomba the modified extended fan sub-equation method and its application to the (2 + 1)-dimensional broer–kaup–kupershmidt equation chaos solitons fract. 27 2006 187 196 [17] z.s. feng the first integral method to study the burgers–kdv equation j. phys. a: math. gen. 35 2002 343 349 [18] z.s. feng x.h. wang the first integral method to the two-dimensional burgers–kdv equation phys. lett. a 308 2003 173 178 [19] z.s. feng traveling wave behavior for a generalized fisher equation chaos solitons fract. 38 2008 481 488 [20] k.r. raslan the first integral method for solving some important nonlinear partial differential equations nonlinear dynam. 53 2008 281 [21] n. bourbaki commutative algebra 1972 addison-wesley paris [22] s.y. lou x.b. hu infinitely many lax pairs and symmetry constraints of the kp equation j. math. phys. 38 1997 6401 6427 [23] h.y. ruan y.x. chen study of a (2 + 1)-dimensional broer–kaup equation acta phys. sinica (overseas ed.) 7 1998 241 248 [24] s.y. lou (2 + 1)-dimensional compacton solutions with and without completely elastic interaction properties j. phys. a: math. gen. 35 2002 10619 10628 [25] h.y. zhi q. wang h.q. zhang a series of new exact solutions to the (2 + 1)-dimensional broer–kau–kupershmidt equation acta phys. sinica 54 3 2005 1002 1007 [26] y. wan l.n. song l. y h.q. zhang generalized method and new exact wave solutions for (2 + 1)-dimensional broer–kaup–kupershmidt system appl. math. comput 187 2007 644 657 [27] k. weierstrass,h ( x ) into (57d) and setting all the coefficients of powers x to be zero,x 2,7],(57b) a 1 ′ ( x ) = a 2 ( x ) g ( x ) - 2 a λ a 2 ( x ) - 2 aba 2 ( x ) x + a 1 ( x ) h ( x ),it is necessary and sufficient that there exists an x in l n such that α is the set of polynomials of k [ x 1,x n ] zero at y,(16b) y ′ ( ξ ) = k 1 x ( ξ ) 3 - k 2 x ( ξ ) 2 - k 3 x ( ξ ) + k 4 . according to the first integral method,u ix 2,g 3 ) = 4 ℘ 3 ( ξ,g 3 ) - 1 24 r ″ ( f 0 ) 2 - 48 r ( f 0 ) r ⁗ ( f 0 ),first integral method,we have (52a) a 1 ′ ( x ) = a 1 ( x ) h ( x ),where g 2,r 2 = 1 2 k λ lc 1,g 3 of elliptic functions ℘ ( ξ,m ) are polynomials of x and a m ( x ) ≠ 0 . eq. (50) is called the first integral to (49),r 2 = 2 lc 1 a 0 a 1 2 . using the conditions (28) in eq. (17),then it can be seen that our solutions are new. remark all solutions presented in this paper have been checked with maple by putting them back into the original eqs. (7) and (45) . 4 conclusion the first integral method is applied successfully for solving the system of nonlinear partial differential equations which are (2 + 1)-dimensional broer–kaup–kupershmidt system and (3 + 1)-dimensional burgers equations exactly. the performance of this method is reliable and effective and gives more solutions. this method has more advantages: it is direct and concise. thus,t ) and dependent variable u i,then we find a 1 ( x ) and a 0 ( x ) (31a) a 1 ( x ) = a 0 x + c 1,we obtain (54) a 1 = 0 . using the conditions (54) in eq. (50),and p ( x,u ix 3 x 3,y ) vanishes at all zero points of p ( x,johnson,periodic solutions according to eq. (36) are determined by (42) x ( ξ ) = f 0 + r ′ ( f 0 ) 4 - e 1 2 + 3 e 1 2 csc 2 3 e 1 2 ξ - 1 24 r ″ ( f 0 ),where ξ 0 is arbitrary integration constant. if deg ( g ( x ) ) = 0,(26b) a 1 ′ ( x ) = a 2 ( x ) g ( x ) + a 1 ( x ) h ( x ),u ix 1 t,g 3 ) are related to the coefficients of r ( x ) by [29] (38a) g 2 = - c 2 k 2 + 32 λ r 2 k 4 l + 3 ( λ 2 l - 2 r 1 ) 2 k 4,y ),in the case of (21b),riccati equation rational expansion method [10],then the abundant travelling wave solutions to (2 + 1)-dimensional broer–kaup–kupershmidt system can be written as (44) u ( x,we obtain (34) ( x ( ξ ) ′ ) 2 = 1 k 2 x 4 - 2 λ k 2 x 3 + λ 2 l - 2 r 1 k 2 x 2 + 4 r 2 k 2 l x - c 2 = r ( x ) . as is well known [27,3],which leads a new system of (5a) x i ′ ( ξ ) = y i ( ξ ),and q ( x,t ) = u ( ξ ),we deduce that the proposed method can be extended to solve many systems of nonlinear partial differential equations which are arising in the theory of solitons and other areas. acknowledgements the project is supported by the state key basic research program of china under grant no. 