Theory of Laminated Turbulence: Open Questions

Theory of Laminated Turbulence: Open Questions Elena Kartashova RISC, J. Kepler University, Altenbergerstr. 69, 4040 Linz, Austria e-mails: [email protected] 1 Introduction The roots of the theory of nonlinear dispersive waves date back to hydrodynamics of the 19th century. It was observed, both experimentally and theoretically, that, under certain circum- stances, the dissipative effects in nonlinear waves become less important then the dispersive ones. In this way, balance between nonlinearity and dispersion gives rise to formation of stable patterns (solitons, cnoidal waves, etc.). Driven by applications in plasma physics, these phe- nomena were widely studied, both analytically and numerically, starting from the middle of the 20th century. The main mathematical break-through in the theory of nonlinear evolutionary PDEs was the discovery of the phenomenon of their integrability that became the starting point of the modern theory of integrable systems with Korteweg-de Vries equation being the first instance in which integrability appeared. But most evolutionary PDEs are not integrable, of course. As a powerful tool for numerical simulations, the method of kinetic equation has been developed in 1960-th and applied to many different types of dispersive evolutionary PDEs. The wave kinetic equation is approximately equivalent to the initial nonlinear PDE: it is an averaged equation imposed on a certain set of correlation functions and it is in fact one limiting case of the quantum Bose-Einstein equation while the Boltzman kinetic equation is its other limit. Some statistical assumptions have been used in order to obtain kinetic equations; the limit of their applicability then is a very complicated problem which should be solved separately for each specific equation. The role of the nonlinear dispersive PDEs in the theoretical physics is so important that the notion of dispersion is used for ”physical” classification of the equations in partial variables. arXiv:math-ph/0607067v2 17 Nov 2006 On the other hand, the only mentioning of the notion ”dispersion relation” in mathematical literature we have found in the book of V.I.Arnold [1] who writes about important physical principles and concepts such as energy, variational principle, the Lagrangian theory, dispersion relations, the Hamiltonian formalism, etc. which gave a rise for the development of large areas in mathematics (theory of Fourier series and integrals, functional analysis, algebraic geometry and many others). But he also could not find place for it in the consequent mathematical presentation of the theory of PDEs and the words ”dispersion relation” appear only in the introduction. In our paper we present wave turbulence theory as a base of the ”physical” classification of PDEs, trying to avoid as much as possible specific physical jargon and give a ”pure” mathemati- cian a possibility to follow its general ideas and results. We show that the main mathematical object of the wave turbulence theory is an algebraic system of equations called resonant manifolds. We also present here the model of the laminated wave turbulence that includes classical 1 statistical results on the turbulence as well as the results on the discrete wave systems. It is shown that discrete characteristics of the wave systems can be described in terms of integer points on the rational manifolds which is the main novelty of the theory of laminated turbulence. Some applications of this theory for explanation of important physical effects are given. A few open mathematical and numerical problems are formulated at the end. Our purpose is attract pure mathematicians to work on this subject. 2 General Notions For the complicity of presentation we began this section with a very brief sketch of the tradi- tional mathematical approach to the classification of PDEs. 2.1 Mathematical Classification Well-known mathematical classification of PDEs is based on the form of equations and can be briefly presented as follows. For a bivariate PDE of the second order aψxx + bψxy + cψyy = F (x,y,ψ,ψx, ψy) its characteristic equation is written as dx b 1 = √b2 4ac dy 2a ± 2a − and three types of PDEs are defined: b2 < 4ac, elliptic PDE: ψ + ψ =0 • xx yy b2 > 4ac, hyperbolic PDE: ψ ψ xψ =0 • xx − yy − x b2 =4ac, parabolic PDE: ψ 2xyψ ψ =0 • xx − y − Each type of PDE demands then special type of initial/boundary conditions for the problem to be well-posed. ”Bad” example of Tricomi equation yψxx + ψyy = 0 shows immediately incompleteness of this classification even for second order PDEs because a PDE can change its type depending, for instance, on the initial conditions. This classification can be generalized to PDEs of more variables but not to PDEs of higher order. 