In this paper, we present the globally conservative solutions to the Cauchy problem for the modified Camassa–Holm (MOCH) equation. First, we transform the equation into an equivalent semi-linear system under new variables. Second, according to the standard ordinary differential equation theory with the aid of the conservation law, we give the global solutions of the semi-linear system. Finally, returning to the original variables, we obtain the globally conservative solutions to the MOCH equation.

1.
H.
Bahouri
,
J. Y.
Chemin
, and
R.
Danchin
,
Fourier Analysis and Nonlinear Partial Differential Equations
(
Springer-Verlag
,
Berlin, Heidelberg
,
2011
).
2.
A.
Bressan
and
A.
Constantin
, “
Global dissipative solutions of the Camassa–Holm equation
,”
Anal. Appl.
05
(
01
),
1
27
(
2007
).
3.
A.
Bressan
and
A.
Constantin
, “
Global conservative solutions of the Camassa–Holm equation
,”
Arch. Ration. Mech. Anal.
183
(
2)
,
215
239
(
2007
).
4.
A.
Constantin
, “
The Hamiltonian structure of the Camassa–Holm equation
,”
Expo. Math.
15
(
1
),
53
85
(
1997
).
5.
A.
Constantin
, “
Existence of permanent of solutions and breaking waves for a shallow water equation: A geometric approach
,”
Ann. Inst. Fourier
50
,
321
362
(
2000
).
6.
A.
Constantin
, “
On the scattering problem for the Camassa–Holm equation
,”
Proc. R. Soc. London, Ser. A
457
,
953
970
(
2001
).
7.
A.
Constantin
and
J.
Escher
, “
Global existence and blow-up for a shallow water equation
,”
Ann. Sc. Norm. Super. Pisa Cl. Sci.
26
,
303
328
(
1998
).
8.
A.
Constantin
and
J.
Escher
, “
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
,”
Commun. Pure Appl. Math.
51
,
475
504
(
1998
).
9.
A.
Constantin
and
J.
Escher
, “
Wave breaking for nonlinear nonlocal shallow water equations
,”
Acta Math.
181
,
229
243
(
1998
).
10.
R.
Camassa
and
D. D.
Holm
, “
An integrable shallow water equation with peaked solitons
,”
Phys. Rev. Lett.
71
,
1661
1664
(
1993
).
11.
A.
Constantin
and
D.
Lannes
, “
The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations
,”
Arch. Ration. Mech. Anal.
192
,
165
186
(
2009
).
12.
A.
Constantin
and
L.
Molinet
, “
Global weak solutions for a shallow water equation
,”
Commun. Math. Phys.
211
,
45
61
(
2000
).
13.
A.
Constantin
and
W. A.
Strauss
, “
Stability of peakons
,”
Commun. Pure Appl. Math.
53
,
603
610
(
2000
).
14.
S. S.
Chern
and
K.
Tenenblat
, “
Pseudo-spherical surfaces and evolution equations
,”
Stud. Appl. Math.
74
,
55
83
(
1986
).
15.
R.
Danchin
, “
A few remarks on the Camassa–Holm equation
,”
Differ. Integr. Equations
14
,
953
988
(
2001
).
16.
R.
Danchin
, “
A note on well-posedness for Camassa-Holm equation
,”
J. Differ. Equations
192
,
429
444
(
2003
).
17.
A.
Fokas
and
B.
Fuchssteiner
, “
Symplectic structures, their Bäcklund transformation and hereditary symmetries
,”
Physica D
4
(
1
),
47
66
(
1981
).
18.
Z.
Guo
,
X.
Liu
,
L.
Molinet
, and
Z.
Yin
, “
Ill-posedness of the Camassa–Holm and related equations in the critical Space
,”
J. Differ. Equations
266
,
1698
1707
(
2019
).
19.
P.
Gorka
and
E. G.
Reyes
, “
The modified Camassa–Holm equation
,”
Int. Math. Res. Not.
2011
,
2617
2649
.
20.
J.
Li
and
Z.
Yin
, “
Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces
,”
J. Differ. Equations.
261
(
11
),
6125
6143
(
2016
).
21.
Z.
Qiao
, “
The Camassa–Holm hierarchy, related N-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold
,”
Commun. Math. Phys.
239
,
309
341
(
2003
).
22.
G.
Rodríguez-Blanco
, “
On the Cauchy problem for the Camassa–Holm equation
,”
Nonlinear Anal.
46
,
309
327
(
2001
).
23.
E. G.
Reyes
, “
Geometric integrability of the Camassa–Holm equation
,”
Lett. Math. Phys.
59
,
117
131
(
2002
).
24.
Z.
Xin
and
P.
Zhang
, “
On the weak solutions to a shallow water equation
,”
Commun. Pure Appl. Math.
53
,
1411
1433
(
2000
).
25.
M.
Zhou
,
The Theory of Function of Real Variables
(
Peking University Press
,
Beijing
,
2008
).
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