Abstract
We study symmetries, invariant solutions, and conservation laws for the dispersionless Veselov–Novikov equation. The emphasis is placed on cases when the odes involved in description of the invariant solutions are integrable by quadratures. Then we find some non-invariant solutions, in particular, solutions that are polynomials of an arbitrary degree N ge 3 with respect to the spatial variables. Finally we compute all conservation laws that are associated to cosymmetries of second order.
Highlights
We consider the dispersionless Veselov-Novikov equation [11] written in the form ut x y =x +y . (1)This equation describes the propagation of the high frequency electromagnetic waves in certain nonlinear media, see [12] and references therein
In the present paper we study exact solutions and conservation laws of dispersionless Veselov-Novikov equation (dVN)
We find some cases when the obtained odes are integrable by quadratures, providing exact solutions for dVN
Summary
We consider the dispersionless Veselov-Novikov equation (dVN) [11] written in the form ut x y = (ux x uxy )x + (uxy u yy )y. Of dVN is the dispersionless reduction of the Lax representation of the Nizhnik– Veselov–Novikov equation. In [2] the Lax representation (2) was used to construct two-dimensional reductions of dVN. We factorize dVN with respect to the symmetries from the optimal system and obtain two-dimensional partial differential equations (pdes) (8) and (50) for the invariant solutions as well as their Lax representations. We find the symmetry algebras and their optimal systems of one-dimensional subalgebras for equations (8) and (50). The factorization with respect to the subalgebras provide the collection of ordinary differential equations (odes) that describe invariant solutions to (8) and (50). We find the whole set of conservation laws that are associated to cosymmetries defined on the second order jets
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