1. Self-similar analysis of various non-linear partial differential equations
Imre Ferenc Barna
1Wigner Research Center of the
Hungarian Academy of Sciences, Budapest
2 ELI-ALPS Nonprofit Kft Szeged

2. Outline Introduction various PDEs, basic solutions self-similar,
travelling wave , an example heat conduction
Systems I already investigated
a list of various Navier-Stokes equations, heat conduction mechanisms, other physical
systems
The Oberbeck-Bousinesq equ chaos versus solutions
Non-linear Maxwell equations plus ELI related idea
Summary & Outlook

3. The very basic of the idea
Initially: a (nonlinear) partial differential
equation (system) for u(x,t) is given
Ansatz :combination from space and time variable x,t to a new variable η(x,t) so u(η)
Result: a (nonlinear) ordinary differential
equation (system) u’ (η)
+ with some tricks analytic solutions can be
found which depend on some free physical
parameter(s), general properties are
interesting e.g. shock, compact support makes
physics

4. The basic linear partial differential equations
1 no-time dependence we do not consider in the
following 
2 there is mathematical maximum principle for the
solutions in time, dispersion, but infinite propagation
speed for the solution, heat conduction or diffusion eq.
3 no-maximum principle for the solution, finite
propagation speed, wave eq.
There are well-behaving basic solutions for 2-3
Idea: is to use these solutions for any kind of non-linear PDEs, to investigate how the wave or diffusive character are conserved, (ther is no theory for nonlin PDEs)

5. Physically important solutions of PDEs
- Travelling waves:
arbitrary wave fronts
u(x,t) ~ g(x-ct), g(x+ct)
- Self-similar in 1D
Sedov, Zeldovich, Barenblatt
in Fourier heat-conduction

6. Physically important solutions of PDEs II
- generalised self-similar:
- one case: blow-up solution:
goes to infinity in finite time
other forms are available too
- generalized travelling waves,
exponential method

7. Physically important solutions of PDEs III
- additional:
additive separable
multiplicative separable
generalised separable eg.
these are all Ansätze, there are methods too,
eg. Lie algebra, Bäcklund transformation, Painleve
analysis, and so

8. First example: ordinary diffusion/heat conduction equation
U(x,t) temperature distribution
Fourier law + conservation law
parabolic PDA, no time-reversal sym.
strong maximum principle ~ solution is smeared out in time
the fundamental solution:
general solution is:
kernel is non compact = inf. prop. Speed paradox of heat cond.
Problem from a long time 
But have self-similar solution 

9. Very first example: how the method works
General remarks:
1) No contradiction among the exponents
= exists a dissipative solution it is not always a
case
2) No extra free parameters, eg. equation of state, all the parameters are fixed, no
different , solutions/physics for
different materials

10. Systems I investigated till now
non-linear various Navier-Stokes non-linear
heat conduction equations in 2d or 3d Maxwell
in solids in 1D equations
generalised
Fourier heat
conduction laws
shock wave solu-
tions, compact
support
2 published papers
1 in progress
compressible, non-compressible, Newtonian and non-Newtonian
3 published papers
field dependent material equations,
analogy came from
Fourier heat cond.
1 published papers
Coupling the two effects, most interesting field 
2 published papers, 1 in progress
I will explain in details
1 work is in progress ELI related
I will explain in details

11. Various hydrodynamical equations
Starting point:
Ideal compressible flow,
continuity equation + Euler equation
based on conservation laws (no heat exchange)
(in 1 dimension)
how to go further?
Incompressibility or + Equation of state or + extra terms,
Bernulli Eq eg viscosity Navier-Stokes

12. Various dissipative fluid equations
Non-compressible and
Newtonian (I)
Compressible and
Newtonian (II)
Non-compressible and Non- Newtonian (III) (an example)
Non-Newtonian
tooo complicated, we will see…
Euler description, v velocity field, p pressure, a external field
viscosity, density, kompressibility, E stress tensor, n behav. ind.

