Stability in the energy space of the sum of $N$ peakons for a Camassa-Holm-type equation with quartic nonlinearity
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Considered herein is the orbital stability in the energy space $H^1(\R)$ of a decoupled sum of $N$ peakons for a Camassa-Holm-type equation with quartic nonlinearity, which admits single peakon and multi-peakons. Based on our obtained result of the stability of a single peakon, then combining modulation argument with monotonicity of local energy $H^1$-norm, we get the stability of the sum of $N$ peakons.
Abstract
Considered herein is the orbital stability in the energy space $H^1(\R)$ of a decoupled sum of $N$ peakons for a Camassa-Holm-type equation with quartic nonlinearity, which admits single peakon and multi-peakons. Based on our obtained result of the stability of a single peakon, then combining modulation argument with monotonicity of local energy $H^1$-norm, we get the stability of the sum of $N$ peakons.
참고문헌 (33)
- R. Beals, "Acoustic scattering and the extended Korteweg-de Vries hierarchy", Adv. Math, 140(2), pp.190~206 (1998)
- R. Beals, "Multi-peakons and a theorem of Stieltjes", Inverse Problems, 15(1), pp.L1~L4 (1999)
- R. Camassa, "An integrable shallow water equation with peaked solitons", Phys. Rev. Lett, 71(11), pp.1661~1664 (1993)
- R. Camassa, "A new integrable shallow water equation", Adv. Appl. Mech, 31, pp.1~33 (1994)
- R. M. Chen, "Oscillation-induced blow-up to the modied Camassa-Holm equation with linear dispersion", Adv. Math, 272, pp.225~251 (2015)
- A. Constantin, "Existence of permanent and breaking waves for a shallow water equation:a geometric approach", Ann. Inst. Fourier (Grenoble), 50(2), pp.321~362 (2000)
- A. Constantin, "On the scattering problem for the Camassa-Holm equation", R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci, 457(2008), pp.953~970 (2001)
- A. Constantin, "The trajectories of particles in Stokes waves", Invent. Math, 166(3), pp.523~535 (2006)
- A. Constantin, "Global existence and blow-up for a shallow water equation", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26(2), pp.303~328 (1998)
- A. Constantin, "Well-posedness, global existence, and blowup phenomena for a periodic quasilinear hyperbolic equation", Comm. Pure Appl. Math, 51(5), pp.475~504 (1998)
- A. Constantin, "Wave breaking for nonlinear nonlocal shallow water equations", Acta Math, 181(2), pp.229~243 (1998)
- A. Constantin, "Analyticity of periodic traveling free surface water waves with vorticity", Ann. of Math, 173(1), pp.559~568 (2011)
- A. Constantin, "The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations", Arch. Ration. Mech. Anal, 192(1), pp.165~186 (2009)
- A. Constantin, "Stability of peakons", Comm. Pure Appl. Math, 53(5), pp.603~610 (2000)
- K. El Dika, "Stability of multipeakons", Ann. Inst. H. Poincare Anal. Non Lineaire, 26(4), pp.1517~1532 (2009)
- K. El Dika, "Stability of multi antipeakon-peakons prole", Discrete Contin. Dyn. Syst. Ser. B, 12(3), pp.561~577 (2009)
- A. S. Fokas, "The Korteweg-de Vries equation and beyond", Acta Appl. Math, 39(1-3), pp.295~305 (1995)
- B. Fuchssteiner, "Symplectic structures, their Backlund transformations and hereditary symmetries", Phys. D, 4(1), pp.47~66 (1982)
- Y. Fu, "On the Cauchy problem for the integrable modied Camassa-Holm equation with cubic nonlinearity", J. Dierential Equations, 255(7), pp.1905~1938 (2013)
- B. Fuchssteiner, "Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation", Phys. D, 95(3-4), pp.229~243 (1996)
- G. Gui, "Wave-breaking and peakons for a modied Camassa-Holm equation", Comm. Math. Phys, 319(3), pp.731~759 (2013)
- A. A. Himonas, "The Cauchy problem for the Novikov equation", Non-linearity, 25(2), pp.449~479 (2012)
- A. N. W. Hone, "Integrable peakon equations with cubic nonlinearity", J. Phys. A, 41(37), pp.10~ (2008)
- R. S. Johnson, "Camassa-Holm, Korteweg-de Vries and related models for water waves", J. Fluid Mech, 455, pp.63~82 (2002)
- X. Liu, "Stability of peakons for the Novikov equation", J. Math. Pures Appl. (9), 101(2), pp.172~187 (2014)
- X. Liu, "Orbital stability of the train of peakons for an integrable modied Camassa-Holm equation", Adv. Math, 255, pp.1~37 (2014)
- X. X. Liu, "Stability of peakons for a Camassa-Holm-type equation with quartic nonlinearity", ~
- Y. Liu, "On the blow-up of solutions to the integrable modied Camassa-Holm equation", Anal. Appl. (Singap.), 12(4), pp.355~368 (2014)
- Y. Martel, "Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations", Comm. Math. Phys, 231(2), pp.347~373 (2002)
- V. Novikov, "Generalizations of the Camassa-Holm equation", J. Phys. A, 42(34), pp.14~ (2009)
- P. J. Olver, "Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support", Phys. Rev. E (3), 53(2), pp.1900~1906 (1996)
- Z. Qiao, "A new integrable equation with cuspons and W", M-shape-peaks solitons, 2006, pp.47(11), 9~
- Z. Qiao, "An integrable equation with nonsmooth solitons", Theoret. and Math. Phys, 167(2), pp.584~589 (2011)
