Abstract
This paper is devoted to a new integrable two-component Camassa–Holm system with peaked solitons (peakons) and weak-kink solutions. It is the first integrable system that admits weak kink and kink–peakon interactional solutions. In addition, the new system includes both standard (quadratic) and cubic Camassa–Holm equations as two special cases. In the paper, we first establish the local well-posedness for the Cauchy problem of the system, and then derive a precise blow-up scenario and a new blow-up result for strong solutions to the system with both quadratic and cubic nonlinearity. Furthermore, its peakon and weak kink solutions are discussed as well.
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Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der MathematischenWissenschaften, vol. 343. Springer, Berlin (2011)
Bressan A., Constantin A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)
Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Camassa R., Holm D., Hyman J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
Chen M., Liu S-Q., Zhang Y.: A 2-component generalization of the Camassa–Holm equation and its solutions. Lett. Math. Phys. 75, 1–15 (2006)
Coclite G.M., Karlsen K.H.: On the well-posedness of the Degasperis–Procesi equation. J. Funct. Anal. 233, 60–91 (2006)
Constantin A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)
Constantin A., Escher J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Sup. Pisa 26, 303–328 (1998)
Constantin A., Escher J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504 (1998)
Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)
Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. (2) 173, 559–568 (2011)
Constantin A., Ivanov R.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372, 7129–7132 (2008)
Constantin A., Ivanov R., Lenells J.: Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity 23, 2559–2575 (2010)
Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Commun. Math. Helv. 78, 787–804 (2003)
Constantin A., Lannes D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)
Constantin A., Molinet L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61 (2000)
Constantin, A. Strauss W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)
Dai H.H.: Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod. Acta Mech. 127, 193–207 (1998)
Danchin R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14, 953–988 (2001)
Degasperis A., Holm D.D., Hone A.N.W.: A new integral equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)
Degasperis A., Procesi M.: Asymptotic Integrability. Symmetry and Perturbation Theory (Rome, 1998), pp. 23–37. World Scientific Publishing, River Edge (1999)
Dullin H.R., Gottwald G.A., Holm D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504 (2001)
Escher J., Kolev B.: The Degasperis–Procesi equation as a non-metric Euler equation. Math. Z. 269, 1137–1153 (2011)
Escher J., Lechtenfeld O., Yin Z.: Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst. 19, 493–513 (2007)
Escher J., Liu Y., Yin Z.: Global weak solutions and blow-up structure for the Degasperis–Procesi equation. J. Funct. Anal. 241, 457–485 (2006)
Escher J., Liu Y., Yin Z.: Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation. Indiana Univ. Math. J. 56, 87–117 (2007)
Escher J., Yin Z.: Initial boundary value problems for nonlinear dispersive wave equations. J. Funct. Anal. 256, 479–508 (2009)
Fokas A.: On a class of physically important integrable equations. Phys. D 87, 145–150 (1995)
Fokas A., Fuchssteiner B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D 4, 47–66 (1981)
Fuchssteiner B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation. Phys. D 95, 229–243 (1996)
Guan C., Yin Z.: Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. J. Differ. Equ. 248, 2003–2014 (2010)
Guan C., Karlsen K.H., Yin Z.: Well-posedness and blow-up phenomena for a modified two-component Camassa–Holm equation. Contemp. Math. 526, 199–220 (2010)
Guan C., Yin Z.: Global weak solutions for a two-component Camassa–Holm shallow water system. J. Funct. Anal. 260, 1132–1154 (2011)
Guan C., Yin Z.: Global weak solutions for a modified two-component Camassa–Holm equation. Ann. Inst. Henri Poincare (C) Non Linear Anal. 28, 623–641 (2011)
Gui G., Liu Y.: On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Funct. Anal. 258, 4251–4278 (2010)
Gui G., Liu Y., Olver P., Qu C.: Wave-breaking and peakons for a modified Camassa–Holm equation. Commun. Math. Phys. 319, 731–759 (2013)
Himonas A.A., Holliman C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)
Holm D., Naraigh L., Tronci C.: Singular solution of a modified two-component Camassa–Holm equation. Phys. Rev. E (3) 79, 1–13 (2009)
Hone A.N., Wang J.P.: Integrable peakon equations with cubic nonlinearity. J. Phys. A 41, 372002 (2008)
Hou Y., Zhao E., Fan P., Qiao Z.: Algebro-geometric solutions for the Degasperis–Procesi hierarchy. SIAM J. Math. Anal. 45, 1216–1266 (2013)
Johnson R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 457, 63–82 (2002)
Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Mathematics, Vol. 448, pp. 25–70. Springer Verlag, Berlin (1975)
Liu Y., Yin Z.: Global existence and blow-up phenomena for the Degasperis–Procesi equation. Commun. Math. Phys. 267, 801–820 (2006)
Lundmark H.: Formation and dynamics of shock waves in the Degasperis–Procesi equation. J. Nonlinear Sci. 17, 169–198 (2007)
Novikov V.: Generalizations of the Camassa–Holm equation. J. Phys. A 42, 342002 (2009)
Olver P.J., Rosenau P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)
Popowicz Z.: A two-component generalization of the Degasperis–Procesi equation. J. Phys. A Math. Gen. 39, 13717–13726 (2006)
Qiao Z.: The Camassa–Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold. Commun. Math. Phys. 239, 309–341 (2003)
Qiao Z.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006)
Qiao Z., Xia B.: Integrable peakon systems with weak kink and kink–peakon interactional solutions. Front. Math. China 8, 1185–1196 (2013)
Qiao Z.: Integrable hierarchy (the DP hierarchy), 3 by 3 constrained systems, and parametric and stationary solutions. Acta Appl. Math. 83, 199–220 (2004)
Tan W., Yin Z.: Global periodic conservative solutions of a periodic modified two-component Camassa–Holm equation. J. Funct. Anal. 261, 1204–1226 (2011)
Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)
Whitham G.B.: Linear and Nonlinear Waves. Wiley, New York (1980)
Wu X., Yin Z.: Global weak solutions for the Novikov equation. J. Phys. A 44, 055202 (2011)
Xia, B., Qiao, Z.: A new two-component integrable system with peakon and weak kink solutions. arXiv:1211.5727v3
Xin Z., Zhang P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53, 1411–1433 (2000)
Yan K., Yin Z.: Analytic solutions of the Cauchy problem for two-component shallow water systems. Math. Z. 269, 1113–1127 (2011)
Yan K., Yin Z.: On the Cauchy problem for a two-component Degasperis–Procesi system. J. Differ. Equ. 252, 2131–2159 (2012)
Yan K., Yin Z.: Well-posedness for a modified two-component Camassa–Holm system in critical spaces. Discrete Contin. Dyn. Syst. 33, 1699–1712 (2013)
Yan K., Yin Z.: Initial boundary value problems for the two-component shallow water systems. Rev. Mat. Iberoam. 29, 911–938 (2013)
Yin Z.: On the Cauchy problem for an integrable equation with peakon solutions. Illinois J. Math. 47, 649–666 (2003)
Yin Z.: Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal. 212, 182–194 (2004)
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Yan, K., Qiao, Z. & Yin, Z. Qualitative Analysis for a New Integrable Two-Component Camassa–Holm System with Peakon and Weak Kink Solutions. Commun. Math. Phys. 336, 581–617 (2015). https://doi.org/10.1007/s00220-014-2236-1
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DOI: https://doi.org/10.1007/s00220-014-2236-1