Exceptionality Condition and Linearization Procedure for a Third Order Nonlinear PDE

We determine a third order hyperbolic nonlinear partial differential equation possessing the property of being completely exceptional; i.e., every admissible discontinuity wave never evolves into nonlinear shock wave. A special member of this class describes the Riemannian geometric theory of critical phenomena by requiring the scalar thermodynamic curvature to be proportional to the inverse of free energy. By using reciprocal transformations it is shown that certain subclasses may be reduced either to linear canonical forms or to an equation which is linear in the third order derivatives.