TruncatedPainleveexpansionandAuto-B¨acklundtransformationforakindofgeneralizedvariable-coefficientKadomtsev-Petviashviliequations.doc

精品论文大集合truncated painleve expansion and auto-backlund transformation for a kind of generalized variable-coefficient kadomtsev-petviashvili equationshe tian-juna , chen yonga,bnonlinear science center and department of ningbo universitye-mail:abstract among the topics attracting much attention in mathematical physics are the variable-coefficient generalizations of the well-known kadomtsev-petviashvili equation (gvckp) . we make use of both of truncated painleve expansion and symbolic compu- tation leads to a new class of analytical solutions to a kind of gvckps, and we obtain an auto-backlund transformation .pacs numbers: 02.30.jrkey words:truncated painleve expansion; variable coefficient ; symbolic computation; auto- backlund transformation;a kind of generalized variable-coefficient kadomtsev- petviashvili equations. left1 introductionwith its many physical applications from water waves to plasma physics and field theories, the well-known kadomtsev-petviashvili (kp) equation is a completely integrable soliton equation which also has some impressive properties. however, the physical situa- tions in which the kp equation arises tend to be highly idealized, owing to the assumption of constant coefficients, e.g., in the propagation of small-amplitude surface waves in a fluid of constant depth1 . in recent years, much attention has been paid on the study of non- linear partial differential equations with variable coefficient, especially, variable coefficient burger and kdv equations25 the variable-coefficient generalizations of the kadomtsev- petviashvili equation (gvckp) , however, are able to provide more realistic models in physical situations such as in the canonical and cylindrical cases, propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity, etc. re- cently, much progress has been made in the studies of certain gvckps obtaining their solitary wave solutions, soliton interactions, complete integrability, lax pairs, etc. (see refs.611 and references therein.) the development of symbolic computation enables us, in this paper, to solve analytically the following a kind of variable-coefficient generaliza- tions of kadomtsev-petviashvili equation (gvckp)11xutx + u2 + uuxx + uxxxx + uyy + (b1y + b0)uxy + (c1y + c0)uxx = 0(1) where c1 = c1 (t), c0 = c0(t), b1 = b1(t), b0 = b0(t), = 1 are arbitrary functions withrespect to twhich show (1) has an infinite-dimensional symmetry group and the existence of an infinite dimensional symmetry group makes it possible to use lie group theory to obtain large classes of solutions.5this paper is organized as follows. in section.2, we introduce truncated painleve ex- pansion,in section.3, painleve backlund equations and an auto-backlund transformation are give,some calculations are presented in section 4.2 truncated painleve expansionthe sufficient condition for a partial differential equation (pde) to be completely in- tegrable is that it possesses the painleve property , the solutions to the pde, written asu(x, y, t) = (x, y, t)j x un (x, y, t)(x, y, t)n (2)n=0are single-valued in the neighbourhood of a noncharacteristic, movable singularity mani- fold,m = (x, y, t)|(x, y, t) = 0(3)where j is a natural number to be determined; un (x, y, t) and (x, y, t) are analytic func- tions with u0(x, y, t) = 0 at present, we do not require eq. (1) to be completely integrable,but truncate the painleve expansion (i.eeq. (2) ) at the constant-level term,ju(x, y, t) = (x, y, t)j x un (x, y, t)(x, y, t)n (4)n=0to investigate the backlund transformation as well as the analytical solutions to eq. (1)the leading-order analysis for eq. (1) yields j = 2, so thatu(x, y, t) = u0(x, y, t) + u1(x, y, t) + u (x, y, t) (5)(x, y, t)2(x, y, t) 23 painleve backlund equations and an auto-backlund trans- formationwhen substituting eq(5) into eq.