The dispersionless Veselov–Novikov equation: symmetries, exact solutions, and conservation laws

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.

1 Introduction

We consider the dispersionless Veselov-Novikov equation (dVN) [11] written in the form

$$\begin{aligned} u_{txy} = (u_{xx}u_{xy})_x+(u_{xy}u_{yy})_y. \end{aligned}$$

(1)

This equation describes the propagation of the high frequency electromagnetic waves in certain nonlinear media, see [12] and references therein. Nontrivial t-independent solutions of dVN are related to the problem of existence of the first integrals for the geodesic or magnetic geodesic flows on a two-torus, [23]. Equation (1) is the dispersionless reduction of the Nizhnik–Veselov–Novikov equation [17, 24]. The Lax representation [21]

$$\begin{aligned} \left\{ \begin{array}{lcl} q_t &{}=&{} \displaystyle {\frac{1}{3}\,\left( q_x^3-\frac{u_{xy}^3}{q_x^3}\right) +u_{xx}\,q_x-\frac{u_{xy}\,u_{yy}}{q_x}}, \\ q_y &{}=&{} \displaystyle {-\frac{u_{xy}}{q_x}}. \end{array} \right. \end{aligned}$$

(2)

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.

In the present paper we study exact solutions and conservation laws of dVN. We find the contact symmetry algebra and the explicit form for the transformations from the contact symmetry pseudogroup of dVN. Then we employ the pseudogroup to find the optimal system of one-dimensional subalgebras of the symmetry algebra. 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. Then 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 some cases when the obtained odes are integrable by quadratures, thus providing exact solutions for dVN. Further, we study solutions that are not invariant with respect to contact symmetries. In particular, we find a class of solutions to dVN that are polynomials in x and y of arbitrary degree. Finally we find the whole set of conservation laws that are associated to cosymmetries defined on the second order jets.

2 Preliminaries

The presentation in this section closely follows [13, 15, 25]. Let $\pi :{\mathbb {R}}^n \times {\mathbb {R}} \rightarrow {\mathbb {R}}^n$, $\pi :(x^1, \dots , x^n, u) \mapsto (x^1, \dots , x^n)$, be a trivial bundle, and $J^\infty (\pi )$ be the bundle of its jets of infinite order. The local coordinates on $J^\infty (\pi )$ are $(x^i,u,u_I)$, where $I=(i_1, \dots , i_n)$ are multi-indices, and for every local section $f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n \times {\mathbb {R}}$ of $\pi$ the corresponding infinite jet $j_\infty (f)$ is a section $j_\infty (f) :{\mathbb {R}}^n \rightarrow J^\infty (\pi )$ such that $u_I(j_\infty (f)) =\displaystyle {\frac{\partial ^{\#I} f}{\partial x^I}} =\displaystyle {\frac{\partial ^{i_1+\dots +i_n} f}{(\partial x^1)^{i_1}\dots (\partial x^n)^{i_n}}}$. We put $u = u_{(0,\dots ,0)}$. Also, we will simplify notation in the following way, e.g., in the case of $n=3$: we denote $x^1 = t$, $x^2= x$, $x^3= y$, and $u_{(i,j,k)}=u_{{t \dots t}{x \dots x}{y \dots y}}$ with i times t, j times x, and k times y.

The vector fields

$$\begin{aligned} D_{x^k} = \frac{\partial }{\partial x^k} + \sum \limits _{\# I \ge 0} u_{I+1_{k}}\,\frac{\partial }{\partial u_I}, \qquad k \in \{1,\dots ,n\}, \end{aligned}$$

$(i_1,\dots , i_k,\dots , i_n)+1_k = (i_1,\dots , i_k+1,\dots , i_n)$, are called total derivatives. They commute everywhere on $J^\infty (\pi )$ and are annihilated by the ideal of contact forms $\langle du_I - \sum _{k=1}^{n} u_{I+1_{k}}\,dx^k \,\,\vert \,\, \#I \ge 0 \rangle$.

The evolutionary vector field associated to an arbitrary smooth function $\varphi :J^\infty (\pi )$ $\rightarrow$ ${\mathbb {R}}$ is defined as

$$\begin{aligned} {\mathbf {E}}_{\varphi } = \sum \limits _{\# I \ge 0}D_I(\varphi )\,\frac{\partial }{\partial u_I} \end{aligned}$$

with $D_I=D_{(i_1,\dots \,i_n)} =D^{i_1}_{x^1} \circ \dots \circ D^{i_n}_{x^n}$.

A pde $F(x^i,u_I) = 0$ of order $s \ge 1$ with $\# I \le s$ defines the submanifold $\mathcal {E}=\{(x^i,u_I)\in J^\infty (\pi )\,\,\vert \,\,D_K(F(x^i,u_I))=0,\,\,\# K\ge 0\}$ in $J^\infty (\pi )$.

A function $\varphi :J^\infty (\pi ) \rightarrow {\mathbb {R}}$ is called a (generator of an infinitesimal) symmetry of equation $\mathcal {E}$ when ${\mathbf {E}}_{\varphi }(F) = 0$ on $\mathcal {E}$. The symmetry $\varphi$ is a solution to the defining system

$$\begin{aligned} \ell _{\mathcal {E}}(\varphi ) = 0, \end{aligned}$$

(3)

where $\ell _{\mathcal {E}} = \ell _F \vert _{\mathcal {E}}$ with the differential operator

$$\begin{aligned} \ell _F = \sum \limits _{\# I \ge 0} \frac{\partial F}{\partial u_I}\,D_I. \end{aligned}$$

The symmetry algebra ${\mathrm {Sym}} (\mathcal {E})$ of equation $\mathcal {E}$ is the linear space of solutions to (3) endowed with the structure of a Lie algebra over ${\mathbb {R}}$ by the Jacobi bracket $\{\varphi ,\psi \} = {\mathbf {E}}_{\varphi }(\psi ) - {\mathbf {E}}_{\psi }(\varphi )$. The algebra of contact symmetries ${\mathrm {Sym}}_0 (\mathcal {E})$ is the Lie subalgebra of ${\mathrm {Sym}} (\mathcal {E})$ defined as ${\mathrm {Sym}} (\mathcal {E}) \cap C^\infty (J^1(\pi ))$. The point symmetries are the contact symmetries whose generators are polynomials in $u_{x^i}$ of degree 1.

The contact symmetry pseudogroup of a pde $\mathcal {E} \subset J^\infty (\pi )$ is the collection of all the local diffeomorphisms $\varGamma _\infty :J^\infty (\pi ) \rightarrow J^\infty (\pi )$ that preserve the submanifold $\mathcal {E}$ as well as the ideal of contact forms.

Let $\phi$ be a symmetry of $\mathcal {E}$, then the $\phi$–invariant solution of $\mathcal {E}$ is the solution of the compatible system $F= 0$, $\phi =0$.

For a pde $\mathcal {E}$ in three independent variables t, x, y a conservation law, [18, § 4.3], [25, Ch. 5], is a horizontal two-form

$$\begin{aligned} \varOmega = P_1 \,dx \wedge dy + P_2 \,dy \wedge dt + P_3 \, dt \wedge dx, \end{aligned}$$

closed with respect to the horizontal differential $d_h$, which means that

$$\begin{aligned} d_h\varOmega = \left( D_t (P_1) + D_x (P_2) + D_y (P_3)\right) \,dt \wedge dx \wedge dy = 0 \end{aligned}$$

on $\mathcal {E}$. Functions $P_i$ are smooth functions on $\mathcal {E}$. The conservation law is referred to as a trivial conservation law when there exists a horizontal one-form $\omega$ such that $\varOmega = d_h\omega$. Nontrivial conservation laws are associated with cosymmetries of equation $\mathcal {E}$, see discussion in [14, Ch. 1]. Let ${\tilde{P}}_i$ be arbitrary extensions of $P_i$ on $J^{\infty }(\pi )$, then for ${\tilde{\varOmega }} = {\tilde{P}}_1 \,dx \wedge dy + {\tilde{P}}_2 \,dy \wedge dt + {\tilde{P}}_3 \, dt \wedge dx$ there holds $d_h{\tilde{\varOmega }} = \psi \cdot F\,dt \wedge dx \wedge dy$ for some function $\psi$ on $J^\infty (\pi )$. The restriction $\psi \vert _{\mathcal {E}}$ depends on $\varOmega$ only and is called the generating function or characteristic of the conservation law $\varOmega$. The conservation law is trivial if and only if its generating function vanishes. Generating functions are solutions to equation

$$\begin{aligned} \ell _{\mathcal {E}}^{*}(\psi ) = 0 \end{aligned}$$

(4)

with $\ell _{\mathcal {E}}^{*} = \ell _{F}^{*}\vert _{\mathcal {E}}$, where the adjoint operator to $\ell _F$ is

$$\begin{aligned} \ell _F^{*} = \sum \limits _{\# I \ge 0} (-1)^{\vert I\vert }\,D_I\circ \frac{\partial F}{\partial u_I}. \end{aligned}$$

A solution of (4) is referred to as a cosymmerty of equation $\mathcal {E}$.

3 Symmetries of dVN

3.1 The symmetry algebra

Direct computations¹ show that the contact symmetry algebra ${\mathrm {Sym}}_0({\mathrm {dVN}})$ is generated by functions

$$\begin{aligned} \phi _0(A)= & {} -A\,u_t-\textstyle {\frac{1}{3}}\,A^{\prime }\,(x\,u_x+y\,u_y) -\textstyle {\frac{1}{18}}\,A^{\prime \prime }\,(x^3+y^3), \\ \phi _{1,1}(A)= & {} -A\,u_x-\textstyle {\frac{1}{2}}\,A^{\prime }\,x^2, \\ \phi _{2,1}(A)= & {} -A\,u_y-\textstyle {\frac{1}{2}}\,A^{\prime }\,y^2, \\ \phi _{1,2}(A)= & {} -A\,x, \\ \phi _{2,2}(A)= & {} -A\,y, \\ \phi _{3}(A)= & {} A, \\ \psi= & {} 3\,u-x\,u_x-y\,u_y, \end{aligned}$$

where $A= A(t)$ and $B=B(t)$ below are arbitrary smooth functions of t. Actually, all the contact symmetries of (1) turn out to be point symmetries. The structure of the Lie algebra ${\mathrm {Sym}}_0({\mathrm {dVN}})$ is defined by the commutator table

$$\begin{aligned} \{\phi _0(A), \phi _0(B)\}= & {} \phi _0(A\,B^{\prime } - B\,A^{\prime }), \\ \{\phi _0(A)\, \phi _{i,j}(B)\}= & {} \phi _{i,j}\left( A\,B^{\prime } - \left( 1- \textstyle {\frac{2}{3}}\,j\right) \, B\,A^{\prime }\right) ,\qquad i, j \in \{1, 2\}, \\ \{\phi _0(A), \phi _3(B)\}= & {} \phi _3(A\,B^{\prime }), \\ \{\phi _{i,1}(A), \phi _{j,1}(B)\}= & {} \delta _{ij}\,\phi _{i,2}(A\,B^{\prime } - B\,A^{\prime }), \\ \{\phi _{i,1}(A), \phi _{j,2}(B) \}= & {} \delta _{ij}\,\phi _{3}(A\,B^{\prime }), \\ \{\phi _{i,1}(A), \phi _3(B)\}= & {} 0, \\ \{\phi _{i,2}(A), \phi _{j,2}(B)\}= & {} 0, \\ \{\phi _{i,2}(A), \phi _{3}(B)\}= & {} 0, \\ \{\phi _3(A), \phi _3(B)\}= & {} 0, \\ \{\psi , \phi _0(A)\}= & {} 0, \\ \{\psi , \phi _{i,j}(A)\}= & {} -j\,\phi _{i,j}(A), \\ \{\psi , \phi _3(A)\}= & {} -3\,\phi _3(A). \end{aligned}$$

