Travelling Wave Group-Invariant Solutions and Conservation Laws for θ-Equation

We study a class of nonlinear dispersive models called the theta-equations from the Lie group-theoretic point of view. The Lie point symmetry generators of the class of equations are derived. Using the optimal system of one-dimensional subalgebras constructed from these symmetry generators, we obtain symmetry reductions and travelling wave group-invariant solutions for the underlying equation. Moreover, we construct conservation laws for the class of equations by making use of the partial Lagrangian approach and the multiplier method. The underlying equation is of odd order and thus not variational. To apply the partial variational method a nonlocal transformation u = v(x) used to raise the order of the given class of equations. We show that the existence of nonlocal conservation laws for underlying equation is possible only if theta = 1/3. In the multiplier approach, we obtain conservation laws for the given class of equations and a special case of the equation when theta = 1/3 in which the first-order multipliers are considered.