Nonlinear Fourier Analysis

Transcription

1 Nonlinear Fourier Analysis The Direct & Inverse Scattering Transforms for the Korteweg de Vries Equation Ivan Christov Code 78, Naval Research Laboratory, Stennis Space Center, MS 99, USA Supported by ONR/NRL funding. Nonlinear Fourier Analysis p./

2 Background & Introduction (I) 89: Korteweg and de Vries derived the simplest known equation that models both nonlinearity and dispersion, as the leading order approximation of the eq ns of fluid dynamics, for long waves in shallow-water. 96: Zabusky and Kruskal conducted numerical experiments of continuum limit of the Fermi-Pasta-Ulam lattice (= the KdV equation!); discovered soliton interactions. 967: Greene et al. showed the equivalence of the initial value problems for the KdV on the infinite line to the classical Schrödinger eigenvalue problem. 97: Extending the latter, Ablowitz et al. (AKNS) formulated a general method for solving nonlinear evolution equations known as the Scattering Transform. Nonlinear Fourier Analysis p./

3 Background & Introduction (II) 97-6: Matveev et al., concurrently with Flaschka and McLaughlin, used methods from algebraic geometry to develop the theory of the periodic Scattering Transform; crucial work for the application of the S.T. to data analysis. 98s 99s: Osborne extended the technique of the Scattering Transform to the analysis of oceanographic data; develops the first (and only) practical approach to the nonlinear Fourier analysis of physical data. Present day: Apply Osborne s nonlinear Fourier analysis framework to the analysis of internal solitons in the ocean, since (linear) Fourier analysis has proven ineffective for the analysis of nonlinear phenomena. Nonlinear Fourier Analysis p./

4 Fourier Analysis of Linear PDEs Consider the linearized KdV equation with periodic IC: η t + c η x + βη xxx =, η(x + L, ) = η(x, ), x L, where c = gh, β = c h /6. The exact solution to the preceding is a Fourier Series:. Discrete Fourier Transform (DFT): Find ( the spectrum {c j ;φ j }: c j = a j + b j, φ j = tan b j a j ), where a j and b j are the usual Fourier coefficients.. Inverse Discrete Fourier Transform (IDFT): Construct the solution to the linear PDE from the spectrum {c j ;φ j }: η(x,t) = a + N j= c j cos(k j x ω j t + φ j ), ω j = c k j βk j. Nonlinear Fourier Analysis p.4/

5 Integrable Nonlinear PDEs Consider the full KdV equation with periodic IC: η t + c η x + αηη x + βη xxx =, η(x + L, ) = η(x, ), x L, α = c /(h). We can find the exact solution to the preceding via the Scattering Transform:. Direct Scattering Transform (DST): Solve the Schrödinger eigenvalue problem Ψ xx + [λη(x, ) + E]Ψ =, where λ = α/(6β), to get the discrete spectrum, aka scattering data, {E j ;µ j }.. Inverse Scattering Transform (IST): Construct the nonlinear Fourier series from the spectrum {E j ;µ j }. Nonlinear Fourier Analysis p./

6 The Nonlinear Fourier Series In terms of the hyperelliptic (aka Abelian) functions: λη(x,t) = E + N j= µ j (x,t) E j E j+. All nonlinear waves and their interactions are obtained from this linear superposition. In the small amplitude limit we have µ j (x,t) cos(x ω j t + φ j ), i.e., we get the ordinary Fourier series! If there are no interactions, e.g., a single wave (N = ), we have µ(x,t) cn (x ωt + φ m), which is a Jacobian elliptic function with modulus m (a cnoidal wave). Nonlinear Fourier Analysis p.6/

7 The Θ-function Representation In terms of the N-degree-of-freedom Riemann Θ-function: η(x,t) = λ x ln Θ N(η B) = η cn + η int, where η = (η,...,η N ), η j = k j x ω j t + φ j and B is the interaction matrix. The solution is a linear superposition of cnoidal waves, and their nonlinear interactions. Nonlinear Fourier Analysis p.7/