Direct computations show that the contact symmetry pseudogroup of equation (1) is generated by the infinite prolongations of the (local) diffeomorphisms $\varGamma _0 :(t,x,y,u) \mapsto ({\tilde{t}},{\tilde{x}},{\tilde{y}},{\tilde{u}})$ of the form

$$\begin{aligned} \left\{ \begin{array}{lcl} {\tilde{t}}&{}=&{}B_0, \qquad {\tilde{x}} =\varepsilon \,\left( B_0^{\prime }\,x +B_1\right) , \qquad {\tilde{y}} =\varepsilon \,\left( B_0^{\prime }\,y +B_2\right) , \\ {} {\tilde{u}} &{}=&{} \displaystyle { \varepsilon ^3\,\left( u-\frac{B_0^{\prime \prime }}{18 B_0^{\prime }}\,(x^3+y^3) -\frac{1}{2(B_0^{\prime })^{1/3}}\,(B_1^{\prime }\,x^2+B_2^{\prime }\,y^2) \right. } \\ &{}&{} \displaystyle { \left. +B_3\,x+B_4\,y+B_5\right) }, \end{array} \right. \end{aligned}$$

(5)

where $\varepsilon \ne 0$, $B_i=B_i(t)$ are arbitrary functions, $B_i^{\prime } =\displaystyle {\frac{dB_i}{dt}}$, and $B_0^{\prime }(t) \ne 0$. In other words, substitution for (5) into dVN written in the tilded variables yields (1).

3.2 The optimal system of one-dimensional subalgebras

Since the symmetry algebra of dVN is infinite-dimensional and depends on 6 arbitrary functions of one variable, the problem of examining all invariant solutions is very complicated. To overcome the difficulty, we use the following observation: transformations from the symmetry pseudogroup (5) preserve equation (1), while changing the symmetry generators. Therefore we can classify the orbits of the action of (5) on the ${\mathrm {Sym}}_0({\mathrm {dVN}})$. In order to use symmetries for computing invariant solutions we consider symmetries whose generators depend explicitly on at least one of the variables u, $u_t$, $u_x$, or $u_y$.

Proposition 1

Each symmetry

$$\begin{aligned} \varPhi = \phi _0(A_0)+\phi _{1,1}(A_{1,1})+\phi _{2,1}(A_{2,1})+\phi _{2,1}(A_{2,1})+\phi _{2,2}(A_{2,2}) +\phi _3(A_3) + \mu \,\psi \end{aligned}$$

from ${\mathrm {Sym}}_0({\mathrm {dVN}})$ with $A_0^2 +\mu ^2 +A_{1,1}^2+A_{2,1}^2 \not \equiv 0$ is equivalent with respect to the action of the pseudogroup (5) to one of symmetries

$$\begin{aligned}&\chi _1=\phi _0(1)+\mu \,\psi = -u_t+\mu \,(3\,u-x\,u_x-y\,u_y), \\&\chi _2 = \psi = 3\,u-x\,u_x-y\,u_y, \\&\chi _3 = \phi _{1,1}(1)+\phi _{2,1}(A) = - u_x -A\,u_y-\textstyle {\frac{1}{2}}\,A^{\prime }\,y^2, \\&\chi _4 = \phi _{2,1}(1)+\phi _{1,2}(A) = -u_y-A\,x. \end{aligned}$$

Proof

Let $A_0(t) \ne 0$. Put $\varepsilon =1$ and consider solutions $B_0(t)$, ... , $B_5(t)$ to the system of odes

$$\begin{aligned} \left\{ \begin{array}{lcl} B_0^{\prime } &{}=&{} A_0^{-1}, \\ B_1^{\prime } &{}=&{} \mu \,A_0^{-1}\,B_1-A_0^{-4/3}\,A_{1,1}, \\ B_2^{\prime } &{}=&{} \mu \,A_0^{-1}\,B_2-A_0^{-4/3}\,A_{2,1}, \\ B_3^{\prime } &{}=&{} \frac{1}{3}\,A_0^{-1}\,(6\,\mu -A_0^{\prime })\,B_3+\mu \,A_0^{-5/3}\,A_{1,1}\,B_1-A_0^{-2}\,(A_0\,A_{1,2}+A_{1,1}^2), \\ B_4^{\prime } &{}=&{} \frac{1}{3}\,A_0^{-1}\,(6\,\mu -A_0^{\prime })\,B_4 +\mu \,A_0^{-5/3}\,A_{2,1}\,B_2-A_0^{-2}\,(A_0\,A_{2,2}+A_{2,1}^2), \\ B_5^{\prime } &{}=&{} A_0^{-1}\,(3\,\mu \,B_5-A_{1,1}\,B_3-A_{2,1}\,B_4-A_3). \end{array} \right. \end{aligned}$$

Direct computations show that for ${\tilde{\chi }}_1 =-{\tilde{u}}_{{\tilde{t}}}+\mu \,(3\,{\tilde{u}}-{\tilde{x}}\,{\tilde{u}}_{{\tilde{x}}} -{\tilde{y}}\,{\tilde{u}}_{{\tilde{y}}})$ there holds $\varGamma _1^{*}({\tilde{\chi }}_1) =\varPhi$, where $\varGamma _1$ is the first prolongation of (5). Therefore the inverse diffeomorphism maps $\varPhi$ to ${\tilde{\chi }}_1$.

When $A_0(t) \equiv 0$ and $\mu \ne 0$, put $\varepsilon = \mu ^{1/3}$ and define (5) by $B_0=t$, $B_1 = \mu ^{-1}\,A_{1,1}$, $B_2 = \mu ^{-1}\,A_{2,1}$, $B_3 = \frac{1}{2}\,\mu ^{-2}\,(\mu \,A_{1,2}-A_{1,1}\,A_{1,1}^{\prime })$, $B_4 = \frac{1}{2}\,\mu ^{-2}\,(\mu \,A_{2,2}-A_{2,1}\,A_{2,1}^{\prime })$, $B_5 = \frac{1}{6}\,\mu ^{-3}\,(2\,\mu ^2\,A_3+\mu \,(A_{1,1}\,A_{1,2}+A_{2,1}\,A_{2,2}) -A_{1,1}^2\,A_{1,1}^{\prime } -A_{2,1}^2\,A_{2,1}^{\prime })$. Then we have $\varGamma _1^{*}({\tilde{\chi }}_2) =\varPhi \vert _{A_0=0}$.

Suppose now that $A_0(t) \equiv 0$, $\mu =0$, $A_{1,1}(t) \ne 0$, and $A_{2,1}(t) \ne 0$. Put $\varepsilon =1$ and define functions $B_i$, $A({\tilde{t}})$ by equations $B_0^{\prime } = A_{1,1}^{-3}$, $B_1^{\prime } = A_{1,2}A_{1,1}^{-2}$, $B_2^{\prime } = (A_{2,2}+(A_{1,1}A_{2,1}^{\prime }-A_{2,1}A_{1,1}^{\prime })\,B_2)\,A_{1,1}^{-1}A_{2,1}^{-1}$, $B_3 = \frac{1}{2}\,(A_{1,1}A_{2,1}^{\prime }-A_{2,1}A_{1,1}^{\prime })\,B_2^2-A_3\,A_{1,1}^{-1}$, $B_4 = 0$, and $A(B_0(t))= A_{2,1}(t)/A_{1,1}(t)$. Then $\varGamma _1^{*}({\tilde{\chi }}_3) =\varPhi \vert _{A_0=0,\mu =0}$.

Finally, when $A_0(t) \equiv 0$, $\mu =0$, $A_{1,1}(t) \equiv 0$, and $A_{2,1}(t) \ne 0$, put $\varepsilon =1$ and define functions $B_i$, $A({\tilde{t}})$ by equations $B_0^{\prime } = A_{2,1}^{-3}$, $B_1 \equiv 0$, $B_2 = A_{2,2}A_{2,1}^{-2}$, $B_3 \equiv 0$, $B_4 = -A_{3}\,A_{2,1}^{-1}$, and $A(B_0(t))= A_{1,2}(t)\,A_{2,1}(t)$. Then $\varGamma _1^{*}({\tilde{\chi }}_4) =\varPhi \vert _{A_0=0,\mu =0, A_{1,1} =0}$. $\square$

4 Invariant solutions

In this section we analyze reductions of dVN with respect to the symmetries $\chi _1$, ... , $\chi _4$.

