# American Institute of Mathematical Sciences

December  2017, 37(12): 6471-6485. doi: 10.3934/dcds.2017280

## Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation

 1 Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China 2 Faculty of Information Technology, Macau University of Science and Technology, Macau, China

1Corresponding author

Received  June 2017 Published  August 2017

Fund Project: This work was partially supported by NNSFC (No.11671407), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004).

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in $H^s (\mathbb{S}), s > \frac{3}{2},$ by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

Citation: Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
##### References:
 [1] R. Beals, D. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter--Saxton equations, Appl. Anal., 78 (2001), 255-269.  doi: 10.1080/00036810108840938. [2] A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.  doi: 10.1137/090748500. [3] J. Boyd, Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: Matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves), European J. Appl. Math., 16 (2005), 65-81.  doi: 10.1017/S0956792504005625. [4] 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. [5] A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.  doi: 10.1137/050623036. [6] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z. [7] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857. [8] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [9] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701. [10] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757. [11] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [12] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033. [13] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [14] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [15] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7. [16] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12. [17] A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017. [18] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [19] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [20] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801. [21] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [22] H. H. Dai and M. Pavlov, Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. P. Soc. Japan, 67 (1998), 3655-3657.  doi: 10.1143/JPSJ.67.3655. [23] R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (1001), 953-988. [24] J. M. Delort, Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup., 34 (2001), 1-61.  doi: 10.1016/S0012-9593(00)01059-4. [25] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [26] R. Grimshaw and D. Pelinovsky, Global existence of small-norm solutions in the reduced Ostrovsky equation, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 557-566.  doi: 10.3934/dcds.2014.34.557. [27] N. Hayashi and P. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.  doi: 10.1007/s00033-007-7008-8. [28] A. Hone, V. Novikov and J. Wang, Generalizations of the short pulse equation, arXiv preprint, arXiv: 1612.02481 (2016). [29] J. Hunter, Numerical solutions of some nonlinear dispersive wave equations, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., AMS, Providence, RI, 26 (1990), 301-316. [30] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075. [31] J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D, 79 (1994), 361-386.  doi: 10.1016/S0167-2789(05)80015-6. [32] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., Springer, Berlin, 448 (1975), 25-70. [33] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967. [34] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Academic Press, 8 (1983), 93-128. [35] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704. [36] J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064.  doi: 10.1016/j.geomphys.2007.05.003. [37] J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.  doi: 10.1016/j.jde.2016.08.031. [38] M. Li and Z. Yin, Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity, Nonlinear Anal., 151 (2017), 208-226.  doi: 10.1016/j.na.2016.12.003. [39] 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. [40] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.  doi: 10.4310/DPDE.2009.v6.n4.a1. [41] T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107.  doi: 10.3934/dcds.2014.34.3095. [42] 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. [43] P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitions and solitary wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900. [44] E. J. Parkes, Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals, 31 (2007), 181-191.  doi: 10.1016/j.chaos.2005.10.028. [45] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [46] D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.  doi: 10.1080/03605300903509104. [47] T. Schäfter and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.  doi: 10.1016/j.physd.2004.04.007. [48] 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. [49] Y. A. Stepanyants, On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos Solitons Fractals, 28 (2006), 193-204.  doi: 10.1016/j.chaos.2005.05.020. [50] H. Sunagawa, Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494. [51] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001. [52] A. M. Wazwaz, The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Applied Mathematics and Computation, 167 (2005), 1196-1210.  doi: 10.1016/j.amc.2004.08.005. [53] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [54] Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.  doi: 10.1137/S0036141003425672.

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##### References:
 [1] R. Beals, D. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter--Saxton equations, Appl. Anal., 78 (2001), 255-269.  doi: 10.1080/00036810108840938. [2] A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588.  doi: 10.1137/090748500. [3] J. Boyd, Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: Matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves), European J. Appl. Math., 16 (2005), 65-81.  doi: 10.1017/S0956792504005625. [4] 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. [5] A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026.  doi: 10.1137/050623036. [6] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  doi: 10.1007/s00205-006-0010-z. [7] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857. [8] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [9] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701. [10] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757. [11] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [12] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033. [13] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [14] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [15] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7. [16] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12. [17] A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017. [18] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [19] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [20] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801. [21] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [22] H. H. Dai and M. Pavlov, Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. P. Soc. Japan, 67 (1998), 3655-3657.  doi: 10.1143/JPSJ.67.3655. [23] R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (1001), 953-988. [24] J. M. Delort, Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup., 34 (2001), 1-61.  doi: 10.1016/S0012-9593(00)01059-4. [25] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [26] R. Grimshaw and D. Pelinovsky, Global existence of small-norm solutions in the reduced Ostrovsky equation, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 557-566.  doi: 10.3934/dcds.2014.34.557. [27] N. Hayashi and P. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.  doi: 10.1007/s00033-007-7008-8. [28] A. Hone, V. Novikov and J. Wang, Generalizations of the short pulse equation, arXiv preprint, arXiv: 1612.02481 (2016). [29] J. Hunter, Numerical solutions of some nonlinear dispersive wave equations, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math., AMS, Providence, RI, 26 (1990), 301-316. [30] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.  doi: 10.1137/0151075. [31] J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D, 79 (1994), 361-386.  doi: 10.1016/S0167-2789(05)80015-6. [32] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., Springer, Berlin, 448 (1975), 25-70. [33] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967. [34] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Academic Press, 8 (1983), 93-128. [35] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704. [36] J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064.  doi: 10.1016/j.geomphys.2007.05.003. [37] J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.  doi: 10.1016/j.jde.2016.08.031. [38] M. Li and Z. Yin, Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity, Nonlinear Anal., 151 (2017), 208-226.  doi: 10.1016/j.na.2016.12.003. [39] 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. [40] Y. Liu, D. Pelinovsky and A. Sakovich, Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.  doi: 10.4310/DPDE.2009.v6.n4.a1. [41] T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107.  doi: 10.3934/dcds.2014.34.3095. [42] 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. [43] P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitions and solitary wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900. [44] E. J. Parkes, Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals, 31 (2007), 181-191.  doi: 10.1016/j.chaos.2005.10.028. [45] A. 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Sunagawa, Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494. [51] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001. [52] A. M. Wazwaz, The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Applied Mathematics and Computation, 167 (2005), 1196-1210.  doi: 10.1016/j.amc.2004.08.005. [53] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [54] Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283.  doi: 10.1137/S0036141003425672.
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