Dispersive nonlinear partial differential equations

Size: px

Start display at page:

Download "Dispersive nonlinear partial differential equations"

Archibald Dixon
5 years ago
Views:

1 Dispersive nonlinear partial differential equations Elena Kartashova Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

2 Mathematical Classification of PDEs based on the FORM of EQUATIONS Method of Characteristics: a 2 ψ x 2 + b 2 ψ x y + ψ ψ c 2 = F(x, y,ψ, y 2 x, ψ y ) dx dy = b 2a ± 1 b 2a 2 4ac Three Types of PDEs: b 2 < 4ac, elliptic PDE: ψ xx + ψ yy = 0 b 2 > 4ac, hyperbolic PDE: ψ xx ψ yy xψ x = 0 b 2 = 4ac, parabolic PDE: ψ xx 2xyψ y ψ = 0 Bad example - Tricomi equation: yψ xx + ψ yy = 0 Each type of PDE demands special type of initial/boundary conditions for the problem to be well-posed. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

3 Physical Classification of PDEs, based on the FORM of SOLUTIONS Zero Step: Two Types of Variables Time variable t and space variable x or x = (x 1,..., x n ) PDE is then said to be of (1+1)-order or (1+n)-order. First Step: Linear PDE, constant coefficients, arbitrary order Suppose that this LPDE has a wave-like solution ψ(x, t) = A exp i(kx ωt) with amplitude A, wave-number k and wave frequency ω. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

4 Physical Interpretation ω = 2π T is wave frequency and k = 2π λ is wave number, where T is wave period and λ is wave length. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

5 Second Step: computation of frequency (I) Let us regard a linear PDE ψ tt + α 2 ψ xxxx = 0 and make some preliminary calculations: ψ t = ψ = ω( i)a exp i(kx ωt), t ψ tt = t ( t ψ) = (ω( i))2 A exp i(kx ωt) = ω 2 A exp i(kx ωt), ψ x = ψ = kia exp i(kx ωt), x..., ψ xxxx = 4 x 4ψ = (ki)4 A exp i(kx ωt) = k 4 A exp i(kx ωt). Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

6 Second step: computation of frequency (II) After substituting these results into initial PDE one gets: 0 = ψ tt + α 2 ψ xxxx = = ω 2 A exp i(kx ωt) + α 2 k 4 A exp i(kx ωt), which yields equation for frequency ω(k) = ±αk 2. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

7 Substitution of t = iω, x = ik into LPDE transforms it into a POLYNOMIAL on ω and k. Examples ψ t + αψ x + βψ xxx = 0 ω(k) = αk βk 3 ψ tt + α 2 φ xxxx = 0 ω 2 (k) = α 2 k 4 ψ tttt α 2 ψ xx + β 2 ψ = 0 ω 4 (k) = α 2 k 2 + β 2 Definitions Real-valued function ω = ω(k) : d 2 ω/dk 2 0 is called dispersion function A linear PDE with wave-like solutions are called evolutionary dispersive LPDE A nonlinear PDE with dispersive linear part are called evolutionary dispersive NPDE. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

8 Famous evolutionary dispersive NPDEs Boussinesq Equation: ψ tt + (ψ xx + ψ 2 ) xx = 0 Korteweg-de Vries Equation: ψ t + ψ xxx 6ψψ x = 0 Kadomtsev-Petviashlili Equation: (ψ t + 6ψψ x + ψ xxx ) x + 3ψ yy = 0 Schrödinger Equation: iψ t + ψ xx + f( ψ )ψ = 0 Zakharov s System of Equations: iψ t + ψ xx ψϕ = 0, ϕ tt ϕ xx ψ 2 xx = 0 many others Great discovery of 20th century - exact soliton solutions Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

9 Small nonlinearity: L(ψ) = εn(ψ) Linear Waves A 1 sin [ k 1 x ω( k 1 )t], A 2 sin [ k 2 x ω( k 2 )t] are solutions of a LPDE A 1 sin [ k 1 x ω( k 1 )t] + A 2 sin [ k 2 x ω( k 2 )t] is also a solution. Important: Amplitudes A 1, A 2 are constants. Resonance conditions (depend only on geometry): ω( k 1 ) + ω( k 2 ) = ω( k 3 ), k1 + k 2 = k 3 Nonlinear Waves - a Suggestion A 1 sin [ k 1 x ω( k 1 )t], A 2 sin [ k 2 x ω( k 2 )t] are solutions of a NPDE A 1 sin [ k 1 x ω( k 1 )t] + A 2 sin [ k 2 x ω( k 2 )t] is also a solution. Important: Amplitudes A 1 = A 1 (T), A 2 = A 2 (T) are not constants any more! Resonance conditions: as above Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

