In the thesis, we carried out extended symmetry analysis of the (real symmetric potential) dispersionless Nizhnik equation
$u_{txy}=(u_{xx}u_{xy})_x + (u_{xy}u_{yy})_y,$ (1)
which is also called as the dispersionless Nizhnik–Novikov–Veselov equation or even the dispersionless Novikov–Veselov equation. This equation is the dispersionless counterpart of the real symmetric potential Nizhnik equation.
Simultaneously with the equation (1), we considered its nonlinear representation Lax representation
$v_t=\frac13\left(v_x^3-\frac{u_{xy}^3}{v_x^3}\right)+u_{xx}v_x-\frac{u_{xy}u_{yy}}{v_x}, v_y=-\frac{u_{xy}}{v_x},$ (2)
and the dispersionless counterpart
$p_t=(h^1p)_x+(h^2p)_y, h^1_y=p_x, h^2_x=p_y$ (3)
of the symmetric Nizhnik system.
In Chapter (1), we studied symmetry properties of the equation (1) and the systems (2) and (3). In particular, we found their maximal Lie invariance algebras $\mathfrak g$, $\mathfrak g_{\rm L}$ and $\mathfrak g_{\rm dN}$ and the maximal contact-symmetry algebra $\mathfrak g_{\rm c}$ of the equation (1).
Applying an original megaideal-based version of the algebraic method, we computed the point-symmetry pseudogroups $G$, $G_{\rm L}$ and $G_{\rm dN}$ of the equation (1) and the systems (2) and (3), respectively, as well as the contact-symmetry pseudogroup $G_{\rm c}$ of the equation (1). It turned out that the necessary algebraic condition, which is the base of the method, completely defines the pseudogroup $G$, and therefore there is no need to use the direct method for completing the computation. This is the first example of this kind in the literature. In addition, we proved that the pseudogroup $G$ contains exactly three independent discrete elements, and the pseudogroup $G_{\rm c}$ is the first prolongation of $G$. The computation of the pseudogroup $G_{\rm c}$ is the first example of applying the megaideal-based version of the algebraic method to finding the contact-symmetry pseudogroup of a differential equation.
We described all the third-order partial differential equations in three independent variables that are invariant with respect to the algebra $\mathfrak g$. We also find a set of geometric properties of the equation (1) that singles out it from the entire class of third-order partial differential equations with three independent variables.
In Chapter 2, the Lie reductions of the equation (1) are exhaustively studied and the wide families of its invariant solutions are constructed. We presented for the first time a precise and formalized description of the complete optimized Lie reduction procedure in the case of a system of partial differential equations with three independent variables, which is relevant to the equation (1).
Using the results of Chapter 1, we classified one- and two-dimensional subalgebras of the algebra $\mathfrak g$ and one-dimensional subalgebras of the algebra $\mathfrak g_{\rm L}$ up to the $G$- and $G_{\rm L}$-equivalences, respectively. Instead of the standard approach, which is based on finding and using inner automorphisms of Lie algebras, we considered the action of the pseudogroup $G$ on the algebra $\mathfrak g$, which was found by pushing forward vector fields from $\mathfrak g$ by elements of the pseudogroup $G$.
We computed for the first time the point symmetry groups of reduced equations, including their discrete point symmetries, and it was checked in all the cases whether these symmetries are hidden or induced. Since most of the obtained reduced equations for the equation (1) are quite cumbersome, various versions of the algebraic method are much more efficient in the course of the above computation than the direct method. In addition, some of these reduced equations are not of maximal rank. Therefore, the mentioned analysis of reduced equations is, in particular, the first explicit and systematic study of Lie and general point symmetries of differential equations that are not of maximal rank.
As a result, we constructed wide families of new invariant solutions of the equation (1) in explicit form in terms of elementary, Lambert and hypergeometric functions as well as in parametric or implicit form. Since any function of the form $u=w(t,x)+\tilde w(t,y)$, which corresponds to the additive separation of the variables $x$ and $y$, is a solution of the equation (1), this separation of variables is trivial for (1). Therefore, to look for non-Lie solutions of the equation (1) that generalize some of its invariant solutions, we used the multiplicative separation of the variables $x$ and $y$, the ansatz for which has the form $u=\varphi (t,x)\psi(t,y)$ with $\varphi_x\ne0$ and $\psi_y\ne0$. The obtained results show that more closed-form solutions of (1) can be constructed using other tools of symmetry analysis of differential equations.
