New Travelling-Wave Solutions for Dodd-Bullough Equation

initiated by Dodd and Bullough [1] and Žiber and Šabat [2], plays a significant role in many scientific applications such as solid state physics, nonlinear optics, and quantum field theory. There are many research works for Dodd-Bullough equation in the last decades. It is shown thatDodd-Bulloughdetermines the intrinsic geometry of the two-dimensional affine sphere in the three-dimensional unimodular affine space [3]. It has also been shown that the Dodd-Bullough equation is related to the nonlinear two-dimensional SL(3,R) sigma model in [4]. To solve the Dodd-Bullough equation, many mathematicians have put forward some methods, and different forms of solution formulas have been retrieved. Reference [5] develops two-soliton solution formula and N-soliton solution formula by means of dressing the trivial one φ ≡ 0. The dressing procedure for (1) is also suggested in [6]. By using three Weierstrass functions, [7] proposes some explicit solution formulas. Reference [8, 9] obtain some general ineffective formulas for solutions of (1), but no concrete solutions. The MSE (modified simple equation) method has been proposed in [10–12] to explore the exact traveling solution of nonlinear physical systems, then [13] applies the MSE method to the Tzitzeica-Dodd-Bullough equation for some new solutions. Tanh method is efficient to deal with many kinds of nonlinear equations. Applying the tanh method to the DoddBullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, [14] derives some solitons and periodic solutions. By developing an extended tanh method, [15] builds some new explicit solutions for Dodd-Bullough equation. This paper aims to propose a newmethod to deal with the Dodd-Bullough equation, which is based on an assumption that the first order differential of φ on the travelling wave variable ξ = u + cV. A new family of explicit solutions for (1) is derived.The proposedmethod works efficiently to solve other forms of Dodd-Bullough equations. We also compare some of our results to some previous solutions and find that the solution of theDodd-Bullough-Mikhailov equation given in [14] is only a special case of ours. Moreover, our solutions for Tzitzeica-Dodd-Bullough equations are different from previous results in [13] and other works.