218 - Österreichische Mathematische Gesellschaft

hold while in KAM theory only (24) is considered. This yields additional open
questions first formulated in [10]; they will not be discussed in the present paper.
In the next sections we present briefly the only known method – q-class decomposition
– allowing to compute solutions of resonance conditions (23) for a big class
of dispersion functions arising in real life problems.
2 q-class decomposition
The method of q-class decomposition is presented in [10] in 2010 in many details
and generalizations while its theoretical background and some examples have
been first given about twenty years earlier in [7, 8].
We will illustrate the method with one example of resonance conditions which
arises in geophysics. The system of equations
⎧
⎪⎨ (m 2 1 + n2 1 )1/4 + (m 2 2 + n2 2 )1/4 = (m 2 3 + n2 3 )1/4 + (m 2 4 + n2 4 )1/4 ,
m 1 + m 2 = m 3 + m 4 , m j ,n j ∈ N, ∀ j = 1,2,3,4,
(25)
⎪⎩
n 1 + n 2 = n 3 + n 4 ,
describes 4-wave resonance conditions for surface water waves. Here the unknowns
m j ,n j are indexes of Fourier modes A j (T )exp[i(m j x+n j y)−ω(m j ,n j )t],
x,y are space variables, T = t/ε is a time variable, ω(m j ,n j ) = (m 2 j +n2 j )1/4 is the
corresponding dispersion function and A j (T ) is the time-dependent mode amplitude.
Accordingly, the dynamical system (in a somewhat simplified Hamiltonian
form)
{
iȦ 1 = ZA ∗ 2 A 3A 4 , iȦ 2 = ZA ∗ 1 A 3A 4 ,
iȦ 3 = Z ∗ A ∗ 4 A 1A 2 , iȦ 4 = Z ∗ A ∗ 3 A (26)
1A 2 ,
describes the time evolution of resonant modes. The most important fact is that
the coefficients Z = Z(m 1 ,n 1 ,...,m 4 ,n 4 ) are explicit functions of the solutions of
(25). In other words, in order, say, to predict the appearance of freak waves in
the ocean, we must first solve the system of Diophantine equations (25) and than
proceed to solve the system of nonlinear differential equations (26).
In the absence of any methods for simplifying and/or solving analytically the system
(25), our only hope is the trivial but very general problem-solving technique
of brute-force or exhaustive numerical search. In the physically relevant computational
domain |m|,|n| ≤ D = 1000, this direct approach forces us to perform extensive
computations with integers of the order of 10 12 (computational complexity
D 8 ). Such computations for (25) in the substantially smaller domain |m|,|n| ≤ 128
were performed in 2005 at Warwick University by S. Nazarenko’s group. This
took 3 days using a Pentium 4. The reason why this sort of problem is extremely
22