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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, 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), 056, 18 pp.  Google Scholar [2] J. P. Boyd, Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation, Physics Letters A, 338 (2005), 36-43. 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), 56-62. 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), 321-362. 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), 953-970. 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), 1177-1192.  Google Scholar [7] A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-145. doi: 10.1088/0266-5611/19/1/307.  Google Scholar [8] J. Hunter, Numerical solutions of some nonlinear dispersive wave equations, in "Computational Solution of Nonlinear Systems of Equations" (Fort Collins, CO, 1988), Lectures in Appl. Math., 26, Amer. Math. Soc., Providence, RI, (1990), 301-316.  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), 414-436. 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), 1895-1916. 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), arXiv:1101.5773. Google Scholar [12] F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation, J. Diff. Eqs., 222 (2006), 325-340. 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), 291-310.  Google Scholar [14] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the Ostrovsky-Hunter equation, SIAM J. Math. Anal., 42 (2010), 1967-1985. doi: 10.1137/09075799X.  Google Scholar [15] M. A. Manna and A. Neveu, Short-wave dynamics in the Euler equations, Inverse Problems, 17 (2001), 855-861. 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), 1427-1437. doi: 10.1088/0951-7715/12/5/314.  Google Scholar [17] L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181-191. 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), 613-629. 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), 692-726. 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), 2600-2617. 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), 3163-3180. 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), 955-956. Google Scholar [23] V. A. Vakhnenko, Solitons in a nonlinear model medium, J. Phys. A, 25 (1992), 4181-4187. 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, Solitons and Fractals, 13 (2002), 1819-1826. 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, Solitons and Fractals, 17 (2003), 683-692. 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), 731-753. doi: 10.3934/dcds.2004.10.731.  Google Scholar

show all references

##### 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), 056, 18 pp.  Google Scholar [2] J. P. Boyd, Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation, Physics Letters A, 338 (2005), 36-43. 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), 56-62. 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), 321-362. 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), 953-970. 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), 1177-1192.  Google Scholar [7] A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations, Inverse Problems, 19 (2003), 129-145. doi: 10.1088/0266-5611/19/1/307.  Google Scholar [8] J. Hunter, Numerical solutions of some nonlinear dispersive wave equations, in "Computational Solution of Nonlinear Systems of Equations" (Fort Collins, CO, 1988), Lectures in Appl. Math., 26, Amer. Math. Soc., Providence, RI, (1990), 301-316.  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), 414-436. 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), 1895-1916. 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), arXiv:1101.5773. Google Scholar [12] F. Linares and A. Milanés, Local and global well-posedness for the Ostrovsky equation, J. Diff. Eqs., 222 (2006), 325-340. 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), 291-310.  Google Scholar [14] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the Ostrovsky-Hunter equation, SIAM J. Math. Anal., 42 (2010), 1967-1985. doi: 10.1137/09075799X.  Google Scholar [15] M. A. Manna and A. Neveu, Short-wave dynamics in the Euler equations, Inverse Problems, 17 (2001), 855-861. 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), 1427-1437. doi: 10.1088/0951-7715/12/5/314.  Google Scholar [17] L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologia, 18 (1978), 181-191. 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), 613-629. 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), 692-726. 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), 2600-2617. 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), 3163-3180. 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), 955-956. Google Scholar [23] V. A. Vakhnenko, Solitons in a nonlinear model medium, J. Phys. A, 25 (1992), 4181-4187. 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, Solitons and Fractals, 13 (2002), 1819-1826. 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, Solitons and Fractals, 17 (2003), 683-692. 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), 731-753. doi: 10.3934/dcds.2004.10.731.  Google Scholar
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