2.2 Physical Classification Physical classification of PDEs is based on the form of solution and is almost not known to pure mathematicians. In this case, a PDE is regarded in the very general form, without any restrictions on the number of variables or the order of equation. On the other hand, the necessary preliminary step in this classification is the division of all the variables into two groups - time- and space-like variables. This division originated from the special relativity theory where time and three-dimensional space are treated together as a single four-dimensional Minkowski space. In Minkowski space a metrics allowing to compute an interval s along a curve between two events is defined analogously to distance in Euclidean space: ds2 = dx2 + dy2 + dz2 c2dt2 − 2 where c is speed of light, x, y, z and t denote respectively space and time variables. Notice that though in mathematical classification all variables are treated equally, obviously its results can be used in any applications only after similar division of variables have been done. Suppose now that linear PDE with constant coefficients has a wave-like solution ψ(x, t)= A exp i[kx ωt] or ψ(x, t)= A sin(kx ωt) − − with amplitude A, wave-number k and wave frequency ω. Then the substitution of ∂t = iω, ∂ = ik transforms LPDE into a polynomial on ω and k, for instance: − x ψ + αψ + βψ =0 ω(k)= αk βk3, t x xxx ⇒ − ψ + α2φ =0 ω2(k)= α2k4, tt xxxx ⇒ ψ α2ψ + β2ψ =0 ω4(k)= α2k2 + β2 tttt − xx ⇒ where α and β are constants. Definition Real-valued function ω = ω(k) : d2ω/dk2 = 0 is called dispersion relation or dispersion function. A linear PDE with wave-like solutions6 are called evolutionary dispersive LPDE. A nonlinear PDE with dispersive linear part are called evolutionary dispersive NPDE. This way all PDEs are divided into two classes - dispersive and non-dispersive [2]. This classification is not complementary to a standard mathematical one. For instance, though hyperbolic PDEs normally do not have dispersive wave solutions, the hyperbolic equation ψtt 2 2 − α ψxx + β ψ = 0 has them. Given dispersion relation allows to re-construct corresponding linear PDE. All definitions above could be easily reformulated for a case of more space variables, namely x1, x2, ..., xn. Linear part of the initial PDE takes then form ∂ ∂ ∂ P ( , , ..., ) ∂t ∂x1 ∂xn and correspondingly dispersion relation can be computed from P ( iω, ik , ..., ik )=0 − 1 n with the polynomial P . In this case we will have not a wave number k but a wave vector ~k =(k1, ..., kn) and the condition of non-zero second derivative of the dispersion function takes a matrix form: ∂2ω =0. |∂ki∂kj | 6 2.3 Perturbation technique Perturbation or asymptotic methods (see, for instance, [3]) are much in use in physics and are dealing with equations having some small parameter ε> 0. To understand the results presen- tated in the next Section one needs to have some clear idea about the perturbation technique and this is the reason why we give here a simple algebraic example of its application. The main idea of a perturbation method is very straightforward - an unknown solution, depending on ε, 3 is written out in a form of infinite series on different powers of ε and coefficients in front of any power of ε are computed consequently. Let us take an algebraic equation x2 (3 2ε)x +2+ ε =0 (1) − − and try to find its asymptotic solutions. If ε =0 we get x2 3x +2=0 (2) − with roots x = 1 and x = 2. Eq.(1) is called perturbed and Eq.(2) - unperturbed. Natural suggestion is that the solutions of perturbed equation differ only a little bit from the solutions of unperturbed one. Let us look for solutions of Eq.(1) in the form 2 x = x0 + εx1 + ε x2 + ... where x0 is a solution of Eq.(2), i.e. x0 = 1 or x0 = 2. Substituting this infinite series into Eq.(1), collecting all the terms with the same degree of ε and consequent equaling to zero all coefficients in front of different powers of ε leads to an algebraic system of equations 0 2 ε : x0 3x0 +2=0, 1 − ε :2x0x1 3x1 2x0 +1=0, 2 − 2 − (3) ε :2x0x2 + x1 3x2 2x1 =0, − − ... with solutions x =1, x = 1, x =3, ... and x =1 ε +3ε2 + ...; 0 1 − 2 − x =2, x =3, x = 3, ... and x =2+3ε 3ε2 + ... 0 1 2 − − Notice that exact solutions of Eq.(1) are 1 x = [3 + 2ε √1+8ε +4ε2] 2 ± and the use of binomial representation for the expression under the square root 1 −1 1 ( ) (1+8ε +4ε2) 2 =1+(8ε +4ε2)+ 2 2 (8ε +4ε2)2 + ... =1+4ε 6ε2 + ... 2! − gives finally 1 x = (3+2ε 1 4ε +6ε2 + ...)=1 ε +3ε2 + ... 2 − − − and 1 x = (3+2ε +1+4ε 6ε2 + ...)=2+3ε 3ε2 + ... 2 − − as before. This example was chosen because the exact solution in this case is known and can be compared to the asymptotic one.

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