10 Wave turbulence theory Main object to study: L(ψ) = εn(ψ) with wave linear part The problem is reduced to studying resonance conditions ω( k 1 ) ± ω( k 2 ) ± ω( k 3 ) = 0, k1 ± k 2 ± k 3 = 0 (1) dynamical system for (real-valued) wave amplitudes Ȧ 1 = α 1 A 2 A 3, Ȧ 2 = α 2 A 1 A 3, Ȧ 3 = α 3 A 1 A 2 (2) where α i depend on solutions (1). Energy E i A 2 i. Two types of wave systems: Infinite domain: (1) to be solved in real numbers (Kolmogorov 1941, Zakharov 1961, Hasselman, Phillips, etc. till 1981); Finite domain: (1) to be solved in integers (Kartashova 1998). Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

11 Form of dispersion function: examples ω 2 = k 3 capillary waves (water) ω 2 = k gravity waves (water) ω = m/(m 2 + n 2 + 1) drift waves in plasma ω = 1/k ocean planetary waves... with k = m 2 + n 2 and m, n Z. Form of resonance conditions: examples 4-wave interactions of gravity waves (k i = m 2 i + n 2 i ): k 1/4 1 + k 1/4 2 = k 1/4 3 + k 1/4 4, m 1 + m 2 = m 3 + m 4, n 1 + n 2 = n 3 + n 4 3-wave interactions of ocean planetary waves: 1/(m n2 1 )1/2 + 1/(m n2 3 )1/2 = 1/(m n2 3 )1/2, m 1 ± m 2 = m 3 Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

12 Main example - ocean planetary waves: ω = 1/ m 2 + n 2 The norm of any vector (m, n) : m, n N can always be represented as (m, n) = m 2 + n 2 = γ q, γ, q N (3) with square-free q. Number q is called index of a vector (m, n). The set of all vectors with the same index q is called class of index q and is notated as Cl q. Lemma 1. If three vectors (m i, n i ), i = 1, 2, 3 give a solution of 1/ m m 21 + n21 + 1/ 22 + n22 = 1/ m3 2 + n2 3 (4) then they belong to the same class: q N : (m i, n i ) Cl q, i = 1, 2, 3. Statement of this Lemma is equivalent to irrationality of the square root of a product of different primes. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

13 Properties of classes Lemma 2. The following properties of classes keep true 1) Cl q1 Cl q2 {0} iff Cl q1 = Cl q2 (intersection of two classes is not empty iff these classes coincide); 2) card {Cl q } = (there exists an infinite number of classes); 3) Cl q card Cl q = (every non-empty class consists of infinite number of elements); 4) Cl q q = p 1 p 2 p n where p i P are different primes such that p i 3 ( mod 4); 5) Cl q & q > 1 q = a 2 + b 2 for some integers a, b Z (in each non-empty class, the minimal element has norm q); 6) Cl 1 = {m, n : m 2 + n 2 = k 2, m = a 2 b 2, n = 2ab, k = a 2 + b 2, a > b} (elements of class Cl 1 can be parameterized by two natural parameters.) Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

14 Complete solution Let us omit index in (4), rewrite it as 1 γ 1 = 1 γ γ 3, γ i N, i = 1, 2, 3 (5) Now γ 2 γ 3 = γ 1 (γ 2 + γ 3 ) and introduce new variables x = γ 2 /γ 1, y = γ 3 /γ 1, then xy = x + y. Setting y = tx with some t Q one gets tx 2 = x + tx tx = 1 + t, x = 1 + t, y = 1 + t. t The rational number t has form t = r/s with some integers r, s, i.e γ 2 = r + s, γ 1 r γ 3 = r + s γ 1 s (r + s) (r + s) γ 2 = γ 1, γ 3 = γ 1 r s and we can get the complete set of solutions for (5). Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

15 Indeed, let r, s N, (r, s) = 1, then general form of solution is γ 1 = wrs, γ 2 = ws(r + s), γ 3 = wr(r + s) for some w N. (6) The general form of solution does not depend on the class index and is the same for arbitrary q. Now let us fix some non-empty class Cl q and consider all representations of numbers γ1 2q, γ2 2 q, γ2 3q as the sums of two squares (they exist due to Lagrange theorem), then m1,j 2 + n2 1,j = γ1 2q = (wrs)2 q m2,i 2 + n2 2,i = γ2 2q = [ws(r + s)]2 q m3,l 2 + n2 3,l = γ3 2q = [wr(r + s)]2 q The indices i, j, l show that in each case there may exist several possibilities of partitioning. (7) Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

16 Structure of the solution set From mathematical point of view the problem is completely solved. From physical point of view the crucial question now is to study the structure of the solution set because it gives complete information about energy transfer along the wave spectrum: Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

17 Corresponding dynamical systems Each triangle corresponds to the system Ȧ 1 = α 1 A 2 A 3, Ȧ 2 = α 2 A 1 A 3, Ȧ 3 = α 3 A 1 A 2, each butterfly corresponds to two triangle systems Ȧ 1 = α 1 A 2 A 3, Ȧ 2 = α 2 A 1 A 3, Ȧ 3 = α 3 A 1 A 2, Ȧ 4 = α 4 A 5 A 6, Ȧ 5 = α 5 A 4 A 6, Ȧ 6 = α 6 A 4 A 5, and if, say, A 3 = A 4, that is we have Ȧ 1 = α 1 A 2 A 3, Ȧ 2 = α 2 A 1 A 3, Ȧ 3 = 1 2 (α 3A 1 A 2 + α 4 A 5 A 6 ), Ȧ 5 = α 5 A 3 A 6, Ȧ 6 = α 6 A 3 A 5, Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