4.1 Reduction w.r.t. $\chi _1$

The $\chi _1$–invariant solutions of dVN satisfy equation (1) and

$$\begin{aligned} \chi _1=-u_t+\mu \,(3\,u-x\,u_x-y\,u_y) =0, \end{aligned}$$

so they are of the form

$$\begin{aligned} u=\mathrm {e}^{3 \mu t} \,{\hat{U}}(s,w), \qquad s = x\,\mathrm {e}^{-\mu t}, \qquad w = y\,\mathrm {e}^{-\mu t}. \end{aligned}$$

(6)

We introduce function

$$\begin{aligned} {\hat{U}} = U-\frac{\mu }{6}\,(s^3+w^3) \end{aligned}$$

(7)

for convenience of the further computations. Substitution for (6), (7) into (1) shows that U(z, w) is a solution to equation $\mathcal {E}_\mu$ defined by

$$\begin{aligned} U_{www} = -U_{sss} -\frac{U_{ss}U_{ssw}+U_{ww}U_{sww}}{U_{sw}}+3\,\mu . \end{aligned}$$

(8)

This equation admits a Lax representation. Indeed, symmetry $\chi _1$ has the lift $(\chi _1,{\hat{\chi }}_1)$ with ${\hat{\chi }}_1= -q_t+\mu \,\left( \frac{3}{2}\,q-x\,q_x-y\,q_y\right)$ to the Lax representation (2). Solutions to equation ${\hat{\chi }}_1=0$ are of the form $q=\mathrm {e}^{\frac{3}{2}\,\mu \,t}Q(s,w)$. Substituting this into (2) yields the Lax representation

$$\begin{aligned} \left\{ \begin{array}{lcl} Q_s^6 &{}=&{}\displaystyle { -3\,U_{ss}\,Q_s^4+\frac{9}{2}\,\mu \,Q\,Q_s^3+3\,U_{sw}\,U_{ww}\,Q_s^2+U_{sw}^3}, \\ Q_w &{}=&{} \displaystyle {-\frac{U_{sw}}{Q_s}} \end{array} \right. \end{aligned}$$

for equation (8). We put $Q_s=S$ and obtain another Lax representation for (8):

$$\begin{aligned} \left\{ \begin{array}{lcl} S_s &{}=&{} \displaystyle { -\frac{S}{2}\,\frac{ (3\,\mu +2\,U_{sss})\,S^4 -2\,(U_{ww}U_{ssw}+U_{sw}U_{sww})\,S^2 -2\,U_{sw}^2U_{ssw} }{S^6+U_{ss}\,S^4+U_{sw}U_{ww}\,S^2+U_{sw}^3} }, \\ S_w &{}=&{} \displaystyle {-\frac{S}{2}\,\frac{ 2\,U_{ssw} S^4+(2\,U_{ss} U_{ssw}+U_{sw}\,(2\,U_{sss}+3\,\mu )\, S^2-2\,U_{sw}U_{sww} }{S^6+U_{ss}\,S^4+U_{sw}U_{ww}\,S^2+U_{sw}^3}}. \end{array} \right. \end{aligned}$$

Now we analyze invariant solutions to equation (8). We consider cases $\mu \ne 0$ and $\mu =0$ separately.

4.1.1 Case $\mu \ne 0$.

When $\mu \ne 0$, the symmetry algebra ${\mathrm {Sym}}_0(\mathcal {E}_\mu )$ of equation (8) is generated by functions $\eta _1 = 3\,U-s\,U_s-w\,U_w$, $\eta _2 = -U_s$, $\eta _3 = -U_w$, $\eta _4 = s$, $\eta _5 = w$, $\eta _6 =1$. The structure of this Lie algebra is defined by the commutators in Table 1.

Table 1 Commutator table of the Lie algebra ${\mathrm {Sym}}_0(\mathcal {E}_\mu )$

The adjoint representation of the Lie group $G_\mu$ associated with the finite-dimensional Lie algebra ${\mathrm {Sym}}_0(\mathcal {E}_\mu )$ is defined by the Lie series

$$\begin{aligned} {\mathrm {Ad}}_{\tau \,\eta _i} (\eta _j) = \exp (\tau \,{\mathrm {ad}}\,\eta _i) (\eta _j) = \sum \limits _{k\ge 0} \frac{\tau ^k}{k!} ({\mathrm {ad}}\,\eta _i)^k (\eta _j), \end{aligned}$$

where ${\mathrm {ad}}\,\eta _i (\eta _j) = \{\eta _i, \eta _j\}$. This representation is given by Table 2. In this table the (i, j)-th entry is the expression for ${\mathrm {Ad}}_{\tau \eta _i} (\eta _j)$. Using this table one can classify the orbits of action of the adjoint representation of the Lie group $G_\mu$ on its Lie algebra ${\mathrm {Sym}}_0(\mathcal {E}_\mu )$:

Table 2 The adjoint reperesentation of the Lie group $G_\mu$

Proposition 2

Each symmetry of equation $\mathcal {E}_\mu$ with $\mu \ne 0$ is equivalent under the action of the adjoint representation of $G_\mu$ to one of the symmetries

$$\begin{aligned}&\zeta _1= \eta _1 =3\,U-s\,U_{s}-w\,U_{w}, \\&\zeta _2 =\eta _3+\alpha \, \eta _2 +\beta \,\eta _4+\gamma \,\eta _5 = -U_w-\alpha \,U_s+\beta \,s+\gamma \,w,\\&\zeta _3= \eta _2 +\beta \,\eta _4+\gamma \,\eta _5 = -U_s+\beta \,s+\gamma \,w, \end{aligned}$$

where $\alpha , \beta , \gamma \in {\mathbb {R}}$.

Proof

is obtained by the standard computation, see, e.g., [18, § 3.3]. $\square$

Thus each invariant solution to equation (8) can be obtained by an action of appropriate superposition of transformations ${\mathrm {Ad}}_{\tau \,\eta _i} (\eta _j)$ from $\zeta _k$–invariant solutions. Below we examine such solutions.

4.1.1.1. Solutions invariant w.r.t. $\zeta _1$.

The $\zeta _1$–invariant solutions satisfy (8) and $\zeta _1 =0$, therefore they are of the form

$$\begin{aligned} U=s^3\,W(z), \qquad z = w\,s^{-1}, \end{aligned}$$

(9)

where W is a solution to the ode

$$\begin{aligned} W_{zzz}= & {} \frac{(5\,z^3-1)\,W_{zz}^2-z\,(22\,z\,W_z-18\,W+3\,\mu )\,W_{zz} }{2\,\left( (z^3-1)\,W_{zz} -(3\,z^3-1)\,W_z+3\,z^2\,W\right) } \nonumber \\&\qquad \qquad +\frac{20\,z\,W_z^2-6\,(4\,W-\mu )\,W_z}{2\,\left( (z^3-1)\,W_{zz} -(3\,z^3-1)\,W_z+3\,z^2\,W\right) }. \end{aligned}$$

(10)

We could not obtain the general solution to this ode. Instead, we find a family of particular solutions given by

$$\begin{aligned} W=\frac{1}{18\,a_1\,a_2}\,\left( (3\,\mu \,a_1 a_2-4\,a_1^3+2\,a_2^3)\,z^3+3\,\mu \,a_1 a_2-4\,a_2^3+2\,a_1^3\right) +a_2\,z^2+a_1\,z, \end{aligned}$$

where $a_1 a_2 \ne 0$. Then (6), (7), (9) give the t-independent solution

$$\begin{aligned} u= x\,y\,(a_1\,x+a_2\,y) +\frac{1}{9\,a_1a_2}\,\left( (a_1^3-2\,a_2^3)\,x^3+(a_2^3-2\,a_1^3)\,y^3\right) \end{aligned}$$

(11)

to dVN.

4.1.1.2. Solutions invariant w.r.t. $\zeta _2$.

For $\zeta _2$–invariant solutions there holds $\zeta _2=0$, or

$$\begin{aligned} U_w+\alpha \,U_s-\beta \,s -\gamma \,w=0. \end{aligned}$$

When $\alpha \ne 0$, this equation gives

$$\begin{aligned} U=V(z)+\frac{\beta }{2\,\alpha }\,s^2+\frac{\gamma }{2}\,w^2,\qquad z = s-\alpha \,w. \end{aligned}$$

(12)

Substituting for (12) into (8) yields

$$\begin{aligned} V_{zzz} = -\frac{3\,\alpha \,\mu \,V_{zz}}{2\,\alpha \,(\alpha ^3-1)\,V_{zz}-\beta +\alpha ^2\,\gamma }. \end{aligned}$$

(13)

This equation is integrable by quadratures. When $\beta = \alpha ^2\,\gamma$, $\alpha \ne -1$, the general solution to (13) is

$$\begin{aligned} V = -\frac{\mu }{4\,(\alpha ^3-1)}\,z^3+c_2\,z^2+c_1\,z+c_0, \end{aligned}$$

where $c_0$, $c_1$, $c_2$ are arbitrary constants. This function produces solution

$$\begin{aligned} u= & {} \frac{\mu }{12\,(\alpha ^3-1)}\,\left( (\alpha ^3+2)\,y^3 - (2\,\alpha ^3+1)\,x^3\right) +\frac{3\,\alpha \,\mu }{4\,(\alpha ^3-1)}\,x\,y\,(x-\alpha \,y)\nonumber \\&\qquad \qquad +\frac{1}{2}\,\mathrm {e}^{\mu \,t}\left( (\alpha \,\gamma +2\,c_2)\,x^2+(\gamma +2\,c_2\,\alpha ^2)\,y^2-4\,c_2\,\alpha \,x\,y\right) \nonumber \\&\qquad \qquad +c_1\,\mathrm {e}^{2\,\mu \,t}\,(x-\alpha \,y) +c_0\,\mathrm {e}^{3\,\mu \,t} \end{aligned}$$

(14)

of dVN.

Remark 1

If u is a solution to (1), then for arbitrary functions $b_0(t)$, $b_1(t)$, $b_2(t)$ the linear combination $u+b_0(t)+b_1(t)\,x+b_2(t)\,y$ is a solution to dVN as well. So we can put $c_0=c_1=0$ in (14) without loss of generality. In what follows we will ignore the linear in x and y terms in solutions of dVN. $\square$

When $\beta \ne \alpha ^2\,\gamma$, $\alpha \ne -1$, equation (13) has the first integral

$$\begin{aligned} 2\,\alpha \,(\alpha ^3-1)\,V_{zz} +(\alpha ^2\,\gamma -\beta )\,\ln V_{zz} =- 3\,\alpha \,\mu \,z+ c_0, \qquad c_0 \in {\mathbb {R}}. \end{aligned}$$

hence the general solution in this case is of form

$$\begin{aligned} V= \int \left( \int H(-3\,\alpha \,\mu \,z+c_0)\,dz \right) \,dz, \end{aligned}$$

(15)

where function $H(\tau )$ is defined by formula

$$\begin{aligned} 2\,\alpha \,(\alpha ^3-1)\,H(\tau ) +(\alpha ^2\,\gamma -\beta )\,\ln H(\tau ) \equiv \tau . \end{aligned}$$

(16)

This function can be expressed in terms of the Lambert W function, [6], while the expression is too complicated to be written here.

When $\alpha =-1$, $\beta \ne \gamma$, the general solution of (13) reads

$$\begin{aligned} V = c_2\ {\mathrm {exp}} \left( \frac{3\,\mu \,z}{\beta -\gamma }\right) +c_1\,z+c_0, \qquad c_i \in {\mathbb {R}}, \end{aligned}$$

to dVN of the form

$$\begin{aligned} u = c_2\,{\mathrm {exp}}\left( 3\,\mu \,\left( t+\mathrm {e}^{-\mu \,t}\,\frac{x-y}{\beta -\gamma }\right) \right) -\frac{\mu }{6}\,(x^3+y^3) +\frac{1}{2}\,\mathrm {e}^{\mu \,t} (\beta \,x^2+\gamma \,y^2), \nonumber \\ \end{aligned}$$

(17)

cf. Remark 1.

