Multi-soliton Solution and Periodic Solution of the Fractional Lax Equation Using by Darboux Transformation

Through this paper we derived the exact solution of the fractional order Lax equation. We consider the fractional Lax equation which is the generalization version of the Lax equation by replacing the modified Riemann-Liouville fractional derivative from classical derivative. Some important definition and result of modified Riemann-Liouville fractional derivative are discussed. The Darboux transformation and fractional complex transformation are used to compute some exact solution such as one-soliton solution, two-soliton solution, singular soliton solution, singular periodic solution, etc. Also in graphically the dynamical behavior of the solution by changing the fractional parameter \(\rho \) are illustrated.