Short thesis description.

The thesis is an application of the Lax pair and the Riemann-Hilbert problem (RHP) techniques to an old non linear boundary partial differential equation (PDE) problem originated in Einstein's theory of General Relativity.

The Lax Pair technique is roughly the following. Consider a pair of the linear differential equations

y_x=Py, y_y=Qy

where the y=y(x,y) is unknown matrix valued function, P=P(x,y) and Q=Q(x,y) are known regular matrix valued functions. If a regular solution y=y(x,y) exists the we must have

y_xy=y_yx

or

(Py)_y=(Qy)_x.

We compute both left and right hand sides and obtain

P_y+PQ=Q_x+QP            (*)

which is a non-linear PDE if P and Q depend on some unknown function of interest. For example, we may have

P=P(x,y,u(x,y)) and Q=Q(x,y,u(x,y))

where the function u(x,y) is the unknown function of interest and the functions P(...) and Q(...) have known functional form.  Hence, we may regard a non-linear PDE as a solvability condition for some linear system of differential equations.

How this helps? Suppose we introduce a complex spectral parameter into the matrices P and Q: P=P(l,x,y,u(x,y)), Q=Q(l,x,y,u(x,y)). Then the solution y also depends on the l. For particular functions P(...), Q(...) one regularly can prove that the first non trivial term of asymptotic of the solution  y depends on the u(x,y). Hence, if we construct a general solution of the linear PDE system 

y_x=P(l,x,y,u(x,y))y, y_y=Q(l,x,y,u(x,y))y          (**)

then we recover a class of solutions of the non-linear PDE.

It is an elementary fact from complex analysis that if a function of a complex argument has known singularities and it is otherwise analytic then the information about the singularities determines the function. The task of recovering of such a function is called the Riemann-Hilbert problem. 

The solution y(l,x,y) exists on the l-Riemann surface of the functions P(l,...)y, Q(l,...). The spectral analysis of the (**) reveals necessary singularities of the solution. A researcher may be able to add freedom into the problem by discovering a way to introduce more singularities.  This way the problem (**) gives rise to a RHP on some restriction of the Riemann surface of the functions P(l,...)y, y_y=Q(l,...). The construction of a solution of such RHP is a developed area. The technique is called the Backer-Akhiezer function.

If this all has been performed then there are still two remaining questions. First, the solution is specified in terms of the RHP parameters while the original PDE comes from physics. One has to identify the physical meaning of the parameters. Second, the general procedure outlined above provides no statements about boundary conditions. One has to find a particular structure of the RHP which produces the solution u(x,y) with needed boundary conditions.

In the presented thesis all the steps described above have been performed for the following physical problem. Two black holes are rotating around the common axis of symmetry. The situation is stationary. The goal is to find a relativistic correction to the Newton gravity law that is due to the rotation of the bodies.

The axial and time symmetry of the problem allows reduction of the four dimensional space time to a two dimensional non linear PDE

(E+E)(E_zz+r^-1E_r+E_rr)=2(E_r²+E_z²),

E=v(r,z) + i F(r,z),

with some singular boundary conditions on the axis of symmetry between the bodies and conditions of decay at the (r,z)-infinity.