Finally, when $\alpha = 0$ we have solution

$$\begin{aligned} U =\frac{\mu }{2}\,s^3 +\frac{\gamma }{2}\,w^2 + \beta \,s\,w+c_2\,z^2+c_1\,z+c_0 \end{aligned}$$

of equation (8). This produces the following solution to dVN:

$$\begin{aligned} u = \frac{\mu }{6}\,(2\,x^3-y^3) +\mathrm {e}^{\mu \,t}\,(\beta \,x\,y+c_2\,x^2+\gamma \,y^2). \end{aligned}$$

(18)

4.1.1.3. Solutions invariant w.r.t. $\zeta _3$

The $\zeta _3$–invariant solution of equation (8) has the form

$$\begin{aligned} U=\frac{\beta }{2}\,s^2+\gamma \,s\,w+c_0+c_1\,w+c_2\,w^3+\frac{\mu }{2}\,w^3, \qquad c_i \in {\mathbb {R}}, \end{aligned}$$

the corresponding solution of dVN is obtained from (18) by renaming $x \leftrightarrow y$, $\beta \leftrightarrow \gamma$.

Remark 2

For each solution $u=f(t,x,y)$ of dVN expression $u=f(t,y,x)$ defines a solution as well. $\square$

4.1.2 Case $\mu =0$

Now consider equation $\mathcal {E}_0$ obtained by putting $\mu =0$ in (8). Notice that solutions of this equation produce t-independent solutions to dVN, and t-independent solutions are defined up to a nonzero constant multiple. The symmetry algebra ${\mathrm {Sym}}_0(\mathcal {E}_0)$ of this equation has generators $\xi _1 =-s\,U_s-w\,U_w$, $\xi _2 =-U_s$, $\xi _3 =-U_w$, $\xi _4 =s$, $\xi _5 =w$, $\xi _6 =1$, and $\xi _7 =U$, with the commutators given in Table 3.

The adjoint action of the symmetry group $G_0$ of equation $\mathcal {E}_0$ on ${\mathrm {Sym}}_0(\mathcal {E}_0)$ is defined by the Table  4. Then direct computations give the optimal system of one-dimensional subalgebras of ${\mathrm {Sym}}_0(\mathcal {E}_0)$:

Table 3 Commutator table of the Lie algebra ${\mathrm {Sym}}_0(\mathcal {E}_0)$

Table 4 The adjoint action of the Lie group $G_0$

Proposition 3

Each symmetry of equation $\mathcal {E}_0$ is equivalent under the action of the adjoint representation of $G_0$ to one of the following symmetries:

$$\begin{aligned}&\sigma _1 = \xi _1 + \alpha \,\xi _7 = -s\,U_s-w\,U_w+\alpha \,U,\\&\sigma _2 = \xi _1+\alpha \,\xi _6 = -s\,U_s-w\,U_w+\alpha ,\\&\sigma _3 =\xi _1+\xi _7 +\xi _4+\alpha \,\xi _5 = -s\,U_s-w\,U_w+U+s+\alpha \,w,\\&\sigma _4 =\xi _1+\xi _7 +\xi _5 = -s\,U_s-w\,U_w+U+w,\\&\sigma _5 =\xi _2+\alpha \,\xi _3 +\beta \,\xi _7 = -U_s-\alpha \,U_w+\beta \,U, \qquad \alpha \ne 0 \\&\sigma _6 =\xi _2+\alpha \,\xi _3 +\xi _4+ \beta \,\xi _5 = -U_s-\alpha \,U_w+s+\beta \,w, \\&\sigma _7 =\xi _2+\alpha \,\xi _3 +\xi _5 = -U_s-\alpha \,U_w+w,\\&\sigma _8 =\xi _2+\alpha \,\xi _3 = -U_s-\alpha \,U_w,\\&\sigma _9 =\xi _3+\alpha \,\xi _7 =-U_w+\alpha \,U,\\&\sigma _{10} =\xi _3+\alpha \,\xi _4+\beta \,\xi _5 = -U_w+\alpha \,s+\beta \,w. \end{aligned}$$

4.1.2.1. Solutions invariant w.r.t. $\sigma _1$.

The $\sigma _1$–invariant solutions to $\mathcal {E}_0$ have the form

$$\begin{aligned} U=s^\alpha \,W(z), \qquad z = w\,s^{-1}. \end{aligned}$$

Substituting this into (8) gives the ode

$$\begin{aligned} W_{zzz}= (\alpha -2)\,\frac{F(z,W,W_z,W_{zz})}{G(z,W,W_z,W_{zz})}, \end{aligned}$$

(19)

where

$$\begin{aligned}&F(z,W,W_z,W_{zz})=(5\,z^3-1)\,W_{zz}^2-(\alpha -1)\,z\,(11\,z\,W_z-3\,\alpha \,W)\,W_{zz} \\&\quad +(\alpha -1)^2\,(5\,z\,W_z-2\,\alpha \,W)\,W_z \end{aligned}$$

and

$$\begin{aligned} G(z,W,W_z,W_{zz})=2\,z\,(z^3-1)\,W_{zz}-(\alpha -1)\,(3\,z^3-1)\,W_z+\alpha \,(\alpha -1)\,z^2\,W. \end{aligned}$$

We did not find the general solution to this equation for any $\alpha$. For each $\alpha$ equation (19) admits solutions of the form

$$\begin{aligned} W=(z+z_0)^\alpha , \end{aligned}$$

(20)

where $z_0$ is a root of equation $z_0\,(z_0^3+1) =0$. Thus we get solution

$$\begin{aligned} u =(y+z_0\,x)^\alpha \end{aligned}$$

(21)

of equation (1). We found some other solutions of (19) for $\alpha \in \{1, 2, 3, 4\}$.

Equation (19) with $\alpha = 1$

$$\begin{aligned} W_{zzz} = -\frac{1}{2}\,\frac{5\,z^3-1}{z\,(z^3-1)}\,W_{zz}. \end{aligned}$$

is integrable by quadratures, its general solution

$$\begin{aligned} W(z)= c_0+c_1\,z+\int \left( \int z^{-1/2}\,(z^3-1)^{-2/3}\,dz\right) \,dz \end{aligned}$$

(22)

produces solution to dVN of the form

$$\begin{aligned} u = x^4\,W(y\,x^{-1}). \end{aligned}$$

(23)

Notice that integral in (22) can not be expressed in elementary functions, [3, § VIII], [22, Ch. 3, § 14].

When $\alpha =2$, equation (19) reduces to $W_{zzz}=0$, its general solution $W=c_0+c_1\,z+c_2\,z^2$, $c_i \in {\mathbb {R}}$, gives solution of dVN

$$\begin{aligned} u = c_0\,x^2 +c_1\,x\,y+c_2\,y^2. \end{aligned}$$

Remark 3

Expression $u = b_1(t) \, x^2 + c\, x \,y + b_2(t) \, y^2$ with arbitrary functions $b_1(t)$, $b_2(t)$ and constant c defines a trivial solution to dVN. We will ignore such solutions below. $\square$

For $\alpha =3$ we obtain two algebraic solutions

$$\begin{aligned} W = z^{\frac{3}{2}} \end{aligned}$$

(24)

and

$$\begin{aligned} W=(c_2^3-2\,c_1^3)\,z^3 + 9 \,c_1\,c_2\,z\,(c_2\,z+c_1) + c_1^3-2\,c_2^3. \end{aligned}$$

(25)

of (19). They give solutions

$$\begin{aligned} u = x^{\frac{3}{2}}y^{\frac{3}{2}} \end{aligned}$$

(26)

and

$$\begin{aligned} u=(c_2^3-2\,c_1^3)\,y^3 + 9 \,c_1\,c_2^2\,x\,y\,(c_2\,y+c_1\,x) + (c_1^3-2\,c_2^3)\,x^3 \end{aligned}$$

(27)

of dVN.

When $\alpha =4$, equation (19) admits the polynomial solution $W = 17\,z^4-36\,z^3 -90\, z^2 -36 \,z+17$. This corresponds to solution of dVN of the form

$$\begin{aligned} u = 17\,x^4-36\,x^3 y -90\, x^2y^2 -36 \,x\,y^3+17 \,y^4. \end{aligned}$$

(28)

4.1.2.2. Solutions invariant w.r.t. $\sigma _2$.

For $\sigma _2$–invariant solutions of $\mathcal {E}_0$ there holds $\sigma _2 =-U_w+\alpha \,U=0$, therefore these solutions have the form

$$\begin{aligned} U = \mathrm {e}^{\alpha \,w}\, V(s). \end{aligned}$$

(29)

Substituting this into $\mathcal {E}_0$ gives the reduced equation $V_s\,V_{sss} + V_{ss}^2+2\,\alpha ^3\,V\,V_s =0$. Integrating this twice, we obtain the first order ode

$$\begin{aligned} V_s^3+\alpha ^3\,(V^3+\gamma \,V+\delta ) =0 \end{aligned}$$

(30)

with arbitrary constants $\gamma$ and $\delta$. This equation is integrable by quadratures, and the general solution can be expressed in elliptic functions, [10, Ch. V, § 21]. To prove this claim, consider transformation

$$\begin{aligned} \left\{ \begin{array}{lcl} P&{}=&{} \displaystyle {\frac{\gamma ^{1/2} \,(3\,p^2+r-1)}{6\,p}},\\ R&{}=&{} \displaystyle {-\frac{\gamma ^{1/2} \,(3\,p^2-r+1)}{6\,p}}, \end{array} \right. \end{aligned}$$

that maps the curve $R^3=P^3+\gamma \,P+\delta$ onto the curve

$$\begin{aligned} r^2 = 1- 2\,\beta \,p-6\,p^2-3\,p^4 \end{aligned}$$

(31)

with $\beta = 6\,\delta \, \gamma ^{-3/2}$. We have $2\,r\,dr = -2\,(\beta + 6\,p +6\,p^3)\,dp$, hence

$$\begin{aligned} \frac{dP}{R} =\frac{1}{2p}\,\left( \frac{3\,p^2+1}{r}-1\right) \,dp. \end{aligned}$$

Then (29), (6), (7), and $\mu =0$ give solution

$$\begin{aligned} u = \mathrm {e}^{\alpha \,y}\,V(x), \end{aligned}$$

(32)

where function V is defined implicitly by equations

$$\begin{aligned}&V = \frac{\sqrt{\gamma }}{6\,p}\,\left( 3\,p^2-1+\sqrt{1- 2\,\beta \,p-6\,p^2-3\,p^4}\right) , \end{aligned}$$

(33)

$$\begin{aligned}&\int \frac{1}{2\,p}\left( 1 -\frac{3\,p^2+1}{\sqrt{1- 2\,\beta \,p-6\,p^2-3\,p^4}} \right) \,dp = \alpha ^3\,x+C. \end{aligned}$$

(34)

The integral in (34) includes Legendre’s integrals, therefore x can be expressed as an elliptic function of p.