8 More on the Θ-function Repr n The linear superposition of cnoidal waves is given by η cn = λ x ln Θ N(η D) = N j= A j cn [ K(mj ) π ] k j (x C n t) m j. But, their nonlinear interactions are still in terms of Θ-functions: η int = [ λ x ln + Θ ] N(η,B) Θ N (η D). Θ N (η D) Note that B = D + O, i.e., we ve split the interaction matrix into its diagonal and off-diagonal parts. Bad news: in general, η int O(), so we cannot ignore it! Nonlinear Fourier Analysis p.8/

9 On Moduli and Wave Numbers In the hyperelliptic representation, we can let k = πj/l as usual ( commensurable wave numbers). In the Θ-function representation the values of k j,ω j,φ j are forced by the Abelian transformation. Though they are analogous to the Fourier ones, they are physically irrelevant ( incommensurable wave numbers). Furthermore, we can compute the modulus m j of the Jacobian elliptic functions, which we call the soliton index from the discrete spectrum. Then, m j.99 solitons, i.e., cn (x m j = ) = sech (x), m j. Stokes waves (nonlinearly interacting cnoidal waves, less nonlinear than the solitons), m j. radiation, i.e., cn (x m j = ) = cos (x). Nonlinear Fourier Analysis p.9/

10 4 H h H h L m L m A Two-Soliton Initial Condition.. L. x, L x, H c x H c x (Left) η(x +, ) = η(x, ) is the N = case of the well-known infinite line N-soliton solution (i.e., Hirota s construction), taken at t =.s for convenince. (Right) The same -soliton solution but at t =.s, note the amplitudes of the two solitons are.cm and 4.cm, respectively. Nonlinear Fourier Analysis p./

11 4 Comparison of the FFT & DST (I) A j m j a j êl j p êl j p (Left) The DFT finds 6 normal modes. (Right) The DST finds exactly significant waves. N.B.: When the data, i.e., the i.c., is a solution of the KdV eq. (even ), the DST/IST offer a far better interpretation/representation of the data than do the DFT/IDFT. Nonlinear Fourier Analysis p./

12 A Sine Wave I.C. for the KdV Eq. x E E.. h H x, L x R h H x, 6. 9 L x t c h H x, t L x I.C.: η(x +, ) = η(x, ) =.8 cos(πx/) Nonlinear Fourier Analysis p./

13 4 Comparison of the FFT & DST (II). A j. m j.. a j êl j p êl j p (Left) The DFT spectrum shows 4+ normal modes. (Right) The DST finds 8 (well-defined) soliton modes (=Z&K), some mildly nonlinear waves, and radiation. N.B.: The FT is only useful when analyzing a solution. But, the DST gives all the info about an initial condition and its evolution! Nonlinear Fourier Analysis p./

14 Acknowledgments, etc. I would like to thank Dr. Chin-Bing for suggesting this research project and giving me the opportunity to work on it at NRL-SSC this summer, Dr. Jordan for the all the insightful discussions and help on this and other scientific/mathematical topics. I will begin my graduate studies in mathematics at Texas A&M University next week. For correspondence purposes, my there is christov@tamu.edu. Nonlinear Fourier Analysis p.4/

15 Selected Bibliography. Osborne, A. R. (998) Physica D.. Osborne, A. R. (99) Phys. Rev. E 48 ().. Osborne, A. R. (99) Phys. Rev. E (). 4. Osborne, A. R. & Petti, M. (994) Physics of Fluids 6 ().. Osborne, A. R. & Bergamasco, L. (986) Physica 8D. 6. Osborne, A. R. (994) Math. Comp. Sim Osborne, A. R. & Burch, T. L. (98) Science 8 (444). 8. Zabusky, N. J. & Kruskal, M. D. (96) Phys. Rev. Lett. (6). 9. Gardner, C., Greene, J., Kruskal, M. & Miura, R. (967) Phys. Rev. Lett. 9 (9).. Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. (97) Phys. Rev. Lett. ().. Korteweg, D. J. & de Vries, G. (89) Phil. Mag. 9.. Ablowitz, M. J. & Segur, H., (98) Solitons and the Inverse Scattering Transform (Philadelphia: SIAM Studies in Applied Mathematics). Nonlinear Fourier Analysis p./