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D. Levi et al.
We look for a lattice invariant under the vector field ˆ
Q 1 . Details on construction of
invariant lattices can be found in [9, 10, 14]. Introducing the distances and angles of
the discrete net
x n+1,m = x nm + h
x
nm , x n−1,m = x nm + σ
x
nm ,
and of the fields
u n+1,m =u nm + h
x
nm D x u,
u n,m+1 =u nm + σ
x
nm D x u + kD t u,
we get, after some nontrivial calculations, that h x
nm satisfies the polynomial equation
(h
x
nm )
3 (u nm − 1)D x u + (u − 1)(2x nm D x u + u − 3)(h
x
nm )
2
+
2x
2
nm − h(u nm − 1)
D x u + 2x nm (u nm − 3)(u nm − 1)
h
x
nm
− h(u nm − 1)
2
= 0.
This expression will provide h x
nm in terms of x, u and the differences of u in each
point. This a cubic equation for h x
nm and we cannot get explicit expressions for the
point distances simple enough to achieve a complete solution of the problem.
This is a problem we have not found in previous works and it is due to the fact that
the vector field is not projective, that is, the coefficients of the partial derivatives with
respect to x (or t, the independent variables) depend on the dependent variable u.
3 Case ˆ
Q 4
3.1 Symmetry Reduced Equations and Solutions
The symmetry variables corresponding to the vector field ˆ
Q 4 (6) are
v =
x(u − 1)
u
, y = t, v = v(y)
(8)
and then (2) reduces to the ODE
v y = 0,
whose solution is v = c where c is a constant. Then, from (8) a family of solutions
for (2) different from the trivial constant u = 1 is