Potential Symmetries For Nonlinear Telegraph Equation And The Exact Solutions Of H-S Equations

Reducing nonlinear partial differential equation by symmetries is an efficient methodto find exact solutions of the equations. The more symmetries considered differentialequations admit, the more exact solutions can be obtained. In this paper, according toinfinitesimal criterion in symmetry theorem, using differential Wu-method(char-set byMathematica software) and conditions of symmeties of equation u_xx=H(x)u_tt,we givenew potential symmetries for nonlinear telegraph equation u_tt=(F(u)u_x)_x+(G(u))_x.These symmetries can be used to find new exact solutions of nonlinear telegraph equa-tion. The preliminary result shows that we can find new potential symmetries for thescalar equation by different F(u) and G(u). Differential characteristic set method ofdifferential polynomial system is used to overcome the huge difficulties in calculation.Secondly, we study F-expansion method ,consequently, eight exact solitary wavesolutions for nonlinear evolution equation, Hirota-Satsuma equations, are obtained.We make a proper transformation according to the structure of the equation at first,converting the equations into ordinary differential equation, then assume the solution'sform and balance the degree of F between the highest differential term and the highestnonlinear term in the ordinary equation to determine the concrete degree of F insolution form. Finally using computer algbra system mathematica software, we obtainthe exact solutions of Hirota-Satsuma equations . These solutions, varing according tovarious of parameters P,R, Q on which F-method depend, are more genaral solutionsthan those obtained by traditional method. They also play an important role in studingHirota-Satsuma equations in quatity or quality.