Hidden symmetries, hidden conservation laws and exact solutions of dispersionless Nyzhnyk equation
arXiv:2505.02962 · doi:10.1016/j.cnsns.2025.109635
Abstract
Among Lie submodels of the (real symmetric potential) dispersionless Nyzhnyk 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 large family of invariant solutions of the dispersionless Nyzhnyk 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 Nyzhnyk 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 Nyzhnyk 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 Nyzhnyk equation, respectively.
arXiv commentpublished version, 47 pages, 1 table, application of results of arXiv:2211.09759 and arXiv:2308.03744
References in corpus (23)
- Realizations of Real Low-Dimensional Lie Algebras
- Symmetry reduction and exact solutions of the Navier-Stokes equations
- Complete group classification of a class of nonlinear wave equations
- Enhanced preliminary group classification of a class of generalized diffusion equations
- Sufficient set of integrability conditions of an orthonomic system
- Algebraic method for finding equivalence groups
- Symmetries and conservation laws for the Karczewska--Rozmej--Rutkowski--Infeld equation
- Extended symmetry analysis of generalized Burgers equations
- Complete point symmetry group of the barotropic vorticity equation on a rotating sphere
- Enhanced symmetry analysis of two-dimensional Burgers system
- Point and generalized symmetries of the heat equation revisited
- Ordinary differential equations described by their Lie symmetry algebra
- Extended symmetry analysis of isothermal no-slip drift flux model
- Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
- Lie remarkable partial differential equations characterized by Lie algebras of point symmetries
- How to find discrete contact symmetries
- Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system
- Variational symmetries and conservation laws of the wave equation in one space dimension
- Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model
- Generalized symmetries and conservation laws of (1+1)-dimensional Klein-Gordon equation
- Multidimensional integrable systems from contact geometry
- Lie reductions and exact solutions of dispersionless Nizhnik equation
- Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation