zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Persistence properties and unique continuation of solutions to a two-component Camassa-Holm equation. (English) Zbl 1245.35108
Summary: We consider a two-component Camassa-Holm system which arises in shallow water theory. The present work is mainly concerned with persistence properties and unique continuation to this new kind of system, in view of the classical Camassa-Holm equation. Firstly, it is shown that there are three results about these properties of the strong solutions. Then we also investigate the infinite propagation speed in the sense that the corresponding solution does not have compact spatial support for t>0 although the initial data belongs to C 0 ().
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35D35Strong solutions of PDE
References:
[1]Beals, R., Sattinger, D., Szmigielski, J.: Multi-peakons and a theorem of Stieltjes. Inverse Problems 15(1), L1–L4 (1999) · Zbl 0923.35154 · doi:10.1088/0266-5611/15/1/001
[2]Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[3]Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. (Singap.) 5(1), 1–27 (2007) · Zbl 1139.35378 · doi:10.1142/S0219530507000857
[4]Boutet de Monvel, A., Kostenko, A., Shepelsky, D., Teschl, G.: Long-time asymptotics for the Camassa–Holm equation. SIAM J. Math. Anal. 41(4), 1559–1588 (2009) · Zbl 1204.37073 · doi:10.1137/090748500
[5]Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50(2), 321–362 (2000) · Zbl 0944.00010 · doi:10.5802/aif.1757
[6]Constantin, A.: Finite propagation speed for the Camassa–Holm equation. J. Math. Phys. 46(2), 023506, 4 pp. (2005) · Zbl 1076.35109 · doi:10.1063/1.1845603
[7]Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166(3), 523–535 (2006) · Zbl 1108.76013 · doi:10.1007/s00222-006-0002-5
[8]Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51(5), 475–504 (1998) · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[9]Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[10]Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. (N.S.) 44(3), 423–431 (2007) · Zbl 1126.76012 · doi:10.1090/S0273-0979-07-01159-7
[11]Constantin, A., Gerdjikov, V., Ivanov, R.: Inverse scattering transform for the Camassa–Holm equation. Inverse Problems 22(6), 2197–2207 (2006) · Zbl 1105.37044 · doi:10.1088/0266-5611/22/6/017
[12]Constantin, A., Ivanov, R.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372(48), 7129–7132 (2008) · Zbl 1227.76016 · doi:10.1016/j.physleta.2008.10.050
[13]Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[14]Constantin, A., McKean, H.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52(8), 949–982 (1999) · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[15]Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211(1), 45–61 (2000) · Zbl 1002.35101 · doi:10.1007/s002200050801
[16]Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53(5), 603–610 (2000) · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[17]Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[18]Chen, M., Liu, S., Zhang, Y.: A two-component generalization of the Camassa–Holm equation and its solutions. Lett. Math. Phys. 75(1), 1–15 (2006) · Zbl 1105.35102 · doi:10.1007/s11005-005-0041-7
[19]Escher, J., Lechtenfeld, O., Yin, Z.: Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst. 19(3), 493–513 (2007) · Zbl 1149.35307 · doi:10.3934/dcds.2007.19.493
[20]Falqui, G.: On a Camassa–Holm type equation with two dependent variables. J. Phys. A 39(2), 327–342 (2006) · Zbl 1084.37053 · doi:10.1088/0305-4470/39/2/004
[21]Fuchssteiner, B., Fokas, A.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981/82) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[22]Guo, Z.: Blow-up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 372(1), 316–327 (2010) · Zbl 1205.35045 · doi:10.1016/j.jmaa.2010.06.046
[23]Guo, Z., Zhou, Y.: On solutions to a two-component generalized Camassa–Holm equation. Stud. Appl. Math. 124(3), 307–322 (2010) · Zbl 1189.35255 · doi:10.1111/j.1467-9590.2009.00472.x
[24]Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlin. Math. Phys. 12(3), 342–347 (2005) · Zbl 1086.35079 · doi:10.2991/jnmp.2005.12.3.3
[25]Henry, D.: Infinite propagation speed for a two component Camassa–Holm equation. Discrete Contin. Dyn. Syst. Ser. B 12(3), 597–606 (2009) · Zbl 1180.35458 · doi:10.3934/dcdsb.2009.12.597
[26]Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271(2), 511–522 (2007) · Zbl 1142.35078 · doi:10.1007/s00220-006-0172-4
[27]Ivanov, R.: Water waves and integrability. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 365(1858), 2267–2280 (2007) · Zbl 1152.76322 · doi:10.1098/rsta.2007.2007
[28]Johnson, R.: Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)
[29]Mustafa, O.G.: On smooth traveling waves of an integrable two-component Camassa–Holm shallow water system. Wave Motion 46(6), 397–402 (2009) · Zbl 1231.76063 · doi:10.1016/j.wavemoti.2009.06.011
[30]Olver, P., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E (3) 53(2), 1900–1906 (1996) · doi:10.1103/PhysRevE.53.1900
[31]Toland, J.: Stokes waves. Topol. Methods Nonlinear Anal. 7(1), 1–48 (1996)
[32]Whitham, G.: Linear and Nonlinear Waves. Reprint of the 1974 Original, xviii+636 pp.. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. Wiley, New York (1999)
[33]Xin, Z., Zhang, P.: On the weak solution to a shallow water equation. Commun. Pure Appl. Math. 53(11), 1411–1433 (2000) · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[34]Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57(1), 137–152 (2004) · Zbl 1106.35070 · doi:10.1016/j.na.2004.02.004
[35]Zhou, Y.: Wave breaking for a periodic shallow water equation. J. Math. Anal. Appl. 290(2), 591–604 (2004) · Zbl 1042.35060 · doi:10.1016/j.jmaa.2003.10.017
[36]Zhou, Y.: Stability of solitary waves for a rod equation. Chaos Solitons Fractals 21(4), 977–981 (2004) · Zbl 1046.35094 · doi:10.1016/j.chaos.2003.12.030
[37]Zhou, Y., Guo, Z.: Blow up and propagation speed of solutions to the DGH equation. Discrete Contin. Dyn. Syst., Ser. B 12(3), 657–670 (2009) · Zbl 1180.35473 · doi:10.3934/dcdsb.2009.12.657