(1) with maple, we make the coefficients of like powers of vanish, so as to obtain the following set of painleve-backlund equations,x6 : u0 = 122(6)5 : 72u02 xx + 12u1 4 48u0x3 4u0 u0xx + 6u02 u1 u2xx = 0(7)x x x x 04 : 6u0xt + 72u0xxxx + 36u0xx2 + 18u02+ 24u0xxxx 36u12 xxxxxx24u1x3 6u0u1xx 6u1u0xx + 6u02 + 6c0u02 + 3u22xy x1 x+u0u0xx + 6u2u02 3u1 u0xx + 6b1yu0xy + 6b0u0xy + 6c1yu02 = 0xx3 : 2b1yu0y x 2b1yu0xy + 2b1u1y xx4c1 yu0xx + 2c1yu12 + 2u1t x 2u0xt 2u0t x 2u0xt 8u0xxx xxxx2u0xxxx + u0u1xx + u1u0xx 12u0xx xx + 6u12 8u0x xxx + 12u1xx 2u2xx + 2u1 2 2b0u0y x 2b1u0x y 4c0 u0x x + 2c0u12 + 24u1x xxx1 y x+8u1xxxx 2u0yy 2b0 u0xy 2c0u0xx 2b0u2xx + 2b1yu1xyx2b1 yu0xy 2c1yu0xx 4u1u1x x 4u2 u0x x + 2u2u124u0u2x x 4u0y y + 2u0xu1x = 0(8)1x2 : 2u2xu1x 2c0 xu1x + 2u2x u0x + u2 b0y u1x b0 xu1y 2y u1y 2u1 xu2x + b1yu0xyu1xt u1xxxx b1yu1y x b1yu1x y + c0u0xx + u0xxxx4u1x xxx + u0xt 2c1yxu1x u1u2xx u1 c0xx u1 b0 yx u1yy + c1yu0xx b1 yu1xyc1yu1xx + b0 u0xy + u2u0xx + u1u1xx + u0yy + u0 u2xx 6u1xx xx 4u1xxx x u1x t u1t x = 01 : u2 u1xx + u1xt + u1xxxx + 2u2x u1x + b1yu1xy + c0 u1xx + u1yy + b0u1xy u1u2xx + c1yu1xx = 0 (9)2x0 : u2tx + u2+ u2u2xx + u2xxxx + u2yy + (b1 y + b0)u2xy + (c1y + c0)u2xx = 0(10)to simplify the above equation,when substituting (6)into (7) we haveu1 = 12xx(11) when substituting (6)and (11)into (8) we haveu2 = (c1 y + c0) 4xxxtx xb1 yy xb0y x3 2xx+2x2y 2x(12)when substituting (6), (11)and (12)into (9) we haveb1y2 xy b0xxxy + b0 2 xy 4xxxxxx xxxt b1yxxy xx+33x xx2 + yy 2 + xt 2 + xxxx 2 = 0(13)xxy xxxwhen substituting (6) ,(11),(12)and (13)into (9), (10) we have22y xx + xyy 2xy xy = 0(14)ywe are able to find a family of exact analytical solutions to eq(1)as follows:2u = 12x + 12xxxxxtb1yyb0 y2 23xxx2 (c1 y + c0) 4 x x+2 + 2xxx(15)with the constraint equation (13) and (14) for , c1, c0, b1, b0, we note that once a backlund transformation discovered, and a set of seed solutions is given, one will be able to find an infinite number of solutions by the repeated applications of the transformation, i.e., to generate a hierarchy of solutions with increasing complexity. in the next section of the paper, we will find a family of the exact analytic solutions to eq(1).4 calculationsseveral comments are in order,eq(1) is larger for special cases of the functions b1, b0 , c1 and c0. f. gungor and p. winternitz11have shown that eq(1)have finite point transfor- mation, when b1 = 0 eq(1) have an infinite-dimensional lie point symmetry group andwhich lie algebra has a kac-moody-virasoro structure.let us now consider the cases 1:sample solution. a trial solutionb1 = 0(16) = 1 + ea(t)x+(t)y+b(t)(17) when substituting eq(16)and eq(17) into eq(13)and eq(14). with maple,symbolic com-putation ,we havea(t) = c ,where c is constant and (t), b(t)are arbitrary functions withrespect to t we find a family of the analytical solutions of eq(1) as follows:442du = (c e 10c ed+ c0 c2+ 2c0c2ed+ c0c2 e2d+ c1 yc2+ 2c1yc2ed+ c1yc2ed+ c4)(c + ced )22(b0(t)c + 2b0(t)ced + b0 (t)ce2d + ct (t)y + 2ct (t)yed + ct (t)yed (c + ced )2(2cbt (t)ed + cbt (t)e2d ct (t)ye2d + (t)2 + 2(t)2ed + (t)2 e2d ) (c + ced )22+ cbt (t)(18)where d = cx + (t)y + b(t)5 summary and conclusionto sum up, we introduce truncated painleve expansion ,with the truncated painleve expansion analysis and the symbolic computation, we have shown that the set of equa- tions(13)and (14),which constitutes an auto-backlund transformation, exists for the gen- eralized kp equation in eq.(1) . leads to a new class of analytical solutions to a kind of gvckps. we obtain an auto-backlund transformation . and note that once a backlund transformation discovered, and a set of seed solutions is given, one will be able to find an infinite number of solutions by the repeated applications of the transformation and wwhen (b1 = 0) .a solitonic solutions(20)is obtain and some explicit solution is given.acknowledgements:the work is supported by the national natural science foundation of china (grantno. 10735030)