18 Example - atmospheric planetary waves: N 1 Ȧ 1 = Z(N 2 N 3 )A 2 A 3, N 2 Ȧ 2 = Z(N 3 N 1 )A 1 A 3, N 3 Ȧ 3 = Z(N 1 N 2 )A 1 A 2 1th conservation law Multiplying 1st equation on A 1 we get N 1 Ȧ 1 = Z(N 2 N 3 )A 2 A 3 A 1 N 1 Ȧ 1 = Z(N 2 N 3 )A 1 A 2 A 3 and analogously 1 2 N 1 A 2 1 = Z(N 2 N 3 )A 1 A 2 A 3, 1 2 N 2 A 2 2 = Z(N 3 N 1 )A 1 A 2 A 3, 1 2 N 3 A 2 3 = Z(N 1 N 2 )A 1 A 2 A 3 N 1 A N 2 A N 3 A 2 3 = 0 N 1A N 2A N 3A 2 3 = const 1. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

19 2nd conservation law Multiplying 1st equation on N 1 A 1 we get N 1 Ȧ 1 = Z(N 2 N 3 )A 2 A 3 A 1 N 2 1Ȧ1 = ZN 1 (N 2 N 3 )A 1 A 2 A 3 and analogously 1 2 N2 1 A 2 1 = ZN 1(N 2 N 3 )A 1 A 2 A 3, 1 2 N2 2 A 2 2 = ZN 2(N 3 N 1 )A 1 A 2 A 3, 1 2 N2 3 A 2 3 = ZN 3(N 1 N 2 )A 1 A 2 A 3 N 2 1 A N2 2 A N2 3 A 2 3 = 0 N2 1 A2 1 + N2 2 A2 2 + N2 3 A2 3 = const 2. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

20 General solution in Jacobian elliptic functions: where A 1 = b 1 cn(t/t 0 λ), A 2 = b 2 dn(t/t 0 λ), A 3 = b 3 sn(t/t 0 λ), b 2 1 = A N 3(N 2 N 3 )/N 1 (N 2 N 1 )A 2 30, b 2 2 = A N 3(N 3 N 1 )/N 2 (N 2 N 1 )A 2 30, b 2 3 = A N 1(N 2 N 1 )/N 3 (N 2 N 3 )A 2 10, t 0 = 1 Z [N 2 N 3 N 1 ( N 3 N 1 N 2 A N 2 N 1 N 3 A 2 20 )]1/2, λ is defined by initial conditions and A i0 are initial values of amplitudes. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

21 Figure: Upper panel: amplitudes (left) and energies (right) of some resonant triad, with µ = 0.3. Lower panel: the same with µ = Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

22 Geometry versus topology (4) (4) (4) (12) Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

23 Uniqueness(?) of graph presentation A 5 A 5 A 1 A 2 A 1 A 2 A 4 A 3 A 6 A 4 A 3 A 6 Two objects above are isomorphic as graphs but not isomorphic as dynamical systems. The first graph represents 4 connected triads with dynamical system (A 1, A 2, A 3 ), (A 1, A 2, A 5 ), (A 1, A 3, A 4 ), (A 2, A 3, A 6 ) while the second - 3 connected triads with dynamical system (A 1, A 2, A 5 ), (A 1, A 3, A 4 ), (A 2, A 3, A 6 ). To discern between the two cases we set a placeholder inside triangle not representing a solution. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

24 What we need to solve problem completely find integer solutions of resonance conditions (done in MATH.), Warwick University group, direct computing, gravity water waves ω = (m 2 + n 2 ) 1/4 (2005): in the domain m, n 128 with Pentium-4 took 3 days. Applying of classes gives 10 minutes in the domain m, n 1000 with Pentium-3. construct unique graph presentation for dynamical systems (in process, with G. Mayrhofer), Hypergraph with weighted edges. write out explicitly coefficients of dynamical systems (done in MATH.). Important to know Just a few simple dynamical systems can be solved analytically. Many of them have to be solved numerically. Results of symbolical computations can be regarded as a preprocessing step for numerical simulations with constructed dynamical systems. Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

25 References (1994) Phys. Rev. Let. 72, pp general physical theory (1998) Amer. Math. Soc., Series Adv. Math. Sci. 2, pp general mathematical theory (2006) J. Low Temp. Phys. 145 (1-4), pp computation method (2006) Int. J. Mod. Phys. C 17 (11), pp (with A. Kartashov) numerical implementations, one class (2007) Commun. in Comput. Phys. 2 (2), pp (with A. Kartashov) numerical implementations, two classes (2007) Two papers submitted to Phys. Rev. Let. (one with V. L vov) physical examples: atmospheric planetary waves and gravity water waves Elena Kartashova (RISC) Dispersive nonlinear PDEs / 25

Generalized bilinear differential equations

Generalized bilinear differential equations Wen-Xiu Ma Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Abstract We introduce a kind of bilinear differential