2004cb318000 and the national natural science foundation of china ( 10771092 ). references [1] m. ablowitz p.a. clarkson soliton nonlinear evolution equations and inverse scattering 1991 cambridge university press new york [2] m. wadati wave propagation in nonlinear lattice j. phys. soc. jpn. 38 1975 673 [3] m.r. miura bäcklund transformation 1978 springer-verlag berlin [4] v.a. matveev m.a. salle darboux transformation and solitons 1991 springer-verlag berlin,we suppose that x ( ξ ) and y ( ξ ) are nontrivial solutions of eq. (49),(9b) - λ v ′ + kv ″ + 2 ( uv ) ′ = 0 . integrating eq. (9a) twice with respect to ξ,δ = 0,we have (19a) a 1 ′ ( x ) = a 1 ( x ) h ( x ),g 3 ) is the weierstrass elliptic function [29] satisfying the nonlinear ordinary differential equation (36) ℘ ′ 2 ( ξ,f -expansion method [13–16],ξ = k ( x + ly - λ t ),y ) in c [ x,we obtain the exact solution to (16) and then the exact solutions to (2 + 1)-dimensional broer–kaup–kupershmidt system can be written as: (23) u 1 ( x,hirota’s bilinear method [6,u ix 2 x 2,this useful method is widely used by many such as in [18–20] and by the reference therein . the present paper investigates for the first time the applicability and effectiveness of the first integral method on high-dimensional nonlinear partial differential system. 2 summary of the first integral method let us consider the nonlinear evolution equations with independent variables x = ( x 1,take a 2 ( x ) = 1 . balancing the degrees of g ( x ) and a 0 ( x ),(38b) g 3 = ( 4 λ 2 - λ 2 l + 2 r 1 ) c 2 k 4 - 16 λ ( λ 2 l - 2 r 1 ) r 2 k 6 l - ( λ 2 l - 2 r 1 ) 3 k 6 - 16 r 2 2 k 6 l 2 . when g 2 and g 3 are real and the discriminant (39) δ = g 2 3 - 27 g 3 2 is positive,(3+1)-dimensional burgers equations,r 1 = - klc 1,not necessarily a real root of r ( x ),we have (57a) a 2 ′ ( x ) = a 2 ( x ) h ( x ),r 2 = - 3 2 aa 1 λ + 1 2 a 1 2 + a 2 λ 2 - 1 2 aba 0 and r 3 = a 0 a 1 - aa 0 λ . substituting a 0 ( x ),bäcklund transformation method [2,where α,new york,then from (26a) we deduce that a 2 ( x ) is constant and h ( x ) = 0 . for simplicity,g 3 ) - 1 24 r ″ ( f 0 ) + 1 24 r ( f 0 ) r ‴ ( f 0 ) 2 ℘ ( ξ,we obtain the exact solution to (16) and then the exact solutions to (2 + 1)-dimensional broer–kaup–kupershmidt system can be written as: (30) u ( x,u itt,λ,u i ″ ( ξ ),t ) = u i ( ξ ),many mathematicians and physical scientists make efforts to seek more exact solutions to them. several powerful methods have been proposed to obtain exact solutions of nonlinear evolution equations,u ix 1,28] the general solution of eq. (34) reads (35) x ( ξ ) = f 0 + r ( f 0 ) d ℘ ( ξ,v ( ξ ) = - λ bl - a 2 λ 2 + 2 aba 0 abl tanh 1 2 a 2 λ 2 + 2 aba 0 ( k ( x + ly + sz - λ t ) + ξ 0 ),we can get (33) y ( ξ ) 2 = 1 k 2 x 4 - 2 λ k 2 x 3 + λ 2 l - 2 r 1 k 2 x 2 + 4 r 2 k 2 l x - c 2 . combining (33) with (16a),and introduce a new independent invariable y i ( ξ ) = x i ′ ( ξ ),assuming that m = 1 and m = 2 in eq. (50) . case 1: suppose that m = 1,g 3 < 0,suppose that g ( x ) = a 1,g 2,we take two different cases,where ξ 0 is an arbitrary constant. similarly,t ) = r 1 2 + λ l 2 u - l 2 u 2 + kl 2 u ′ . example 2 . considering the (3 + 1)-dimensional burgers equations [32–34] that reads: (45) u t = α uu y + β vu x + γ wu x + u xx + u yy + u zz,and then the exact solutions to (2 + 1)-dimensional broer–kaup–kupershmidt system can be written as: (25) u 1 ( x,fluid dynamics,g 3 are real parameters and called invariants. if there exists a simply root f 0 of r ( x ),x n ] zero at x. (4) for a polynomial q of k [ x 1,then from (57a) we deduce that a 1 ( x ) is constant and h ( x ) = 0 . for simplicity,burgers equation
Nonlinear system,Mathematical analysis,Method of characteristics,First-order partial differential equation,Numerical partial differential equations,Adomian decomposition method,Stochastic partial differential equation,Collocation method,Mathematics,Hyperbolic partial differential equation
Journal
Volume
Issue
ISSN
216
4
Applied Mathematics and Computation
Citations 
PageRank 
References 
3
0.58
3
Authors
3
Name
Order
Citations
PageRank
Bin Lu130.92
Hongqing Zhang213848.35
FuDing Xie3716.99