In this paper, for a given conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables.

1.
Bressan
,
A.
and
Chen
,
G.
, “
Generic regularity of conservative solutions to a nonlinear wave equation
,”
Ann. Inst. Henri Poincare, Sect. C
34
(
2
),
335
354
(
2017
).
2.
Bressan
,
A.
,
Chen
,
G.
, and
Zhang
,
Q.
, “
Unique conservative solutions to a variational wave equation
,”
Arch. Ration. Mech. Anal.
217
(
3
),
1069
1101
(
2015
).
3.
Bressan
,
A.
and
Constantin
,
A.
, “
Global conservative solutions of the Camassa-Holm equation
,”
Arch. Ration. Mech. Anal.
183
(
2
),
215
239
(
2007
).
4.
Bressan
,
A.
and
Constantin
,
A.
, “
Global dissipative solutions of the Camassa-Holm equation
,”
Anal. Appl.
05
(
01
),
1
27
(
2007
).
5.
Bressan
,
A.
and
Zheng
,
Y.
, “
Conservative solutions to a nonlinear variational wave equation
,”
Commun. Math. Phys.
266
(
2
),
471
497
(
2006
).
6.
Camassa
,
R.
, “
Characteristic variables for a completely integrable shallow water equation
,” in
Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEDS’79 (Gallipoli, 1999)
(
World Scientific Publishing
,
River Edge, NJ
,
2000
), pp.
65
74
.
7.
Camassa
,
R.
and
Holm
,
D. D.
, “
An integrable shallow water equation with peaked solitons
,”
Phys. Rev. Lett.
71
(
11
),
1661
1664
(
1993
).
8.
Chern
,
S. S.
and
Tenenblat
,
K.
, “
Pseudospherical surfaces and evolution equations
,”
Stud. Appl. Math.
74
(
1
),
55
83
(
1986
).
9.
Constantin
,
A.
, “
Existence of permanent and breaking waves for a shallow water equation: A geometric approach
,”
Ann. Inst. Fourier
50
(
2
),
321
362
(
2000
).
10.
Constantin
,
A.
, “
On the scattering problem for the Camassa-Holm equation
,”
Proc. R. Soc. London, Ser. A
457
,
953
970
(
2001
).
11.
Constantin
,
A.
and
Escher
,
J.
, “
Global existence and blow-up for a shallow water equation
,”
Ann. Sc. Norm. Super. Pisa Cl. Sci.
26
(
2
),
303
328
(
1998
).
12.
Constantin
,
A.
and
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
).
13.
Constantin
,
A.
,
Gerdjikov
,
V. S.
, and
Ivanov
,
R. I.
, “
Inverse scattering transform for the Camassa–Holm equation
,”
Inverse Probl.
22
(
6
),
2197
2207
(
2006
).
14.
Constantin
,
A.
and
McKean
,
H. P.
, “
A shallow water equation on the circle
,”
Commun. Pure Appl. Math.
52
(
8
),
949
982
(
1999
).
15.
Constantin
,
A.
and
Molinet
,
L.
, “
Global weak solutions for a shallow water equation
,”
Commun. Math. Phys.
211
(
1
),
45
61
(
2000
).
16.
Danchin
,
R.
, “
A few remarks on the Camassa-Holm equation
,”
Differ. Integr. Equations
14
(
8
),
953
988
(
2001
).
17.
Danchin
,
R.
, “
A note on well-posedness for Camassa–Holm equation
,”
J. Differ. Equations
192
(
2
),
429
444
(
2003
).
18.
Fuchssteiner
,
B.
and
Fokas
,
A. S.
, “
Symplectic structures, their Bäcklund transformations and hereditary symmetries
,”
Physica D
4
(
1
),
47
66
(
1981/82
).
19.
Górka
,
P.
and
Reyes
,
E. G.
, “
The modified Camassa–Holm equation
,”
Int. Math. Res. Not.
2011
(
12
),
2617
2649
.
20.
Guo
,
Z.
,
Liu
,
X.
,
Molinet
,
L.
, and
Yin
,
Z.
, “
Ill-posedness of the Camassa-Holm and related equations in the critical space
,”
J. Differ. Equations
266
(
2–3
),
1698
1707
(
2019
).
21.
Hernández Heredero
,
R.
and
Reyes
,
E. G.
, “
Geometric integrability of the Camassa-Holm equation. II
,”
Int. Math. Res. Not.
2012
(
13
),
3089
3125
.
22.
Himonas
,
A. A.
and
Holliman
,
C.
, “
The Cauchy problem for a generalized Camassa-Holm equation
,”
Adv. Differ. Equations
19
(
1/2
),
161
200
(
2014
).
23.
Li
,
J.
and
Yin
,
Z.
, “
Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces
,”
J. Differ. Equations
261
(
11
),
6125
6143
(
2016
).
24.
Li
,
M.
and
Zhang
,
Q.
, “
Generic regularity of conservative solutions to Camassa–Holm type equations
,”
SIAM J. Math. Anal.
49
(
4
),
2920
2949
(
2017
).
25.
Lundmark
,
H.
and
Szmigielski
,
J.
, “
A view of the peakon world through the lens of approximation theory
,”
Physica D
440
(
44
),
133446
(
2022
).
26.
Luo
,
Z.
,
Qiao
,
Z.
, and
Yin
,
Z.
, “
On the Cauchy problem for a modified Camassa-Holm equation
,”
Monatsh. Math.
193
(
4
),
857
877
(
2020
).
27.
Luo
,
Z.
,
Qiao
,
Z.
, and
Yin
,
Z.
, “
Globally conservative solutions for the modified Camassa-Holm (MOCH) equation
,”
J. Math. Phys.
62
(
9
),
091506
(
2021
).
28.
Luo
,
Z.
,
Qiao
,
Z.
, and
Yin
,
Z.
, “
Global existence and blow-up phenomena for a periodic modified Camassa-Holm equation (MOCH)
,”
Appl. Anal.
101
(
9
),
3432
3444
(
2022
).
29.
Reyes
,
E. G.
, “
Geometric integrability of the Camassa-Holm equation
,”
Lett. Math. Phys.
59
(
2
),
117
131
(
2002
).
30.
Xin
,
Z.
and
Zhang
,
P.
, “
On the uniqueness and large time behavior of the weak solutions to a shallow water equation
,”
Commun. Partial Differ. Equations
27
(
9–10
),
1815
1844
(
2002
).
31.
Ye
,
W.
,
Yin
,
Z.
, and
Guo
,
Y.
, “
The well-posedness for the Camassa-Holm type equations in critical Besov spaces Bp,11+1p with 1 ≤ p < +
,”
J. Differ. Equations
367
,
729
748
(
2023
).
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