We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this “Kaup-Boussinesq model” for which a flat water surface is modulationally stable, we speak below of “positive dispersion” model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

1.
R.
Courant
and
K. O.
Friedrichs
,
Supersonic Flow and Shock Waves
(
Interscience Publishers, New York
,
1956
).
2.
Y. B.
Zel'dovich
and
Y. P.
Raizer
,
Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena
(
Dover Books on Physics
,
New York
,
2002
)
3.
A. M.
Kamchatnov
,
Nonlinear Periodic Waves and Their Modulations—An Introductory Course
(
World Scientific
,
Singapore
,
2000
).
4.
G. A.
El
and
M. A.
Hoefer
, “
Dispersive shock waves modulation theory
,”
Physica (Amsterdam)
333D
,
11
(
2016
).
5.
D. J.
Korteweg
and
G.
de Vries
, “
On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves
,”
Philos. Mag.
39
,
422
443
(
1895
).
6.
T. B.
Benjamin
and
M. J.
Lighthill
, “
On cnoidal waves and bores
,”
Proc. R. Soc. London, A
224
,
448
(
1954
).
7.
G. B.
Whitham
, “
Non-linear dispersive waves
,”
Proc. R. Soc. London, A
283
,
238
261
(
1965
).
8.
A. V.
Gurevich
and
L. P.
Pitaevskii
, “
Nonstationary structure of a collisionless shock wave
,”
Zh. Eksp. Teor. Fiz.
65
,
590
604
(
1973
)
A. V.
Gurevich
and
L. P.
Pitaevskii
, [
Sov. Phys.-JETP
38
,
291
297
(
1974
)].
9.
J.
Boussinesq
, “
Essai sur la théorie des eaux courantes
,”
Mém. Prés. Div. Sav. Acad. Sci. Inst. