In this work, Laurent series expansion compared with definition of rational function is used to find rational solutions with N simple poles for some systems

Al-Khairullah, N. A.

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    32 On Rational Solutions for some systems  N. A. Al-Khairullah Department of statistics  College of Administration and Economies Sumer University                                                                                                         ABSTRACT: In this work, Laurent series expansion compared with definition of rational function is used to find rational solutions with N simple poles for some systems.          Introduction         During the last three decades, there has been a great deal of interest in rational solution of integrable nonlinear evolution equations: This began with studies of the rational solutions of the Korteweg-de Vries (KdV) and Kadomtsev-Petviashvilli (KP) equations, but soon similar results were obtained concerning the Benjamin-Ono equation and the classical Boussinesq and AKNS systems, [2,3].         Rational solutions were found in the majority of the equations of mathematical physic that possess a rich algebraic structure: an infinite set of symmetries and related commuting flows. It was the rational solutions of such equations are special limiting forms of exponential “multisoliton” solutions and can be obtained from the latter once by long-wave degeneration, [6].         Further applications of rational solutions to soliton equations include the description of explode-decay waves and vortex solutions of the complex sine-Gordon equation, [3].         The direct methods for the construction of rational solutions have great importance in mathematical physics. For instance, such methods are based on the transformation of various forms of the  function to the equation of interest, or on the group theory, or on the analytic properties of the Baker-Akhiezer function, [6].         The goal of this work is to present a method for finding rational solutions with N simple poles for some systems. Description of the Method         Consider a given nonlinear PDE, say in two variables x and t ),...,,,,,(22nnxuxuxuutxFtu         RtCx  ,,          …(1.1)  In order to apply this method, we need the following steps: Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    33  Step 1: We assume that u(x,t) is a rational solution of the equation (1.1) with  N  simple poles, we use 10 1 11R (t)u( , t) (t) (t)( (t))(t)x a a x xx x      …(1.2)    where R1(t) is the residue of u(x,t) near x1(t) and Nk0k 2 1 kR (t)(t) ,(t) (t)ax x  x1(t)  xk(t)  Nk1 2k 21 kR (t)(t)(t) (t)ax x , x1(t)  xk(t).  Step 2: Letting x – x1 to be , then Substitute the following derivatives of u(x,t) with respect the variables  x  and  t  in the equation 11102111 xaaaxRRtu …(1.3a)  111110311211111222222xaaxaaaxRxRxRRtu …(1.3b)      11R (t)u(t)ax   nn1n n( 1) n!Rux           n=2,3,… …(1.3c)   Step 3: Equating the coefficients of i to zero, to have a nonlinear system of algebraic or differential equations and then solve the system.  Remarks:  (i) Our method is based on the comparison of the usual assumption for function with  N  simple poles Nkk 1 kRu( , t) ,x xx x  £  …(1.4) and expand of functions by formal Laurent series near a specified pole (say  x1): nn 1n 1u( , t) ( )x a x x   …(1.5) Denote a – 1  by  R1. We can see that Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    34 Nn kn 1n 0 k 2 kR (t)(t)( (t))(t)a x xx x     …(1.6)  (ii) The coefficients  a0(t), a1(t), … in (1.6) carry information about the poles x2, …, xN.  (iii) The ansatz that we adopt, is:  n1n kn 01R (t)u( , t) + (t) (t)(t)x a x xx x   …(1.7) and because of the difficulty of equating coefficients for (x – x1(t)) of the nonlinear from PDE. Hence we will write the form (1.7) as (1.2) which we have directly in the first step of the method.   Rational Solution for Carleman Mode          In this section, an example illustrating that  u  and  v  may have a common pole has been given        We consider the system of PDEs ut + ux = v2 – u2 …(2.1a)  vt – vx = u2 – v2 …(2.1b) Which was  introduced  by T. Carleman as the two velocity model of the Boltzman equation,[7].  Let x1(t) be the common pole of u  and  v.   To apply the method of finding rational solution with  N  simple poles, the following ansatz will be used: 10 1 11R (t)u( , t) (t) (t)( )x a a x xx x    …(2.2a)  10 1 11S (t)v( , t) (t) (t)( )x b b x xx x    …(2.2b) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    35 where  1 1 2N N Nk k k0 1 02k 2 k 2 k 21 k 1 k1 kR (t) R (t) S (t)(t) , (t) , (t) ,(t) (t) (t) (t)(t) (t)a a bx x x xx x        and   2Nk1 2k 21 kS (t)(t)(t) (t)bx x .          By substituting the functions (2.2) and their derivatives  in equations (2.1) , the following identities have been obtained:  101111022120221101111022120221121111011211222222 aaRaRaaaRbbSbSbbbSaRxaaaRxR …(2.3a) and                 10111102212022110111102212022112111101121122222 bbSbSbbbSaaRaRaaaRbSxbbbSxS                                                                                                       …(2.3b)                                                               From the coefficients of  2  in (2.3a) and (2.3b) we get: 2 21 1 0b a   2 21 1 0a b     then   a1 =  b1.         The coefficients of    give: 10101 22 aabba     10101 22 bbaab  then     11 ba    which means a1 = – b1.          From the coefficients of  0  we get: Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    36 1120112011101120112011102222SbbRaabxbbRaaSbbaxaa Then 100 2aab    and  0 0 1b 3 ca     where 1c  arbitrary constants.          The coefficients of  –1  give: 10101101012222SbRaSRaSbR Then  R1(t) = – S1(t) + 2c  ,  2c  arbitrary constant.          From the coefficients of  –2  we get: 21211112121111SRSxSRSRxR Then      12 11  Sx , and      1 1 32S t t cx      ,  3c  arbitrary constant.            Hence the solution will be: 1 20 1 1 31 310 1 1 1 31 3S cu( , t) a ( 2S t t c )2S t t cSv( , t) 3 + c a ( 2S t t c )2S t t cx a xxx a xx                where  c1 and c3 are arbitrary constants.   Dispersive Long Wave System and Rational Solution  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    37          Consider the following dispersive long wave system:  ut = (u2 – ux + 2w)x …(2.4a)  wt = (2uw + wx )x  …(2.4b)            The analytical solution of (2.4) with initial conditions:  u( ,0)x =2+2 tan h( x )  and  w( ,0) 1 2x    sec h 2x  is given by Bai et al. as follows:[5] u( , t)x =2+2 tan h( 2tx  )   and   w( , t)x 1 2    sec h  22tx  .        The results on analytical and numerical solutions for system of dispersive long wave in (2 + 1) dimension can be found in [1] and [5].         Also, Zhang  Jie and  Fang Guo obtained a nonlinear transformation for simplifying a dispersive long wave equation by using the homogeneous method and found some new multi-Soliton solutions and exact traveling solution of the dispersive long wave equations from the linear partial equation, [4].         In this section, this example shows that one of the functions of a system may have rational solution while the other is not.        In order to obtain the rational solutions with  N  simple Poles for the system (2.4), we assume that the functions u(x,t) and w(x,t) have a common simple pole x1(t) and using the formulas: 10 1 11R (t)u( , t) (t) (t)( )x a a x xx x    …(2.5a)  10 1 11S (t)w( , t) (t) (t)( )x b b x xx x    …(2.5b)  Plugging (2.5) and their derivatives in equations (2.4) gives Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    38 121312110210321111011211222222 bSRaaaRaRxaaaRxR and 21101112112110111110112112))((2))((2 SbbSaRbSaaRxbbbSxS By equating the coefficients of  –3  and  –2  to zero we get: R1(t) = – 1, S2(t) = 0, 01 2ax , and b0(t) = 0           The coefficients of  –1  vanish identically, while the coefficient of 0 give: 10 2ba          From the coefficient of    we get: 11121142babaa  If  1a 0  then 1121 2120 311(t) 2t  c  c(t)(2t  c )2c(t) c2t  caba  and   1 2 1 3 4(t) 2c ln 2t c 2c t cx       Where  c1, c2, c3  and  c4  are arbitrary constants. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    39          Hence the solutions for equations (2.4) will be:  2 1 3 4232 1 3 4 1 12c ln 2t c 2c t c1 2cu( , t) c2c ln 2t c 2c t c 2t c 2t cxxx            …(2.6a) 2 1 3 4212c ln 2t c 2c t cw( , t)(2t c )xx                                               …(2.6b)                                    The pole of (2.6a) is the zero of (2.6b).             When  a1 = 0 then u and w becomes: 1 221 2 31u( , t) 2k t kk t 2k t kxx     21 1 2 3w( , t) k ( 2k t 2k t k )x x     where  k1, k2 and k3  are an arbitrary constants.  Conclusion         In this method for obtaining rational solutions with simple poles using Laurent series expansion, the exact solution for nonlinear PDE using the whole Laurent series expansion may be difficult to be evaluated especially in nonlinear term . We expect there is a relation between the number of poles and the number of terms of  using power series expansion.   References: 1. Chen, Y.,Yong, X., Exact Solutions to the Dispersive Long Wave Equation, Journal of Nonlinear Science, Vol.4, No.2, 147-150, 2007.  2. Clarkson, P.A., Rational Solutions of the Classical Bossinesq System, Nonlinear Analysis: Real World Application, 2008. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.7, No.3, 2017    40 3. Hone, A.N., Crum Transformation and Rational Solutions of the Non-Focusing Nonlinear Schrödinger Equation, J. Phys , 1413-1483, 1997. 4. Jie-Fang, Z., Guan-Ping, G. and Feng-Min, W., New Multi-Soliton solutions and Travelling Wave Solutions of the Dispersive Long-Wave Equations, Chin. Phys. Soc., Vol.11, No.6, 2002. 5. Mohamed, M.A., Comparsion Differential Transformation Technique with Adomian Decomposition Method for Dispersive Long-Wave Equations in (2+1)-Dimensions, Int.Journal (AAM), Vol.5, No.1, 148-166, 2010. 6. Pelinovsky, D., Rational Solutions of the Kadomtsev-Petviashvili Hierarchy and Dynamics of Their Poles.1.New Form of a General Rational Solution, J.Math.Phys., Vol.35, No.11, 1994. 7. Wick, J. and Nunzert , H., “Two Classes of Explicet Solutions of The Carelman Mode”, Mathematical methods in the applied sciences, Vol.6, No.1, 515-519,1984.   

On Rational Solutions for some systems

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On Rational Solutions for some systems

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