Reduction and exact solutions of the eikonal equation (Q1176878)

The authors find here the wide class of the real exact solutions of the eikonal equation \[ (\partial u/\partial x_ 0)^ 2-(\partial u/\partial x_ 1)^ 2-(\partial u/\partial x_ 2)^ 2-(\partial u/\partial x_ 3)^ 2=1. \leqno (1) \] To solve the equation they find the expressions of \(u\) which reduct the equation (1) to ordinary differential equations. They employ 51 C(1,4)-non equivalent subalgebras of AC(1,4) with rank 3 to give the expressions of \(u\). Here AC(1,4) is the algebra with the bases \(P_ \alpha=\partial_ \alpha\), \(J_{\alpha\beta}=g^{\alpha\nu}x_ \nu\partial_ \beta- g^{\beta\nu}x_ \nu\partial_ \alpha\), \(D=-x^ \alpha\partial_ \alpha\), and \(K_ \alpha=-2(g^{\alpha\beta}x_ \beta)D- (g^{\beta\nu}x_ \beta x_ \nu)\partial_ \alpha (\alpha,\beta,\nu=0,1,\ldots,4)\). Their method is the following: Let \(\omega(x)\), \(\omega(x,u)\) and \(\omega'(x,u)\) be invariants of the subalgebra \(L\). They set (i) \(u=f(x)\varphi(\omega)+g(x)\), \(\omega=\omega(x)\), (ii) \(u^ 2=f(x)\varphi(\omega)+g(x)\), \(\omega=\omega(x)\), or (iii) \(\omega'(x,u)=\varphi(\omega(x,u))\), and have the ordinary differential equations of \(\varphi\) and \(L\) invariant solutions \(u=u(x)\). Finally they show the correspondence between the eikonal equation and the Hamilton-Jacobi equation \(u_ t+(\nabla u)/(2m)=0\) through the bases of AC(1,4).