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The Painleve equations and their solutions occur in some areas of pure and applied mathematics and theoretical physics.The Painleve equations have only movable singularities, which mean that their solutions do not have any singularities that are fixed in position.Thus, the Painleve equations are particularly useful in the study of non-linear systems, as they allow for the construction of exact solutions in certain cases.One of the important features of the Painleve equations is their appearance in diverse fields such as random matrix theory, soliton theory, and statistical mechanics.This has led to a wide range of applications, including the study of the behaviour of non-linear waves, random processes, and the dynamics of fluids.In this article, by using the mixed variable transformation = (); = + + , we derived some modified formulas for sinh-Gordon equation reducible to the Painleve equations and we found the exact solutions for them and other equations derived from them.
Nada A. Laabi (Thu,) studied this question.