Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

and Applied Analysis 3 Step 6. Using the results obtained in above steps, we can finally obtain exact solutions of (5). 3. Applications of the Modified Truncated Expansion Method In this section, we apply the proposed modified truncated expansion method to construct the exact solutions for the nonlinear DDEs via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system which are very important in the mathematical physics and have been paid attention by many researchers. 3.1. Example 1: The General Lattice Equation. In this subsection, we use the modified truncated expansion method to find the exact solutions of the general lattice equation. The traveling wave variable (6) permits us converting (1) into the following form: C 1 U 󸀠 (ξ n ) − [α + βU (ξ n ) + γU 2 (ξ n )] × [U (ξ n + d) − U (ξ n − d)] = 0, (12) where (󸀠) = d/dξ n . Considering the homogeneous balance between the highest order derivatives and nonlinear term in (12), we get K = 1. So we look for the solution of (12) in the following form: U (ξ n ) = a 0 + a 1 φ (ξ n ) , (13) where a 0 and a 1 are arbitrary constants to be determined later and φ(ξ n ) satisfies (9) and (10). We substitute (13), (9), and (10) into (12) and collect all terms with the same power in [φ(ξ n )] i , (i = 0, 1, 2, . . .). Setting each coefficient of this polynomial to zero, we derive a set of algebraic equations for a 0 , a 1 , d and C 1 . Solving the set of algebraic equations by using Maple or Mathematica, we have the following results: a 0 = − 1 2 βed + β ± √− (e − 1) 2 (−β 2 + 4αγ)