The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow, which interacts with the edge wave packets or the edge solitons. A conjecture about the existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In the case of localized simple-wave pulses propagating through a quiescent medium, this theory provided a new approach to derivation of an asymptotic formula for the number of solitons eventually produced from such a pulse.

1.
S. C.
Gardner
,
J. M.
Greene
,
M. D.
Kruskal
, and
R. M.
Miura
,
Phys. Rev. Lett.
19
,
1095
(
1967
).
2.
3.
A. V.
Gurevich
and
L. P.
Pitaevskii
,
Zh. Eksp. Teor. Fiz.
65
,
590
(
1973
) [Sov. Phys. JETP 38, 291 (1974)].
4.
G. B.
Whitham
,
Proc. R. Soc. Lond. A
283
,
238
(
1965
).
5.
A. V.
Gurevich
,
A. L.
Krylov
, and
N. G.
Mazur
,
Zh. Eksp. Teor. Fiz.
95
,
1674
(
1989
) [Sov. Phys. JETP 68, 966 (1989)].
6.
C. H.
Su
and
C. S.
Gardner
,
J. Math. Phys.
10
,
536
(
1969
).
7.
A. V.
Gurevich
,
A. L.
Krylov
, and
G. A.
El
,
Zh. Eksp. Teor. Fiz.
98
,
1605
(
1990
) [Sov. Phys. JETP 71, 899 (1990)].
8.
L. D.
Landau
and
E. M.
Lifshitz
,
Fluid Mechanics
(
Pergamon
,
Oxford
,
1987
).
9.
A. V.
Gurevich
and
A. P.
Meshcherkin
,
Zh. Eksp. Teor. Fiz.
87
,
1277
(
1984
) [Sov. Phys. JETP 60, 732 (1984)].
10.
A. V.
Gurevich
and
L. P.
Pitaevskii
,
Zh. Eksp. Teor. Fiz.
93
,
871
(
1987
) [Sov. Phys. JETP 93, 871 (1987)].
11.
A. M.
Kamchatnov
, “
Gurevich-Pitaevskii problem and its development
,”
Phys. Usp.
(published online,
2020
).
12.
C.
Lanczos
,
The Variational Principles of Mechanics
(
University of Toronto Press
,
Toronto
,
1962
).
13.
G. B.
Whitham
,
Linear and Nonlinear Waves
(
Wiley Interscience
,
New York
,
1974
).
14.
15.
G. A.
El
,
R. H. J.
Grimshaw
, and
N. F.
Smyth
,
Phys. Fluids
18
,
027104
(
2006
).
16.
G. A.
El
,
A.
Gammal
,
E. G.
Khamis
,
R. A.
Kraenkel
, and
A. M.
Kamchatnov
,
Phys. Rev. A
76
,
053813
(
2007
).
17.
J. G.
Esler
and
J. D.
Pearce
,
J. Fluid Mech.
667
,
555
(
2011
).
18.
N. K.
Lowman
and
M. A.
Hoefer
,
J. Fluid Mech.
718
,
524
(
2013
).
19.
M. A.
Hoefer
,
J. Nonlinear Sci.
24
,
525
(
2014
).
20.
T.
Congy
,
A. M.
Kamchatnov
, and
N.
Pavloff
,
SciPost Phys.
1
,
6
(
2016
).
21.
M. A.
Hoefer
,
G. A.
El
, and
A. M.
Kamchatnov
,
SIAM J. Appl. Math.
77
,
1352
(
2017
).
22.
X.
An
,
T. R.
Marchant
, and
N. F.
Smyth
,
Proc. R. Soc. Lond. A
474
,
20180278
(
2018
).
23.
T.
Congy
,
G. A.
El
, and
M. A.
Hoefer
,
J. Fluid Mech.
875
,
1145
(
2019
).
24.
M. D.
Maiden
,
D. V.
Anderson
,
N. A.
Franco
,
G. A.
El
, and
M. A.
Hoefer
,
Phys. Rev. Lett.
120
,
144101
(
2018
).
25.
H.
Lamb
,
Hydrodynamics
(
Cambridge University Press
,
Cambridge
,
1994
).
26.
G. G.
Stokes
,
Mathematical and Physical Papers
(
Cambridge University Press
,
Cambridge
,
1905
), Vol.
V
, p.
163
.
27.
O.
Akimoto
and
K.
Ikeda
,
J. Phys. A: Math. Gen.
10
,
425
(
1977
).
28.
S. A.
Darmanyan
,
A. M.
Kamchatnov
, and
M.
Neviére
,
Zh. Eksp. Teor. Fiz.
123
,
997
(
2003
) [Sov. Phys. JETP 96, 876 (2003)].
29.
A. M.
Kamchatnov
,
Phys. Rev. E
99
,
012203
(
2019
).
30.
A. M.
Kamchatnov
,
Teor. Mat. Fiz.
202
,
415
(
2020
) [Theor. Math. Phys. 202, 363 (2020)].
31.
E. A.
Kuznetsov
,
Phys. Lett. A
101
,
314
(
1984
).
32.
J. L.
Bona
,
P. E.
Souganidis
, and
W. A.
Strauss
,
Proc. R. Soc. Lond. A
411
,
395
(
1987
).
33.
R. L.
Pego
and
M. I.
Weinstein
,
Commun. Math. Phys.
164
,
305
(
1994
).
34.
35.
G. A.
El
,
R. H. J.
Grimshaw
, and
N. F.
Smyth
,
Physica D
237
,
2423
(
2008
).
36.
M. D.
Maiden
,
N. A.
Franco
,
E. G.
Webb
,
G. A.
El
, and
M. A.
Hoefer
,
J. Fluid Mech.
883
,
A10
(
2020
).
37.
G. V.
Potemin
,
Russ. Math. Surv.
43
,
39
(
1988
).
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