Complex-valued travelling wave solutions to the Casimir equation for the Ito system via the first integral method

The Ito system was previously been shown to admit a reduction to a single nonlinear Casimir equation. In the present paper, we reduce this nonlinear partial differential equation into an ordinary differential equation governing a travelling wave solution. The ordinary differential equation takes the form of a second order nonlinear equation, and the form of this nonlinearity is a rational function. As such, the nonlinearity can become singular. This makes the problem interesting and somewhat challenging to solve for standard methods. Therefore, we make use of the first integral method in order to obtain exact solutions for this equation of second order and thus obtain exact solutions to the Casimir equation for the Ito system. The first integral method actually allows one to account for multiple solutions, and we demonstrate that there indeed can exist multiple complex-valued solutions for this Casimir equation. The results demonstrate the utility of the first integral method, suggesting that the approach can be used to solve many nonlinear differential equations that may admit multiple solutions.