When polynomial $V^3 +\gamma \,V +\delta$ has multiple roots, the general solution to equation (30) can be expressed in elementary functions. Indeed, the presence of the triple root implies $\gamma = \delta =0$, and then solution to (30) is $V = c\,\mathrm {e}^{-\alpha \,s}$, $c\in {\mathbb {R}}$. This corresponds to solution

$$\begin{aligned} u = \mathrm {e}^{\alpha \,(y-x)} \end{aligned}$$

(35)

of dVN. For a double root we have $V^3+\gamma \,V+\delta =(V+2\,\varepsilon )\,(V-\varepsilon )^2$ for some $\varepsilon \ne 0$. Then we obtain solution to dVN of the form (32), where function V is defined in the implicit form by equations

$$\begin{aligned}&V = \frac{\varepsilon \,(\tau ^3+2)}{t^3-1}, \end{aligned}$$

(36)

$$\begin{aligned}&\frac{1}{2}\,\ln \frac{\tau ^2+\tau +1}{\tau -1} + \sqrt{3}\, \mathrm {arctan}\,\left( \frac{\sqrt{3}}{3}\,(2\,\tau +1)\right) = - \alpha ^3\,x+c \end{aligned}$$

(37)

with $c \in {\mathbb {R}}$.

There are some other values of parameters $\gamma$ and $\delta$ when integral $\int (V^3+\gamma \,V+\delta )^{-\frac{1}{3}}\,dV$ can be expressed by elementary functions. From the results of [4] (see also [9]) we obtain three following cases when function V(x) in (32) admits elementary expressions in implicit form:

(i) for $\gamma = 0$, $\delta \ne 0$ there holds

$$\begin{aligned} \frac{3+i\,\sqrt{3}}{6}\,\left( \ln (V-(V^3+\delta )^{\frac{1}{3}}) +\mathrm {e}^{\frac{\pi i}{3}}\,\ln (V-\mathrm {e}^{\frac{2\pi i}{3}}(V^3+\delta )^{\frac{1}{3}}) \right) = \alpha ^3\,x +c, \nonumber \\ \end{aligned}$$

(38)

(ii) when $\gamma \ne 0$, $\delta = 0$, we get

$$\begin{aligned} \frac{3+i\,\sqrt{3}}{4}\,\left( \ln (V^{\frac{2}{3}}-(V^2+\gamma )^{\frac{1}{3}}) +\mathrm {e}^{\frac{\pi i}{3}}\,\ln (V^{\frac{2}{3}}-\mathrm {e}^{\frac{2\pi i}{3}}(V^2+\gamma )^{\frac{1}{3}}) \right) = \alpha ^3\,x +c,\nonumber \\ \end{aligned}$$

(39)

(iii) conditions $\gamma ^3 =-6\,\delta ^2 \ne 0$ imply

$$\begin{aligned} \ln H(V,1) + \mathrm {e}^{\frac{2 \pi i}{3}} \ln H(V,\mathrm {e}^{\frac{2 \pi i}{3}}) + \mathrm {e}^{\frac{4 \pi i}{3}} \ln H(V,\mathrm {e}^{\frac{4 \pi i}{3}}) = 6\,\alpha ^3\,x +c, \end{aligned}$$

(40)

where

$$\begin{aligned} H(\tau ,\omega ) = \frac{3\,\delta \, \tau - \gamma \,(\tau +\gamma )-\omega \,(\gamma \,\tau +3\,\delta )\,T +2\,\gamma \,\omega ^2\,T^2}{2\,\gamma }\cdot (\tau -\omega \,T) \end{aligned}$$

(41)

with $T=(\tau ^2+\gamma \,\tau +\delta )^{\frac{1}{3}}$.

4.1.2.3. Solutions invariant w.r.t. $\sigma _3$, $\sigma _4$, $\sigma _5$, and $\sigma _6$.

For symmetries $\sigma _3$, $\sigma _4$, $\sigma _5$, and $\sigma _6$ the reduced odes are of the form $W_{zzz}=0$. This gives only trivial solutions to dVN, cf. Remark 3.

4.1.2.4. Solutions invariant w.r.t. $\sigma _7$, $\sigma _8$, $\sigma _9$, and $\sigma _{10}$.

For symmetries $\sigma _7$, ..., $\sigma _{10}$ we found neither general nor particular solutions of the reduced equations. Below we write the forms of the invariant solutions of dVN that include functions W and reduced odes that define these functions.

For $\sigma _7$ we have

$$\begin{aligned} u=\alpha \,\ln x +W(y\,x^{-1}), \end{aligned}$$

(42)

where W(z) satisfies

$$\begin{aligned} W_{zzz}= -2\, \frac{(5\,z^3-1)\,W_{zz}^2+z\,(11\,z\,W_z-3\,\alpha )\,W_{zz}+(5\,z\,W_z-2\,\alpha )\,W_z}{2\,z\,(z^3-1)\,W_{zz}+(3\,z^3-1)\,W_z-\alpha \,z^2}. \end{aligned}$$

(43)

The $\sigma _8$–invariant solutions are

$$\begin{aligned} u=(x+\alpha \,y)\,\ln x+ x\,W(y\,x^{-1}), \end{aligned}$$

(44)

where for W(z) there holds

$$\begin{aligned} W_{zzz}= - \frac{(5\,z^3-1)\,W_{zz}^2-z\,(8\,z-3\,\alpha )\,W_{zz}+3\,z-2\,\alpha }{2\,z\,(z^3-1)\,W_{zz}-2\,z^3+\alpha \,z^2+1}. \end{aligned}$$

(45)

Symmetry $\sigma _9$ produces the invariant solution

$$\begin{aligned} u=y\,(\ln y+W(y\,x^{-1})) \end{aligned}$$

(46)

with the reduced ode

$$\begin{aligned} W_{zzz}= - \frac{z^2 (11 z^3-7)\, W_{zz}^2+(16\, z\, (2 z^3-1)\, W_z-3)\, W_{zz} +4\,(5 z^3-1)\,W_z }{2\,z^3\,(z^3-1)\,W_{zz}+4\,z^2\,(z^3-1)\,W_z-z}.\nonumber \\ \end{aligned}$$

(47)

Finally, for $\sigma _{10}$ we get

$$\begin{aligned} u=\mathrm {e}^{\beta \,x}\,W(y-\alpha \,x). \end{aligned}$$

(48)

with the defining equation for W(z) of the form

$$\begin{aligned}&W_{zzz}= \beta \, \frac{(5\,\alpha ^3-1)\,W_{zz}^2-\alpha \,\beta \,(11\,\alpha \,W_z-3\,\beta \,W)\,W_{zz} }{2\,\alpha \,(\alpha ^3-1)\,W_{zz}-\beta \,(3\,\alpha ^3-1)\,W_z+\alpha ^2\,\beta ^2\,W}\nonumber \\&\qquad \qquad +\beta ^3\, \frac{5\,\alpha \,W_z^2-4\,\beta \,W\,W_z}{2\,\alpha \,(\alpha ^3-1)\,W_{zz}-\beta \,(3\,\alpha ^3-1)\,W_z+\alpha ^2\,\beta ^2\,W}. \end{aligned}$$

(49)

4.2 Reduction of dVN w.r.t. $\chi _2$

The $\chi _2$–invariant solutions of dVN satisfy $\chi _2 = 3\,u-x\,u_x-y\,u_y =0$, therefore they have the form $u=x^3\,U(t,z)$, $z =y\,x^{-1}$, where U is a solution to

$$\begin{aligned}&U_{tzz} = \frac{1}{z}\, (2\,U_{tz} -2\,(z\,(z^3-1)\,U_{zz}-(3\,z^3-1)\,U_z+3\,z^2\,U)\,U_{zzz}\nonumber \\&\quad +(5\,z^3-1)\,U_{zz}^2-2\,z\,(11\,z\,U_z-9\,U)\,U_{zz} +4\,U_z\,(5\,z\,U_z-6\,U) ). \end{aligned}$$

(50)

This equation admits a Lax representation. To show this, consider the lift $(\chi _2, {\hat{\chi }}_2)$ of $\chi _2$ to the Lax representation (2) with ${\hat{\chi }}_2 = \frac{3}{2}\,q-x\,q_x-y\,q_y$. Solutions to ${\hat{\chi }}_2= 0$ have the form $q= x^{\frac{3}{2}}\,Q(t,z)$. Substituting this into (2) yields

$$\begin{aligned} \qquad \left\{ \begin{array}{rcl} Q_t&{}=&{} \displaystyle { \frac{2}{3}\,(z^3-1)\,Q_z^3-\frac{3\,(z^3-1)}{2\,z}\,Q\,Q_z^2 +\frac{2}{z}\,((z^3+1)\,U_z-3\,z^2\,U)\,Q_z} \\ &{}&{}\displaystyle { +\frac{9}{8}\,Q^3- 3\,(3\,z\,U_z-3\,U)\,Q}, \\ Q_z^2&{}=&{} \displaystyle {\frac{1}{2\,z}\,(3\,Q\,Q_z-2\,(z\,U_{zz} -2\,U_z)).} \end{array} \right. \end{aligned}$$

For $S=Q_z$ we obtain another Lax representation

$$\begin{aligned} \qquad \left\{ \begin{array}{rcl} S_t&{}=&{} \displaystyle { \frac{S^6+3\,U_{zz}\,S^4+F(z,U,U_z,U_{zz},U_{zzz})\,S^2+G(z,U,U_z,U_{zz})}{S\,(z\,S^2+2\,U_z-U_{zz})} }, \\ S_z&{}=&{} \displaystyle {\frac{S\,(S^2+2\,U_{zz}-2\,z\,U_{zzz})}{2\,(z\,S^2+2\,U_z-U_{zz})},} \end{array} \right. \end{aligned}$$

where

$$\begin{aligned}&F(z,U,U_z,U_{zz},U_{zzz})=4\,(z\,(z^3-1)\,U_{zz}-(3\,z^3-1)\,U_z+3\,z^2\,U)\,U_{zzz}\\&\quad -(7\,z^2-2)\,U_{zz}^2+2\,z\,(13\,U_z-9\,U)\,U_{zz}-16\,z\,U_z^2+12\,U\,U_z,\\&\quad G(z,U,U_z,U_{zz}) =z^3\,U_{zz}^3-6\,z\,U_z\,U_{zz}\,(z\,U_{zz}-2\,U_z)-8\,U_z^3. \end{aligned}$$

The symmetry algebra of equation (50) is generated by the family $A(t)\,U_t+A^{\prime }(t)\,U+\frac{1}{18}\,A^{\prime \prime }(t)\,(z^3+1)$ with arbitrary function A(t). This symmetry provides invariant solutions to (50) of the form $U=\frac{1}{18}\,(W(z)-A^{\prime }(t)\,(z^3+1))\,A(t)^{-1}$, where W is a solution to ode (19) with $\alpha =3$. Hence solutions (20), (24), (25) of (19) generate the following solutions of dVN:

$$\begin{aligned} u= & {} \frac{1}{A(t)}\,\left( (y+z_0\,x)^3- \frac{A^{\prime }(t)}{18}\,(x^3+y^3)\right) , \qquad z_0\,(z_0^3+1)=0, \end{aligned}$$

(51)

$$\begin{aligned} u= & {} \frac{1}{A(t)}\,\left( x^{\frac{3}{2}}y^{\frac{3}{2}}- \frac{A^{\prime }(t)}{18}\,(x^3+y^3)\right) , \end{aligned}$$

(52)

and

$$\begin{aligned} u =\frac{(c_2^3-2 \,c_1^3)\,y^3 + 9 \,c_1c_2^2 \,x\,y\,(c_2 y+c_1x) + (c_1^3-2c_2^3)\, x^3- A^{\prime }(t)(x^3+y^3)}{A(t)} \nonumber \\ \end{aligned}$$

(53)

with $c_1\,c_2 \ne 0$.

