Analytic Solutions of Nonlinear Partial Differential Equations by the Power Index Method

Abstract

An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or 180 • rotation symmetry, are possible. The method tests whether certain hyperbolic or Jacobian elliptic functions are analytic solutions. The method consists of eight steps. The first six steps are quickly applied; conditions for algebraic equations are more complicated. A few exceptions to the Power Index Method are discussed. The method realizes an aim of Sophus Lie to find analytic solutions of nonlinear differential equations.

Exact Traveling Wave Solutions of Nonlinear Evolution Equations: Indeterminant Homogeneous Balance and Linearizability

“…which is the mKdV equation; the power = 1 a it obeys homogenous balance [1,2]. The solutions of (12) have been known [1,2]. Some solutions are tanh( ̅ ) or coth( ̅ ).…”

“…The homogeneous balance condition is given by the power index and the justification for is discussed in [1,2]. Then for any term in the NLPDE is…”

“…Determining exact, traveling wave solutions of some nonlinear partial differential equations (NLPDEs) has been codified recently where earlier references are noted [1,2]. The approach ties together various methods with new criteria in the PIM.…”