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February  2014, 34(2): 557-566. doi: 10.3934/dcds.2014.34.557

## Global existence of small-norm solutions in the reduced Ostrovsky equation

 1 Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom 2 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1

Received  November 2012 Revised  April 2013 Published  August 2013

We use a novel transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation and prove global existence of small-norm solutions in Sobolev space $H^3(\mathbb{R})$. This scenario is an alternative to finite-time wave breaking of large-norm solutions of the reduced Ostrovsky equation. We also discuss a sharp sufficient condition for the finite-time wave breaking.
Citation: Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557
##### References:
 [1] A. R. Aguirre, T. R. Araujo, J. F. Gomes and A. H. Zimerman, Type-II Bäcklund transformations via gauge transformations,, J. High Energy Phys., (2011). Google Scholar [2] J. P. Boyd, Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation,, Physics Letters A, 338 (2005), 36. doi: 10.1016/j.physleta.2005.02.017. Google Scholar [3] J. C. Brunelli and S. Sakovich, Hamiltonian structures for the Ostrovsky-Vakhnenko equation,, Comm. Nonlin. Sci. Numer. Simul., 18 (2013), 56. doi: 10.1016/j.cnsns.2012.06.018. Google Scholar [4] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Annales de l'Institute Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar [5] A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar [6] R. Iório and D. Pilod, Well-posedness for Hirota-Satsuma's equation,, Diff. Integr. Eqs., 21 (2008), 1177. Google Scholar [7] A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129. doi: 10.1088/0266-5611/19/1/307. Google Scholar [8] J. Hunter, Numerical solutions of some nonlinear dispersive wave equations,, in, 26 (1990), 301. Google Scholar [9] R. H. J. Grimshaw, K. Helfrich and E. R. Johnson, The reduced Ostrovsky equation: Integrability and breaking,, Stud. Appl. Math., 129 (2013), 414. doi: 10.1111/j.1467-9590.2012.00560.x. Google Scholar [10] G. Gui and Y. Liu, On the Cauchy problem for the Ostrovsky equation with positive dispersion,, Comm. Part. Diff. Eqs., 32 (2007), 1895. doi: 10.1080/03605300600987314. Google Scholar [11] R. Kraenkel, H. Leblond and M. A. Manna, An integrable evolution equation for surface waves in deep water,, (2011), (2011). Google Scholar [12] F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation,, J. Diff. Eqs., 222 (2006), 325. doi: 10.1016/j.jde.2005.07.023. Google Scholar [13] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation,, Dynamics of PDE, 6 (2009), 291. Google Scholar [14] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the Ostrovsky-Hunter equation,, SIAM J. Math. Anal., 42 (2010), 1967. doi: 10.1137/09075799X. Google Scholar [15] M. A. Manna and A. Neveu, Short-wave dynamics in the Euler equations,, Inverse Problems, 17 (2001), 855. doi: 10.1088/0266-5611/17/4/317. Google Scholar [16] A. J. Morrison, E. J. Parkes and V. O. Vakhnenko, The $N$ loop soliton solutions of the Vakhnenko equation,, Nonlinearity, 12 (1999), 1427. doi: 10.1088/0951-7715/12/5/314. Google Scholar [17] L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologia, 18 (1978), 181. Google Scholar [18] D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space,, Comm. Part. Diff. Eqs., 35 (2010), 613. doi: 10.1080/03605300903509104. Google Scholar [19] J. Satsuma and D. J. Kaup, A Bäcklund transformation for a higher-order Korteweg-de Vries equation,, J. Phys. Soc. Japan, 43 (1977), 692. doi: 10.1143/JPSJ.43.692. Google Scholar [20] A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation,, J. Diff. Eqs., 249 (2010), 2600. doi: 10.1016/j.jde.2010.05.015. Google Scholar [21] K. Tsugawa, Well-posedness and weak rotation limit for the Ostrovsky equation,, J. Diff. Eqs., 247 (2009), 3163. doi: 10.1016/j.jde.2009.09.009. Google Scholar [22] G. Tzitzeica, Sur une nouvelle classe des surfaces,, C. R. Acad. Sci. Paris, 150 (1910), 955. Google Scholar [23] V. A. Vakhnenko, Solitons in a nonlinear model medium,, J. Phys. A, 25 (1992), 4181. doi: 10.1088/0305-4470/25/15/025. Google Scholar [24] V. O. Vakhnenko and E. J. Parkes, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method,, Chaos, 13 (2002), 1819. doi: 10.1016/S0960-0779(01)00200-4. Google Scholar [25] V. O. Vakhnenko, E. J. Parkes and A. J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,, Chaos, 17 (2003), 683. doi: 10.1016/S0960-0779(02)00483-6. Google Scholar [26] V. Varlamov and Y. Liu, Cauchy problem for the Ostrovsky equation,, Discr. Cont. Dyn. Syst., 10 (2004), 731. doi: 10.3934/dcds.2004.10.731. Google Scholar

