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 & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
References:
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A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036. Google Scholar

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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. Google Scholar

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A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar

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A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar

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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. Google Scholar

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A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

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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. Google Scholar

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R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (1001), 953-988. Google Scholar

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[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. Google Scholar

[30]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007/BF01647967. Google Scholar

[34]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Academic Press, 8 (1983), 93-128. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

Y. LiuD. 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

[40]

Y. LiuD. 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. Google Scholar

[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. Google Scholar

[42]

A. J. MorrisonE. 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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

A. StefanovY. 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

[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. Google Scholar

[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. Google Scholar

[51]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

R. BealsD. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter--Saxton equations, Appl. Anal., 78 (2001), 255-269. doi: 10.1080/00036810108840938. Google Scholar

[2]

A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: 10.1137/090748500. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar

[12]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar

[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. Google Scholar

[14]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta. Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

A. ConstantinV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (1001), 953-988. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

A. Hone, V. Novikov and J. Wang, Generalizations of the short pulse equation, arXiv preprint, arXiv: 1612.02481 (2016).Google Scholar

[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. Google Scholar

[30]

J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007/BF01647967. Google Scholar

[34]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math. Suppl. Stud., Academic Press, 8 (1983), 93-128. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

Y. LiuD. 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

[40]

Y. LiuD. 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. Google Scholar

[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. Google Scholar

[42]

A. J. MorrisonE. 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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[48]

A. StefanovY. 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

[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. Google Scholar

[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. Google Scholar

[51]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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