Hidden symmetries, hidden conservation laws and exact solutions of dispersionless Nizhnik equation

Abstract: Among Lie submodels of the (real symmetric potential) dispersionless Nizhnik equation, we single out a remarkable submodel as such that, despite being the only one, is associated with a family of in general inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra of this equation, which are parameterized by an arbitrary function of the time variable. The wide family of invariant solutions of the dispersionless Nizhnik equation that are related to the above submodel is expressed in terms of an arbitrary function of the time variable and the double quadrature of the well-known (implicit) general solution of the inviscid Burgers equation with respect to a space-like submodel invariant variable. The singled out submodel possesses many other interesting properties. In particular, we show that it is Lie-remarkable, and its maximal Lie invariance algebra completely defines its point symmetry pseudogroup, which provides the second but simpler example of the latter phenomenon in literature. Moreover, only hidden Lie symmetries of the dispersionless Nizhnik equation that are associated with this submodel are essential for finding its exact solutions. Using Lie reductions, we construct new families of exact solutions of the inviscid Burgers equation and the dispersionless Nizhnik equation in closed or parametric form. We also exhaustively described generalized symmetries, cosymmetries and conservation laws of the submodel, which gives the corresponding nonlocal and hidden structures for the inviscid Burgers equation and the dispersionless Nizhnik equation, respectively.