Tamara Grava

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Nonlinear dispersive equations

  1. Highly oscillatory solutions
  2. Long time/small dispersion asymptotics
  3. Dispersive shock waves and rogue waves
  4. Universality and Painlevé equations
I study integrable nonlinear dispersive partial differential equations that are fundamental models of many physical phenomena in nonlinear physics and nonlinear waves. I study solutions in various asymptotic regimes: long time or small dispersion limits. The typical features are the formation of oscillations called dispersive shock waves that now are well understood for some integrable equations, while, beyond integrability, the description remains mostly open. At the transition between various oscillatory zone a universality result in the behaviour of solutions has been conjectured for a large class of nonlinear dispersive equations (6. 7. 11. 13. 16.) and in some cases proved (8. 10. 12. 15.). Open problems I am considering are the modulation instability and the formation of rogue waves against an oscillatory background, the behaviour of solutions of dispersive equations with step-like oscillatory initial data that produce a wide range of unexplored new physical behaviours.

Examples

Relevant publications

  1. On the long-time asymptotic behavior of the modified Korteweg–de Vries equation with step-like initial data Grava, Tamara; Minakov, Alexander SIAM J. Math. Anal. 52 (2020), no. 6, 5892–5993.
  2. Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves Grava, T.; Klein, C.; Pitton, G. Proc. A. 474 (2018), no. 2210, 20170458, 20 pp.
  3. T. Grava, Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations, Chapter in Rogue and Shock Waves in Nonlinear Dispersive Media, Lecture Notes in Physics 926, p. 309-335 2016.
  4. T. Grava, C. Klein, J. Eggers Shock formation in the dispersionless Kadomtsev-Petviashvili equation. Physica D 333, (2016) 157 - 170.
  5. Eggers, J.; Grava, T.; Herrada, M. A.; Pitton, G. Spatial structure of shock formation. J. Fluid Mech. 820 (2017), 208–231.
  6. B. Dubrovin, T.Grava,T.; Klein, On critical behaviour in generalized Kadomtsev--Petviashvili equations. Nonlinearity 29 (2016) 1384 - 1416.
  7. B. Dubrovin, T.Grava, C.Klein, A. Moro, On critical behaviour in systems of Hamiltonian partial differential equations. J. Nonlinear Sci. 25 (2015), no. 3, 631 - 707.
  8. T. Claeys, T. Grava, Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory. Random matrix theory, interacting particle systems, and integrable systems, 71 - 92, Math. Sci. Res. Inst. Publ., 65, Cambridge Univ. Press, New York, 2014.
  9. T. Grava, T.; C.Klein, A numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions. Phys. D 241 (2012), no. 23-24, 2246 - 2264.
  10. T.Claeys, T.Grava The KdV hierarchy: universality and a Painleve transcendent Int. Math. Res. Not. IMRN 2012, no. 22, 5063 - 5099.
  11. B. Dubrovin, T. Grava, C. Klein Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations SIAM J. Appl. Math. 71 (2011), no. 4, 983 - 1008.
  12. T.Claeys, T.Grava, Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit. SIAM J. Math. Analysis, vol 42, p. 2132-2154, 2010.
  13. S. Abenda, T. Grava, C. Klein, Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions. SIAM J. Appl. Math. vol 70, p 2797-2821, 2010.
  14. T. Claeys, T. Grava, Painleve II Asymptotics near the Leading Edge of the Oscillatory Zone for the Korteweg de Vries Equation in the Small-Dispersion Limit. Comm. Pure and Appl. Math. 63, (2010) p. 203- 232.
  15. T. Claeys and T. Grava, Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach. Comm. Math. Phys. 286 (2009), no. 3, 979--1009.
  16. B. Dubrovin, T. Grava, C. Klein, On universality of critical behaviour in the focusing nonlinear Schroedinger equation, elliptic umbilic catastrophe and the tritronquee solution to the Painlevé-I equation. J. Nonlinear Sci. 19 (2009), no. 1, 57--94.
  17. T. Grava, V.U.Pierce, Fei-Ran Tian, Initial value problem of the Whitham equations for the Camassa-Holm equation. Physica D, 238, (2009) 1, 55--66.
  18. T. Grava, C. Klein, Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlevé-II equation, Proc. R. Soc. A (2008) n.464, 733- 757.
  19. T. Grava, C. Klein, Numerical solution of the small dispersion limit of Korteweg - de Vries and Whitham equations, Communications on Pure and Applied Mathematics, 60, (2007), n.11, 1623-1664.
  20. Abenda, Simonetta; Grava, Tamara Modulation of Camassa-Holm equation and reciprocal transformations. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 1803–1834.
  21. Grava, Tamara Whitham equations, Bergmann kernel and Lax-Levermore minimizer. Acta Appl. Math. 82 (2004), no. 1, 1–86.
  22. Grava, Tamara Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type. Comm. Pure Appl. Math. 55 (2002), no. 4, 395–430.
  23. Grava, Tamara; Tian, Fei-Ran The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit. Comm. Pure Appl. Math. 55 (2002), no. 12, 1569–1639.
  24. T. Grava, From the solution of the Tsarev system to the solution of the Whitham equations. Math. Phys. Anal. Geom. 4 (2001), no. 1, 65–96.