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##### References:
 [1] A. R. Aguirre, T. R. Araujo, J. F. Gomes and A. H. Zimerman, Type-II Bäcklund transformations via gauge transformations,, J. High Energy Phys., (2011). Google Scholar [2] J. P. Boyd, Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation,, Physics Letters A, 338 (2005), 36. doi: 10.1016/j.physleta.2005.02.017. Google Scholar [3] J. C. Brunelli and S. Sakovich, Hamiltonian structures for the Ostrovsky-Vakhnenko equation,, Comm. Nonlin. Sci. Numer. Simul., 18 (2013), 56. doi: 10.1016/j.cnsns.2012.06.018. Google Scholar [4] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Annales de l'Institute Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar [5] A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar [6] R. Iório and D. Pilod, Well-posedness for Hirota-Satsuma's equation,, Diff. Integr. Eqs., 21 (2008), 1177. Google Scholar [7] A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129. doi: 10.1088/0266-5611/19/1/307. Google Scholar [8] J. Hunter, Numerical solutions of some nonlinear dispersive wave equations,, in, 26 (1990), 301. Google Scholar [9] R. H. J. Grimshaw, K. Helfrich and E. R. Johnson, The reduced Ostrovsky equation: Integrability and breaking,, Stud. Appl. Math., 129 (2013), 414. doi: 10.1111/j.1467-9590.2012.00560.x. Google Scholar [10] G. Gui and Y. Liu, On the Cauchy problem for the Ostrovsky equation with positive dispersion,, Comm. Part. Diff. Eqs., 32 (2007), 1895. doi: 10.1080/03605300600987314. Google Scholar [11] R. Kraenkel, H. Leblond and M. A. Manna, An integrable evolution equation for surface waves in deep water,, (2011), (2011). Google Scholar [12] F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation,, J. Diff. Eqs., 222 (2006), 325. doi: 10.1016/j.jde.2005.07.023. Google Scholar [13] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation,, Dynamics of PDE, 6 (2009), 291. Google Scholar [14] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the Ostrovsky-Hunter equation,, SIAM J. Math. Anal., 42 (2010), 1967. doi: 10.1137/09075799X. Google Scholar [15] M. A. Manna and A. Neveu, Short-wave dynamics in the Euler equations,, Inverse Problems, 17 (2001), 855. doi: 10.1088/0266-5611/17/4/317. Google Scholar [16] A. J. Morrison, E. J. Parkes and V. O. Vakhnenko, The $N$ loop soliton solutions of the Vakhnenko equation,, Nonlinearity, 12 (1999), 1427. doi: 10.1088/0951-7715/12/5/314. Google Scholar [17] L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologia, 18 (1978), 181. Google Scholar [18] D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space,, Comm. Part. Diff. Eqs., 35 (2010), 613. doi: 10.1080/03605300903509104. Google Scholar [19] J. Satsuma and D. J. Kaup, A Bäcklund transformation for a higher-order Korteweg-de Vries equation,, J. Phys. Soc. Japan, 43 (1977), 692. doi: 10.1143/JPSJ.43.692. Google Scholar [20] A. Stefanov, Y. Shen and P. G. Kevrekidis, Well-posedness and small data scattering for the generalized Ostrovsky equation,, J. Diff. Eqs., 249 (2010), 2600. doi: 10.1016/j.jde.2010.05.015. Google Scholar [21] K. Tsugawa, Well-posedness and weak rotation limit for the Ostrovsky equation,, J. Diff. Eqs., 247 (2009), 3163. doi: 10.1016/j.jde.2009.09.009. Google Scholar [22] G. Tzitzeica, Sur une nouvelle classe des surfaces,, C. R. Acad. Sci. Paris, 150 (1910), 955. Google Scholar [23] V. A. Vakhnenko, Solitons in a nonlinear model medium,, J. Phys. A, 25 (1992), 4181. doi: 10.1088/0305-4470/25/15/025. Google Scholar [24] V. O. Vakhnenko and E. J. Parkes, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method,, Chaos, 13 (2002), 1819. doi: 10.1016/S0960-0779(01)00200-4. Google Scholar [25] V. O. Vakhnenko, E. J. Parkes and A. J. Morrison, A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,, Chaos, 17 (2003), 683. doi: 10.1016/S0960-0779(02)00483-6. Google Scholar [26] V. Varlamov and Y. Liu, Cauchy problem for the Ostrovsky equation,, Discr. Cont. Dyn. Syst., 10 (2004), 731. doi: 10.3934/dcds.2004.10.731. Google Scholar
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