13. Various dissipative fluid equations
I.F. Barna
Commun in Theor. Phys. 56, (2011)
745
I.F. Barna, L. Mátyás
Fluid Dyn. Res. 46, (2014)
055508
I.F. Barna, G.Bognár,K.Hriczó
Mathematical Modelling and
Analysis 21, (2016) 83
Non-compressible and
Newtonian (I)
Compressible and
Newtonian (II)
Non-compressible and Non- Newtonian (III)
Non-Newtonian
tooo complicated, we will see…
Critique: not much modelling, just solving the basic equations

14. Other analytic solutions
all are for non-compressible N-S
Without completeness, usually from Lie algebra studies
Presented 19 various solutions
one of them is:
Ansatz:
Solutions are Kummer functions as well 
“Only” Radial solution
for 2 or 3 D
Ansatz:
Ansatz:
Sedov, stationary N-S, only the angular part

15. Other Analytic solutions
the Russians and the NASA have probably even more…
Sometimes I find scientific reports – for flows - which were confidental 40 years ago
“The results are sometimes difficult to find
because the studies were conducted in the
last century and often published in reports of limited circulation”

16. My 3 dimensional Ansatz A more general function does not work for N-S
Geometrical meaning:
all v components with
coordinate constrain x+y+z=0
lie in a plane = equivalent
The final applied forms

17. The second example - the Oberbeck-Boussinesq pde system
two-dimensional Navier-Stokes equation coupled to heat conduction via Boussinesq approx. (the simplest way)
exhaustively investigated system, the first meteorological model
for the atmosphere
NS. for the x horiz. cor.
NS. for the z vert. cord.
heat conduction eq.
continuity eq.

18. Oberbeck-Boussinesq PDE system
investigated via stream function with Fourier expansion,
can lead to chaotic ODE systems
highly truncated, considering only 1 term
strange attractor

19. Our applied Ansatz, self.-sym. exponents
and the ODE system,
we investigate the original hydrodynamical system (use the evarage temperature)
positive exponents, mean
dissipative solutions
the remaining
ODE system can be
solved analytically
note: ½ is a simple diffusion exponent, pressure and temperature behave differently

20. The temperature field can be expressed with the
Eq. 3 separates from Eq. 1-2
can be expressed with the
help of the Error functions
For fixed t and x
T(x) has the same shape so
any inhomogenity can start the Rayleigh- Bénard convection this is the main result
Chaos solitons and fractals 78 (2015) 249

21. Rayleigh- Bénard convection
in two dimensional heated liquid convection cells are
created
usually calculated with
3D NS solver softwares
some figures are taken from the web

22. The pressure field
The pressure field is coupled to the temperature field in a simple ODE:
The used parameter set:

23. The velocity field The real part has almost a Gaussian shape
The most general formula:
The real part has almost a
Gaussian shape
The presented function:
Similar form of asymptotic normalized velocity autocorrelation functions are available from turbulence studies

24. Additional remarks
From the velocity field the, enstropy (the vorticity) can be calculated , the Fourier transform can be calculated as well to compare to turbulence studies
Self-similar has some connection to: scaling, universality and renor-
malisation , critical phenomena , (but no clear cut understanding for me…)
A generalization of the system is under investigation
power-law temperature dependent viscosity and/or heat conduction can be considered as physical examples for ocean or other liquid media

25. The third example: non-linear Maxwell
equations
The field equations:
The constitutive Eqs + differential Ohm’s Law
Our consideration, power law:
But:
the exponent is a free physical parameter:

26. Geometry & Applied Ansatz

27. The final ordinary differential equations
’ means derivation with
respect to eta
Universality relations among the parameters:
The first equ. is a total difference so can be integrated :
Remaining a simple ODE with one parameter q (material exponent):

28. Different kind of solutions
No closed-forms are available so the direction fields are presented, E(x,t) & H(x,t) are not analytic
Compact solutions
for q < -1
shock-waves
Non-compact solutions
for q > -1

29. Physically relevant solutions
We consider the Poynting vector which gives us the energy flux (in W/m^2) in the field
Unfortunatelly, there are two contraversial definition of the Poynting vector in media, unsolved problem
R.N. Pfeifer et al. Rev. Mod. Phys. 79 (2007) 1197.
published:
I.F. Barna Laser Phys. 24 (2014) &
arXiv :

30. Non-linear electrodynamics
- In 1934 Born-Infeld proposed non-linear ED to
remove point-like-particles induced singularities
in field theories
- above the Schwinger limit W/cm2 intensities
the dielctric response of vacuum itself becomes
non-linear (very long range ELI interest )
- Lagrange density for linear versus non-lin ED
after variation we get the
equation of motion for the fields
we, Polner M., Mati P.
Barna IF. investigate it

31. Summary we presented physically important,
self-similar solutions for various PDEs
as a first example we investigated Gaussian heat-
conduction
as second example a spec. hydrodinamical system was
analysed
as a third example we investigated the 1 dim.
Maxwell equations (instead fo the wave equation)
closing with non-linear (power law) constitutive
equations

32. Thank you for your attention!
Questions, Remarks, Comments?…