Symmetries of differential equations: Example of dispersionless Nizhnik equation

After explaining the notions of symmetry and differential equations, we review possibilities of symmetry methods and advantages of their usage in the theory of differential equations and mathematical physics.

As a specific example, we discuss the history of the (real potential symmetric) dispersionless  Nizhnik equation and its applications and overview its extended symmetry analysis carried out in our papers. More specifically, we construct essential megaideals of the maximal Lie invariance algebra of this equation. Using the original version of the algebraic megaideals-based method, we compute the point- and contact-symmetry pseudogroups of this equation as well as the point-symmetry pseudogroups of its Lax representation and the original real symmetric dispersionless Nizhnik system. This is the first example in the literature, where there is no need to use the direct method for completing the computation.

In addition, we also find geometric properties of the dispersionless Nizhnik equation that completely define it. Lie reductions of this equation are classified, which results in wide families of its new closed-form invariant solutions. We also study hidden generalized symmetries, hidden cosymmetries and hidden conservation laws of this equation.