Maximum Entropy Solutions with Hyperbolic Cosine and Secant Distributions: Theory and Applications

This work explores the hyperbolic cosine and hyperbolic secant functions within the framework of the maximum entropy principle, deriving these probability distribution functions from first principles. The resulting maximum entropy solutions are applied to various physical systems, including the repulsive oscillator and solitary wave solutions of the advection equation, using the method of moments. Additionally, a different moment analysis using experimental and theoretical inputs is employed to address non-linear systems described by the non-linear Schrödinger equation, non-linear diffusion equation, and Korteweg–de Vries equation, demonstrating the versatility of this approach. These findings demonstrate the broad applicability of maximum entropy methods in solving different differential equations, with potential implications for future research in non-linear dynamics and transport physics.