In this article, we study the self-similar solutions of the 2-component Camassa–Holm equations ρt+uρx+ρux=0, mt+2uxm+umx+σρρx=0, with m=uα2uxx. By the separation method, we can obtain a class of blowup or global solutions for σ=1 or −1. In particular, for the integrable system with σ=1, we have the global solutions, ρ(t,x)=f(η)/a(3t)1/3 for η2<α2/ξ, ρ(t,x)=0 for η2α2/ξ, u(t,x)=ȧ(3t)/a(3t)x, ä(s)ξ/3a(s)1/3=0,a(0)=a0>0,ȧ(0)=a1, f(η)=ξ1/ξη2+(α/ξ)2, where η=xa(s)1/3 with s=3t; ξ>0 and α0 are arbitrary constants. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.

1.
Camassa
,
R.
and
Holm
,
D. D.
, “
Intergrable shallow water equation with peaked solitons
,”
Phys. Rev. Lett.
71
,
1661
(
1993
).
2.
Chen
,
M.
,
Liu
,
S. -Q.
, and
Zhang
,
Y.
, “
A 2-component generalization of the Camassa–Holm equation and its solutions
,”
Lett. Math. Phys.
75
,
1
(
2006
).
3.
Constantin
,
A.
, “
On the blow-up solutions of a periodic shallow water equation
,”
J. Nonlinear Sci.
10
,
391
(
2000
).
4.
Constantin
,
A.
, “
Existence of permanent and breaking waves for a shallow water equation: A geometric approach
,”
Ann. Inst. Fourier
50
,
321
(
2000
).
5.
Constantin
,
A.
, “
The trajectories of particles in Stokes waves
,”
Invent. Math.
166
,
523
(
2006
).
6.
Constantin
,
A.
and
Escher
,
J.
, “
Wave breaking for nonlinear nonlocal shallow water equations
,”
Acta Math.
181
,
229
(
1998
).
7.
Constantin
,
A.
and
Escher
,
J.
, “
On the blow-up rate and the blow-up set of breaking waves for a shallow water equation
,”
Math. Z.
233
,
75
(
2000
).
8.
Constantin
,
A.
and
Escher
,
J.
, “
Particle trajectories in solitary water waves
,”
Bull., New Ser., Am. Math. Soc.
44
,
423
(
2007
).
9.
Constantin
,
A.
and
Ivanov
,
R.
, “
On an integrable two-component Camassa–Holm shallow water system
,”
Phys. Lett. A
372
,
7129
(
2008
).
10.
Constantin
,
A.
and
Lannes
,
D.
, “
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
,”
Arch. Ration. Mech. Anal.
192
,
165
(
2009
).
11.
Deng
,
Y. B.
,
Xiang
,
J. L.
, and
Yang
,
T.
, “
Blowup phenomena of solutions to Euler-Poisson equations
,”
J. Math. Anal. Appl.
286
,
295
(
2003
).
12.
Escher
,
J.
,
Lechtenfeld
,
O.
, and
Yin
,
Z.
, “
Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation
,”
Discrete Contin. Dyn. Syst.
19
,
493
(
2007
).
13.
Guan
,
C. X.
and
Yin
,
Z. Y.
, “
Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system
,”
J. Differ. Equations
248
,
2003
(
2010
).
14.
Goldreich
,
P.
and
Weber
,
S.
, “
Homologously collapsing stellar cores
,”
Astrophys. J.
238
,
991
(
1980
).
15.
Guo
,
Z. G.
, “
Blow-up and global solutions to a new integrable model with two components
,”
J. Math. Anal. Appl.
372
,
316
(
2010
).
16.
Guo
,
Z. G.
and
Zhou
,
Y.
, “
On solutions to a two-component generalized Camassa–Holm system
,”
Stud. Appl. Math.
124
,
307
(
2010
).
17.
Ivanov
,
R. I.
, “
Extended Camassa–Holm hierarchy and conserved quantities
,”
Z. Naturforsch., A: Phys. Sci.
61
,
133
(
2006
).
18.
Johnson
,
R. S.
, “
Camassa-Holm, Korteweg-de Vries and related models for water waves
,”
J. Fluid Mech.
455
,
63
(
2002
).
19.
Lakin
,
W. D.
and
Sanchez
,
D. A.
,
Topics in Ordinary Differential Equations
(
Dover
,
New York
,
1982
).
20.
Li
,
T. H.
, “
Some special solutions of the multidimensional Euler equations in RN
,”
Comm. Pure Appl. Anal.
4
,
757
(
2005
).
21.
Makino
,
T.
, “
Blowing up solutions of the Euler-Poission equation for the evolution of the gaseous stars
,”
Transp. Theory Stat. Phys.
21
,
615
(
1992
).
22.
Spiegel
,
M. R.
,
Lipschutz
,
S.
, and
Liu
,
J.
,
Mathematical Handbook of Formulas and Tables
, 3rd ed. (
McGraw-Hill
,
New York
,
2008
).
23.
Toland
,
J. F.
, “
Stokes waves
,”
Topol. Methods Nonlinear Anal.
7
,
1
(
1996
).
24.
Whitham
,
G. B.
,
Linear and Nonlinear Waves, Pure and Applied Mathematics
(
Wiley-Interscience
,
New York
,
1974
).
25.
Yuen
,
M. W.
, “
Blowup solutions for a class of fluid dynamical equations in RN
,”
J. Math. Anal. Appl.
329
,
1064
(
2007
).
26.
Yuen
,
M. W.
, “
Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars
,”
J. Math. Anal. Appl.
341
,
445
(
2008
).
27.
Yuen
,
M. W.
, “
Analytical solutions to the Navier-Stokes equations
,”
J. Math. Phys.
49
,
113102
(
2008
).
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