Abstract
In this paper we consider the following nonlinear half-wave equation:
where \(D = -i \partial _{x}\), both on \(\mathbb{R }\) and \(\mathbb{T }\). On \(\mathbb{R }\), we prove that, if the initial condition is of order \(O({\varepsilon })\) and supported on positive frequencies only, then the corresponding solution can be approximated by the solution of the Szegő equation. The Szegő equation \(i\partial _tu=\Pi _+(|u|^2u)\), where \(\Pi _+\) is the Szegő projector onto non-negative frequencies, is a completely integrable system that gives an accurate description of solutions of (NLW). The approximation holds for a long time \(0\le t\le C{\varepsilon }^{-2}\big [\log (1/{\varepsilon }^{\delta })\big ]^{1-2\alpha }\), \(0\le \alpha \le 1/2\). The proof is based on the renormalization group method. As a corollary, we give an example of a solution of (NLW) on \(\mathbb{R }\) whose high Sobolev norms grow over time, relative to the norm of the initial condition. An analogous result of approximation was proved by Gérard and Grellier (Anal PDEs, arXiv:1110.5719v1) on \(\mathbb{T }\) using Birkhoff normal forms. We improve their result by finding a second order approximation with the help of an averaging method. We show, in particular, that the effective dynamics is no longer given by the Szegő equation.
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References
Abou Salem, W.K.: On the renormalization group approach to perturbation theory for PDEs. Ann. Henri Poincaré 11(6), 1007–1021 (2010)
Bogolyubov, N.N., Mitropol’skii, YuA: Asymptotic Methods in the Theory of Nonlinear Oscillations. Hindustan, Delhi (1958)
Chen, L.-Y., Goldenfeld, N., Oono, Y.: Renormalization group theory for global asymptotic analysis. Phys. Rev. Lett. 73(10), 1311–1315 (1994)
Chen, L.-Y., Goldenfeld, N., Oono, Y.: Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 543(1), 376–394 (1996)
De Ville, R., Harkin, A., Holzer, M., Josic, K., Kaper, T.: Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations. Physica D 237, 1029–1052 (2008)
Gérard, P., Grellier, S.: The cubic Szegö equation. Annales Scientifiques de l’Ecole Normale Supérieure, Paris, \(4^e\) série, t. 43, 761–810 (2010)
Gérard, P., Grellier, S.: Invariant tori for the cubic Szegö equation. Inventiones Mathematicae 187, 707–754 (2012)
Gérard, P., Grellier, S.: Effective integrable dynamics for some nonlinear wave equation. to appear in Analysis and PDEs, arXiv:1110.5719v1
Germain, P.: Space-time resonances, arXiv:1102.1695
Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 2D quadratic Schrödinger equations, arXiv:1001.5158
Germain, P.: Global existence for coupled Klein-Gordon equations with different speeds, arXiv:1005.5238
Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. C. R. Math. Acad. Sci. Paris 347(15–16), 897–902 (2009)
Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 3D quadratic Schrödinger equations. Int. Math. Res. Not. IMRN 3, 414–432 (2009)
Gustafson, S., Nakanishi, K., Tsai, T.-P.: Scattering theory for the Gross-Pitaevskii equation in three dimensions. Commun. Contemp. Math. 11(4), 657–707 (2009)
Krieger, J., Lenzmann, E., Raphael, P.: Non dispersive solutions to the \(L^2\) critical half wave equation, arXiv:1203.2476
Lax, P.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968)
Moise, I., Temam, R.: Renormalization group method. Applications to Navier Stokes equation. Discret. Cont. Dyn. Syst. 6, 191–200 (2000)
Moise, I., Ziane, M.: Renormalization group method. Applications to partial differential equations. J. Dyn. Differ. Equ. 13, 275–321 (2001)
Shatah, J.: Space-time resonances. Quart. Appl. Math. 68(1), 161–167 (2010)
Petcu, M., Temam, R., Wirosoetisno, D.: Renormalization group method applied to the primitive equations. J. Differ. Equ. 208, 215–257 (2005)
Pocovnicu, O.: Traveling waves for the cubic Szegö equation on the real line. Anal. PDE 4(3), 379–404 (2011)
Pocovnicu, O.: Explicit formula for the solution of the Szegő equation on the real line and applications. Disc. Cont. Dyn. Sys. A 31(3), 607–649 (2011)
Temam, R., Wirosoetisno, D.: Averaging of differential equations generating oscillations and an application to control. Special issue dedicated to the memory of Jacques-Louis Lions. Appl. Math. Optim. 46(2–3), 313–330 (2002)
Ziane, M.: On a certain renormalization group method. J. Maths. Phys., 41(5), (2000)
Acknowledgments
The author would like to thank her Ph.D. advisor Prof. Patrick Gérard for suggesting this problem and for interesting discussions. She is also grateful to Tadahiro Oh for giving her the reference [1], from which she learned about the renormalization group method. Finally, she would like to thank the referee for his comments that helped improve significantly the quality of the exposition of this paper.
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Pocovnicu, O. First and Second Order Approximations for a Nonlinear Wave Equation. J Dyn Diff Equat 25, 305–333 (2013). https://doi.org/10.1007/s10884-013-9286-5
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DOI: https://doi.org/10.1007/s10884-013-9286-5