4.3 Reduction of dVN w.r.t. $\chi _3$

For $\chi _3$–invariant solutions we have $u_x+A\,u_y+\frac{1}{2}\,A^{\prime }\,y^2 =0$, thus such solutions have the form

$$\begin{aligned} u=W(t,z)-\frac{A^{\prime }(t)\,y^3}{6\,A(t)}, \qquad z = y -A(t)\,x. \end{aligned}$$

(54)

Substituting this into (1) and denoting $V= W_{zz}$, we get the nonlinear pde of first order

$$\begin{aligned} V_t + \left( 2\,(A^3-1)\,V+\frac{A^{\prime }}{A}\,z\right) \,V_z + \frac{2\,A^{\prime }}{A}\,V = 0. \end{aligned}$$

(55)

We did not find the general solution to (55), instead we obtain three families of particular solutions for this pde.

First, we note that substitution $V= - \frac{1}{2}\,A^{\prime }A^{-1}\,(A^3-1)^{-1}\,z$ reduces (55) to the ode $A^{\prime \prime }= (2\,A^3+1)\,(A^{\prime })^2\,A^{-1}\,(A^3-1)^{-1}$ for function A. Integrating this when $A \ne {\mathrm {const}}$, we get solution of dVN of the form

$$\begin{aligned} u = - \frac{1}{12}\,\frac{A^{\prime }}{A\,(A^3-1)}\,(y-A\,x)^3-\frac{1}{6}\,\frac{A^{\prime }}{A}\,y^3, \end{aligned}$$

(56)

where function A(t) is defined implicitly by equation

$$\begin{aligned} \ln \frac{(A-1)^2}{A^2+A+1} +2\,\sqrt{3}\,\mathrm {arctan}\,\left( \frac{\sqrt{3}}{3}\,(2\,A+1)\right) =c_1\,t+c_0, \end{aligned}$$

(57)

and $c_0$, $c_1\ne$ are constants.

When $A (t) \equiv \varepsilon \in {\mathbb {R}} \backslash \{1\}$, equation (55) gets the form $V_t +2\,(\varepsilon ^3-1)\,V\,V_z =0$. The (multi-valued) solutions of this equation have the form $2\,(\varepsilon ^3-1)\,t\,V + G(V) = z$, where G is an arbitrary function of one variable. They produce the family of solutions

$$\begin{aligned} u=W(t,z), \qquad z= y-\varepsilon \,x, \end{aligned}$$

(58)

where function W is defined implicitly by ode

$$\begin{aligned} 2\,(\varepsilon ^3-1)\,t\,W_{zz} + G(W_{zz}) = z. \end{aligned}$$

(59)

When $A (t) \equiv 1$, we have $W_{tzz} = 0$, and hence

$$\begin{aligned} u= H(y-x) +p_1(t) (y-x)+ p_0(t) \end{aligned}$$

(60)

with arbitrary functions H, $p_0$, $p_1$ of one variable. We can put $p_0 \equiv p_1 \equiv 0$ without loss generality in accordance with Remark 3.

4.4 Reduction of dVN w.r.t. $\chi _4$

Solutions of dVN that are invariant with respect to $\chi _4$ satisfy equation $u_y = A(t) \,x$ and therefore they have the form $u=A(t)\,x\,y+W(t,x)$. When $A \ne 0$, substituting this into (1) gives $W_{xxx} = A^{\prime }A^{-1}$, so we have solutions of dVN in the form

$$\begin{aligned} u = \frac{1}{6}\,\frac{A^{\prime }(t)}{A(t)}\,x^3+A(t)\,x\,y+p_2(t)\,x^2 \end{aligned}$$

(61)

with arbitrary function $p_2$.

Finally, for $A \equiv 0$, equation dVN is satisfied identically, so

$$\begin{aligned} u=W(t,x) \end{aligned}$$

(62)

is a solution to dVN for arbitrary function W of two variables.

5 Non-invariant solutions

Some of the solutions obtained in Sect. 4 admit natural generalizations that are not invariant with respect to symmetries from ${\mathrm {Sym}}_0({\mathrm {dVN}})$.

For example, solution (62) is a particular case of the family of solutions $u = f(t,x)+g(t,y)$ with arbitrary functions f and g. The $\sigma _2$– invariant solutions from subsection 4.1.2.2 are included is the set of separable solutions of the form $u=X_{\alpha }(x)\,Y_{-\alpha }(y)$, where $X_{\alpha }$ and $Y_{\alpha }$ are independently defined by either one of systems (33)–(34), (36)–(37), (40)–(41), or equation (38), or (39).

Nontrivial polynomial solutions (11), (14), (18), (27), (28), (51), (53), (56)–(57), (61) lead to idea to consider solutions of the form

$$\begin{aligned} u = \sum \limits _{1 \le i+j\le N} T_{ij}(t)\,x^i\,y^j. \end{aligned}$$

(63)

Below we present such solutions for $N \ge 3$.

5.1 $N=3$

Substituting (63) into (1) yields system of odes

$$\begin{aligned} \left\{ \begin{array}{lcl} T_{21}^{\prime } &{}=&{} 6\,(2\,T_{30}+T_{03})\,T_{21}+2\,T_{12}^2, \\ T_{12}^{\prime }&{}=&{} 6\,(T_{30}+2\,T_{03})\,T_{12}+2\,T_{21}^2, \\ T_{11}^{\prime }&{}=&{} 6\,(T_{30}+T_{03})\,T_{11}+4\,(T_{20} T_{21}+T_{02}T_{12}). \end{array} \right. \end{aligned}$$

(64)

for unknown functions $T_{21}$, $T_{12}$, $T_{11}$ and arbitrary functions $T_{30}$, $T_{03}$, $T_{20}$, $T_{02}$ of t.

Solutions (11), (14), (18), (27), (51), (53), (56)–(57), (61) are particular cases of solutions given by system (63)–(64) for appropriate choices of functions $T_{30}$, $T_{03}$, $T_{20}$, $T_{02}$.

Analysis of system (64) gives two families of solutions to dVN. The first one reads

$$\begin{aligned}&u = -\frac{1}{18}\,\left( \frac{T_{21}^{\prime }}{T_{21}}\,\left( y^3-2\,x^3+\frac{3}{2}\,\frac{T_{11}}{T_{21}}\,x^2\right) + \frac{T_{12}^{\prime }}{T_{12}}\,\left( x^3-2\,y^3+\frac{3}{2}\,\frac{T_{11}}{T_{21}}\,x^2\right) \right) \nonumber \\&\quad \qquad +\frac{1}{4}\,\frac{T_{11}^{\prime }}{T_{11}}\,x^2+T_{02}\,y^2 +\frac{1}{9\,T_{21}T_{12}}\,\left( (T_{21}^3-2\,T_{12}^3)\,x^3+(T_{12}^3-2\,T_{21}^3)\,y^3 \right) \nonumber \\&\quad \qquad +x\,y\,(T_{21}\,x+T_{12}\,y+T_{11}), \end{aligned}$$

(65)

where $T_{21}$, $T_{12}$, $T_{11}$, and $T_{02}$ are arbitrary functions of t such that $T_{21} \ne 0$ and $T_{12} \ne 0$. When $T_{21} \equiv T_{12} \equiv 0$, we obtain solution

$$\begin{aligned} u = T_{30} \,(x^3-y^3)+\frac{1}{6}\,\frac{T_{11}^{\prime }}{T_{11}}\,y^3+T_{20}\,x^2+ T_{11}\,x\,y +T_{02}\,y^2 \end{aligned}$$

(66)

with arbitrary functions $T_{11}$, $T_{30}$, $T_{20}$, $T_{02}$ of t such that $T_{11} \ne 0$.

5.2 $N=4$

Substituting (63) with $N=4$ into (1) and analyzing the resulting system we get four families of solutions of dVN:

$$\begin{aligned} u= & {} T_{1}\,\left( 4\,x^3\,y-3\,y^4\right) +\frac{T_1^\prime }{6\,T_1}\,y^3+T_{2}\,y^2, \quad T_{1} \ne 0, \end{aligned}$$

(67)

$$\begin{aligned} u= & {} T\,\left( 17\,x^4-36 \,x^3\,y-90\,x^2\,y^2-36\,x\,y^3+17\,y^4\right) +\frac{1}{6}\,\frac{T^\prime }{T}\,(x^3+y^3), \nonumber \\&\quad T \ne 0, \end{aligned}$$

(68)

$$\begin{aligned} u= & {} T\,\left( 17\,x^4-36 \,x^3\,y-90\,x^2\,y^2-36\,x\,y^3+17\,y^4\right) +\frac{1}{6}\,\frac{T^\prime }{T}\,(x^3+y^3), \nonumber \\&\quad T \ne 0,\quad c \in {\mathbb {R}}, \end{aligned}$$

(69)

$$\begin{aligned} u= & {} \frac{4\,T^2\,T^\prime \,T^{\prime \prime \prime } -8\,T^2\,(T^{\prime \prime })^2+3\,(T^\prime )^4}{1728\,T^2\,(T^\prime )^{16/3}}\,\left( 2\,T\,T^{\prime \prime }\,(x^2+y^2)-3\,(T^\prime )^2\,y^2\right) \nonumber \\&\qquad +(T^\prime )^{4/3}\,(x-y)^4 -\frac{1}{18}\,\frac{T^{\prime \prime }}{T^\prime }\,y\,(3\,x^2-3\,x\,y+2\,y^2) \nonumber \\&\qquad +\frac{1}{12}\,\frac{T^\prime }{T}\,y\,(3\,x^2+y^2), \qquad T \ne 0, \,\,T^{\prime } \ne 0. \end{aligned}$$

(70)

Solution (68) with $T \equiv 1$ coincides with solution (28).

