The first two condition of (11) say that the origin is an
intersection point of two isolines, that of 
and 
, and the
third one says that the isoline of is tangent to the x-axis.
Function 
is just function 
moved somehow (by
) along the plane. Intersection of two isolines is often
used as a definition of the tip of the spiral wave. So, in other words,
conditions (11) say that function v is function u
considered in a frame of reference with its origin at the tip of the
spiral wave and with y axis along the gradient of 
at the tip
(see Fig. 3).
Figure 3:
Frame of reference 
related to the tip of the spiral. The
tip is the intersection of isoline 
(dashed) with isoline
(dotted). The origin of is at the tip, shifted
by vector R from (x,y)-origin, 
-axis is tangent to the
u-isoline, rotated by angle 
from x-axis.
Coordinates , 
in the tip frame are related to those x,
y of the laboratory frame by
Performing this change of variables in the original system (1),
with X, Y and varying with time, after elementary though
tedious calculations we can directly obtain Eqs. (14, 12).
To conclude, the physical interpretation of the newly obtained equations is: (11) is a definition of the spiral tip, (14) is its motion equation, and (12) is an equation for the field in the tip's frame of reference.
Now we can easily interpret the assumption made in Sec. 3 that any group orbit crosses the manifold only once. In terms of this application, this simply means that we consider only solutions with one wave tip.
