New solitary and optical wave structures to the (1 + 1)-dimensional combined KdV–mKdV equation

In this article, we employ the sine-Gordon expansion method to construct a new solitary and optical wave structures to the (1 + 1)-dimensional combined KdV–mKdV equation. Sine-Gordon expansion method is developed based on the travelling wave transformation u(x, y) = U(ξ), ξ = μ(x − ct) and the sine-Gordon equation uxx − utt = m2 sin(u). We find the travelling wave solutions to (1 + 1)-dimensional combined KdV–mKdV equation in the form of complex, hyperbolic, trigonometric and rational function. Our results show that all the travelling wave solutions we obtain are solitary and optical wave structures solutions. All the computations and graphics plot in this article are carried out with the help of Wolfram Mathematica 9.