5.3 $N=5$

For $N =5$ we obtain the family of solutions

$$\begin{aligned} u= & {} T_{50}\,(x-y)^5+T_{30}\,x\,(x^2+3\,y^2)+T_{03}\,y\,(y^2+3\,x^2)\nonumber \\&\qquad +\frac{1}{30}\,\frac{T_{50}^\prime }{T_{50}}\,y^2\,(y-3\,x)+T_{11}\,x\,y, \end{aligned}$$

(71)

where $T_{50}$ is an arbitrary nonzero function of t, while $T_{30}$, $T_{03}$, and $T_{11}$ satisfy the following system of odes:

$$\begin{aligned} \left\{ \begin{array}{lcl} T_{30}^\prime &{}=&{} \displaystyle { \frac{1}{150}\, \left( \frac{5\,T_{50}^{\prime \prime }}{T_{50}}-\frac{7\,(T_{50}^\prime )^2}{T_{50}^2}+\frac{30\,T_{50}^\prime }{T_{50}}\right) \,(T_{30}-2\,T_{03}) } \\ &{}&{} \displaystyle { +6\,(T_{30}+T_{03})^2}, \\ T_{03}^\prime &{}=&{}\displaystyle { \frac{(T_{50}^\prime ) ^2}{150\,T_{50}}+\frac{T_{50}^\prime }{5\,T_{50}}\,(T_{03}-2\,T_{30}) +6\,(T_{30}+T_{03})^2}, \\ T_{11}^\prime &{}=&{}\displaystyle {\left( \frac{T_{50}^\prime }{5\,T_{50}}+6\,(T_{30}+T_{03})\right) \,T_{11}}. \end{array} \right. \end{aligned}$$

(72)

5.4 $N \ge 6$

For each $N \ge 6$ we find the family of solutions

$$\begin{aligned} \begin{array}{lcl} u&{}=&{} \displaystyle { \sum \limits _{k=4}^{N} T_{k}\,(x-y)^k +T_{30}\,(x^3+3\,x^2\,y)+T_{03}\,(y^3+3\,x\,y^2) } \\ &{}&{} \displaystyle { +\frac{1}{2\,N^2}\,\frac{T_N^{\prime }}{T_N}\,y\,\left( (N-1)\,T_{N-1}\,y-N\,T_N\,x\,(x+y)\right) } \\ &{}&{} \displaystyle { -\frac{1}{2\,N}\,\frac{T_{N-1}^{\prime }}{T_N}\,y^2 +T_{20}\,(x^2+y^2)+T_{11}\,x\,y, } \end{array} \end{aligned}$$

(73)

where $T_N$, $T_{N-1}$, and $T_{20}$ are arbitrary functions of t such that $T_N \ne 0$, $T_{N-1} \ne 0$, while functions $T_k$ with $k \in \{4, \dots N-2\}$, $T_{30}$, $T_{03}$, and $T_{11}$ obey the following system of odes:

$$\begin{aligned} \left\{ \begin{array}{lcl} T_k^\prime &{}=&{} \displaystyle {\frac{1}{N^2\,T_N^2}} ((k\,N\,T_k\,T_N-(k+1)\,(N-1)\,T_{k+1}\,T_{N-1})\,T_N^{\prime } \\ &{}&{} +(k+1)\,N\,T_{k+1}T_N T_{N-1}^{\prime }), \qquad k \in \{4, \dots , N-2\}, \\ T_{30}^\prime &{}=&{} \displaystyle { \frac{1}{6\,N^2\,T_N^2}}\,( N\,T_N\,T_N^{\prime \prime }-(N-1)\,(T_N^{\prime })^2 \\ &{}&{}\displaystyle { -24\,T_4\,((N-1)\,T_{N-1}T_N^{\prime }-N\,T_N\,T_{N-1}^{\prime }) } \\ &{}&{}\displaystyle { +36\,N^2\,T_N^2\,(T_{30}+T_{03})^2- 6\,N\,(T_{30}+4\,T_{03})\,T_N\,T_N^{\prime } ) }, \\ T_{03}^\prime &{}=&{} \displaystyle { \frac{1}{6\,N^2\,T_N^2}}\,( N\,T_N\,T_N^{\prime \prime }-(N-1)\,(T_N^{\prime })^2 \\ &{}&{} -24\,T_4\,((N-1)\,T_{N-1}T_N^{\prime }-N\,T_N\,T_{N-1}^{\prime }) \\ &{}&{}\displaystyle { +36\,N^2\,T_N^2\,(T_{30}+T_{03})^2 - 6\,N\,(4\,T_{30}+T_{03})\,T_N\,T_N^{\prime } ) }, \\ T_{11}^\prime &{}=&{} \displaystyle { \frac{1}{N^3\,T_N^3}\,\left( (N\,T_N\,T_{N-1}^{\prime }-(N-1)\,T_{N-1}\,T_N^{\prime })\,T_N^{\prime } \right. } \\ &{}&{} +6\,N^3\,T_N^3\,(T_{30}+T_{03})\,(T_{11}+2\,T_{20}) -6\,N^2\,T_N^2T_{30}T_{N-1}^{\prime } \\ &{}&{}\displaystyle { \left. +6\,N\,\left( (N-1)\,T_N T_{30} -\frac{2}{3}\,N\,T_N\,T_{20}\right) \,T_NT_N^{\prime } \right) }. \\ \end{array} \right. \end{aligned}$$

(74)

6 Cosymmetries and conservation laws

Cosymmetries of equation (1) are solutions $\psi$ to equation (4), which is of the form

$$\begin{aligned} D_tD_xD_y(\psi )= & {} u_{xy}\,(D_x^3(\psi )+D_y^3(\psi ))+ u_{xx}\,D_x^2D_y(\psi )+u_{yy}D_xD_y^2(\psi )\nonumber \\&\qquad +3\,u_{xxy}\,D_x^2(\psi ) +(u_{xxx}+u_{yyy})\,D_xD_y(\psi ) +3\,u_{xyy}\,D_y^2(\psi )\nonumber \\&\qquad +2\,u_{xxxy}\,D_x(\psi ) +2\,u_{xyyy}\,D_y(\psi ). \end{aligned}$$

(75)

Direct computations show that each solution $\psi \in C^\infty (J^2(\pi ))$ of equation (75) is a linear combination of the following cosymmetries:

$$\begin{aligned} \psi _1= & {} \ln u_{xy},\\ \psi _2= & {} A,\\ \psi _3= & {} 2\,A\,u_{xx}+A^\prime \,x,\\ \psi _4= & {} 2\,A\,u_{yy}+A^\prime \,y,\\ \psi _5= & {} 4\,A\,(2\,(u_{tx}-u_{xy}\,u_{yy})+u_{xx}^2)+2\,A^\prime \,(2\,x\,u_{xx}+u_x)+A^{\prime \prime }\,x^2,\\ \psi _6= & {} 4\,A\,(2\,(u_{ty}-u_{xy}\,u_{xx})+u_{yy}^2)+2\,A^\prime \,(2\,y\,u_{yy}+u_y)+A^{\prime \prime }\,y^2,\\ \psi _7= & {} 12\,A\,( 3\,u_{tt} +6\,(u_{tx}\,u_{xx}+u_{ty}\,u_{yy}) -18\,u_{xx}\,u_{xy}\,u_{yy} -2\,u_{xy}^3 )\\&\qquad -6\,A^\prime \,( 3\,u_t +4\,(x\,(u_{tx}-u_{xy}\,u_{yy}) +y\,(u_{ty}-u_{xy}\,u_{xx}))\\&\qquad +2\,(u_{xx} \,(x\,u_{xx}+u_x) +u_{yy} \,(y\,u_{yy}+u_y))) -A^{\prime \prime \prime }\,(x^3+y^3)\\&\qquad -6\,A^{\prime \prime }\,((x\,u_x+y\,u_y+y^2\,u_{yy}+x^2\,u_{xx}). \end{aligned}$$

Here $A=A(t)$ are arbitrary (smooth) functions.

The conservation laws $\varOmega _1$, ... , $\varOmega _7$ associated to cosymmetries $\psi _1$, ... , $\psi _7$ are given by the formulas

$$\begin{aligned}&\varOmega _1 = u_{xy}\,(\ln u_{xy} -1)\,dx\wedge dy + \left( \frac{1}{2}\,u_{yy}^2-u_{xx}\,u_{xy}\,\ln u_{xy}\right) \,dy\wedge dt\\&\qquad + \left( \frac{1}{2}\,u_{xx}^2-u_{xy}\,u_{yy}\,\ln u_{xy}\right) \,dt\wedge dx,\\&\quad \varOmega _2 = 2\,A\,u_{xy}\,(dx\wedge dy -u_{xx}\,dy\wedge dt -u_{yy}\,dx\wedge dt ) -A^{\prime } (u_y\,dy-u_x\,dx) \wedge dt,\\&\quad \varOmega _3 = A\,(3\,u_{xx}\,u_{xy}\,dx\wedge dy -3\,u_{xy}\,(u_{tx}+u_{xx}^2 +u_{xy}\,u_{yy})\,dy\wedge dt\\&\quad \qquad +(3\,u_{xx}\,(u_{tx}- 2\,u_{xy}\,u_{yy})+u_{xx}^3+u_{xy}^3)\,dt\wedge dx)\\&\quad \qquad -3\,A^{\prime }\,x\,(u_{xx}\,u_{xy}\,dy +(u_{tx}-u_{xy}\,u_{yy})\,dx)\wedge dt,\\&\quad \varOmega _4 = A\,(3\,u_{yy}\,u_{xy}\,dx\wedge dy -3\,u_{xy}\,(u_{ty}+u_{yy}^2 +u_{xy}\,u_{xx})\,dx\wedge dy\\&\quad \qquad +(3\,u_{yy}\,(u_{ty}- 2\,u_{xy}\,u_{xx}) +u_{yy}^3+u_{xy}^3)\,dy\wedge dt)\\&\quad \qquad +3\,A^{\prime }\,y\,(u_{yy}\,u_{xy}\,dx +(u_{ty}-u_{xy}\,u_{xx})\,dy)\wedge dt,\\&\quad \varOmega _5 = u_{xy}\,(4\,A\,u_{xx}^2+2\,A^{\prime }\,x\,u_{xx} +A^{\prime \prime }\,x^2)\,dx \wedge dy\\&\qquad +\left( A\,(8\,u_{xy}\,u_{xx}\,(u_{yy}\,u_{xy}-u_{tx}) -\textstyle {\frac{2}{3}}\,u_{xy}\,(u_{xx}^3-u_{xy}^3))\right. \\&\qquad \left. +2\,A^{\prime }\,( x\,u_{xy}(u_{xy}u_{yy}-u_{tx}-u_{xx}^2) -u_x u_{xx}u_{xy} ) -A^{\prime \prime }\,x^2\,u_{xx}u_{xy}\right) \,dy\wedge dt\\&\qquad +\left( A\,(4\, ( (u_{tx}-u_{yy}\,u_{xy})^2-u_{xy}\,u_{yy}\,u_{xx}^2) -\textstyle {\frac{2}{3}}\,u_{xx}\,(4\,u_{xy}^3+u_{xx}^3) \right. \\&\qquad +A^{\prime }\,( 2\,u_{tx}\,(x\,u_{xx}+u_x) -2\,u_{yy}\,u_{xy}\,(2\, x\,u_{xx}+u_x) -\textstyle {\frac{2}{3}}\,x\,(u_{xy}^3+u_{xx}^3) )\\&\qquad \left. -A^{\prime \prime }\,x^2\,u_{xy}\,u_{yy} -A^{\prime \prime \prime }\,x^2\,u_x\right) \,dt \wedge dx,\\&\quad \varOmega _6 = u_{xy}\,(4\,A\,u_{yy}^2+2\,A^{\prime }\,y\,u_{yy} +A^{\prime \prime }\,y^2)\,dx \wedge dy\\&\qquad +\left( A\,(4\, ( (u_{ty}-u_{xx}\,u_{xy})^2-u_{xx}\,u_{xy}\,u_{yy}^2) -\textstyle {\frac{2}{3}}\,u_{yy}\,(4\,u_{xy}^3+u_{yy}^3) \right. \\&\qquad +A^{\prime }\,( 2\,u_{ty}\,(y\,u_{yy}+u_y) -2\,u_{xx}\,u_{xy}\,(2\, y\,u_{yy}+u_y) -\textstyle {\frac{2}{3}}\,y\,(u_{xy}^3+u_{yy}^3) )\\&\qquad \left. -A^{\prime \prime }\,y^2\,u_{xx}\,u_{xy} -A^{\prime \prime \prime }\,y^2\,u_y\right) \,dy \wedge dt\\&\qquad +\left( A\,(8\,u_{xy}\,u_{yy}\,(u_{xx}\,u_{xy}-u_{ty}) -\textstyle {\frac{2}{3}}\,u_{xy}\,(u_{yy}^3-u_{xy}^3))\right. \\&\qquad \left. +2\,A^{\prime }\,( y\,u_{xy}\,(u_{xx}\,u_{xy}-u_{ty}-u_{yy}^2) -u_y\,u_{xy}\,u_{yy} ) -A^{\prime \prime }\,y^2\,u_{xy}\,u_{yy}\right) \,dt\wedge dx,\\&\quad \varOmega _7 = \sum \limits _{k=0}^{3} \frac{d^k A}{dt^k}\,(P_k\,dx\wedge dy +Q_k\,dy\wedge dt+R_k\,dt\wedge dx), \end{aligned}$$

where

$$\begin{aligned} P_0= & {} \frac{1}{12}\,( 2\,u_{xy}\,( 6\,(u_{tx}\,u_{xx}+u_{ty}\,u_{yy})+u_{xx}^3-3\,u_{xx}\,u_{xy}\,u_{yy}+u_{yy}^3) -3\,u_{tx}\,u_{ty}),\\ P_1= & {} \frac{1}{12}\, u_{xy}\,(u_x\,u_{xx}+u_y\,u_{yy}+3\,(y\,u_{ty}+u_t)+2\,(x\,u_{xx}^2+y\,u_{yy}^2) -3\,y\,u_{tx}\,u_{yy}),\\ P_2= & {} \frac{1}{24}\,( 6\,t\,((x\,u_{xy}+y\,u_{yy})\,u_{tx} +u_{xy}\,(x^2\,u_{xx}+y^2\,u_{yy}+2\,(x\,u_x+y\,u_y)) )\\&\qquad -(x\,u_{xx}+y\,u_{xy})\,u_{ty}),\\ Q_0= & {} \frac{1}{12}\, (3\,u_{ty}^2\,u_{yy} +u_{xy}\,(6\,u_{xy}\,u_{yy}\,(6\,u_{tx}+3\,u_{xx}^2 -u_{xy}\,u_{yy}) -u_{tx}\,(u_{tx}+6\,u_{xx}^2) )\\&\qquad +(3\,u_{tt}-12\,u_{xx}\,u_{xy}\,u_{yy}-2\,u_{yy}^3 -2\,u_{xy}^3)\,u_{ty} +2\,u_{xy}\,(2\,u_{xy}^3-3\,u_{tt})\,u_{xx}),\\ Q_1= & {} \frac{1}{36}\, (u_{xy}^3\,(x\,u_{xy}-u_y) -3\,u_{xy}\,(4\,x\,u_{xx}+3\,u_x)\,u_{tx}-3\,y\,u_{ty}^2 -2\,x\,u_{xx}^3\,u_{xy}\\&\qquad -3\,(4\,y\,u_{xx}\,u_{xy}-u_y\,u_{yy})\,u_{ty} +3\,u_{xy}\,(2\,y\,u_{xy}-u_x)\,u_{xx}^2 -u_{yy}^3\,(y\,u_{yy}+u_y)\\&\qquad -3\,u_{xy}\,(3\,u_t+2\,y\,u_{yy}^2-2\,(2\,x\,u_{xy}-u_y)\,u_{yy})\,u_{xx}\\&\qquad +(3\,u_x\,u_{xy}^2+9\,y\,u_{tt}-4\,y\,u_{xy}^3)\,u_{yy}),\\ Q_2= & {} \frac{1}{72}\, ( 3\,(y^2\,u_{ty}\,u_{yy} +6\,t\,(y\,u_{ty}^2+x\,u_{tt}\,u_{xy}) -(x^2\,u_{xy}+6\,t\,x\,u_{ty})\,u_{tx})\\&\qquad -3\,(x^2\,u_{xy}u_{xx}^2 +2\,u_{xy}(x\,u_x+y\,u_y+y^2\,u_{yy})\,u_{xx} +(6\,t\,y\,u_{tt}-x^2\,u_{xy}^2)\,u_{yy})\\&\qquad -y^2\,(u_{xy}^3+u_{yy}^3)-18\,x\,u_t\,u_{xy}),\\ Q_3= & {} \frac{1}{72}\, ((18\,t\,(x\,u_x+y\,u_y-u)+y^3)\,u_{ty}-(x^3+y^3)\,u_{xx}\,u_{xx}-3\,y\,u_y^2),\\ R_0= & {} \frac{1}{12}\, (3\,(u_{tx}^2\,u_{xx}-u_{xy}\,u_{ty}^2) +(3\,u_{tt}-12\,u_{xx}\,u_{xy}\,u_{yy}-2\,(u_{xx}^3+u_{xy}^3))\,u_{tx}\\&\qquad -6\,(u_{xx}^2\,u_{xy}^3-u_{xy}\,(u_{xx}\,u_{xy}-u_{yy}^2)\,u_{ty}-3\,u_{xx}\,u_{xy}^2\,u_{yy}^2)\\&\qquad +2\,u_{xy}\,(4\,u_{xy}^4-3\,u_{tt})\,u_{yy}), \end{aligned}$$

$$\begin{aligned} R_1= & {} \frac{1}{36}\, (3\,(2\,x\,u_{tx}^2+(3\,y\,u_{ty}-4\,x\,u_{xy}\,u_{yy}+2\,u_x\,u_{xx})\,u_{tx}) -u_{xx}^3\,(x\,u_{xx}+u_x)\\&\qquad -3\,u_{xy}\,((4\,y\,u_{yy}+3\,u_y)\,u_{ty} -(2\,x\,u_{xy}-u_y)\,u_{yy}^2+3\,(y\,u_{tt}-u_t\,u_{yy}))\\&\qquad +2\,u_{xy}\,u_{yy}\,((2\,y\,u_{xy}-u_x) -3\,x\,u_{xx}^2\,u_{yy} -y\,u_{yy}^2)+u_{xy}^3\,(y\,u_{xy}-u_x)\\&\qquad +u_{xx}\,u_{xy}^2\,(3\,u_y-4\,x\,u_{xy})),\\ R_2= & {} \frac{1}{72}\, (18\,t\,(y\,u_{tt}\,u_{xy}-x\,u_{tx}^2) +(3\,x^2\,u_{xx}-18\,t\,y\,u_{ty})\,u_{tx} -x^2\,(u_{xx}^3+u_{xy}^3)\\&\qquad -3\,(y^2\,u_{xy}\,(u_{ty}+u_{yy}^2) -(6\,x\,(t\,u_{tt}+u_t)-u_{xy}\,(2\,x^2\,u_{yy}+y^2\,u_{xy}))\,u_{xx})\\&\qquad -6\,u_{xy}\,(x\,u_x+y\,u_y)\,u_{yy}),\\ R_3= & {} \frac{1}{72}\, (18\,t\,(u-x\,u_x-y\,u_y)+x^3)\,u_{tx}-(x^3+y^3)\,u_{xy}\,u_{yy}-3\,x\,u_x^2). \end{aligned}$$

7 Concluding remarks

The results of the paper can be summarized as follows. Employing the methods of the Lie symmetry group analysis we have found a number of exact solutions for the dispersionless Veselov–Novikov equation, including solutions in elementary or elliptic functions (17), (33)–(34), (36)–(37), (38), (39), (40)–(41), (52), or functions represented by quadratures (15)–(16), (22)–(23), (58)–(59). We have indicated ordinary differential equations (10), (19), (43), (45), (47), (49) that describe all other invariant solutions. We have studied some non-invariant solutions and found a broad set of polynomial solutions (65)—(74). Furthermore, we have presented all the local conservation laws of order up to two.

While odes (10), (19), (43), (45), (47), (49) are not integrable by quadratures in general, their origin as reductions of Lax-integrable pdes (8) and (50) allows one to hope that the method of prolongation structures in the version implemented in [16] could be applicable to examine these odes. Likewise, the methods of weak symmetries [19, 20], nonclassical symmetry reductions, see [5] and references therein, conditional symmetries [7, 8], or stable-range approach [26], can be useful to generate new non-invariant solutions of the dispersionless Veselov–Novikov equation. We intend to address these issues in our future work.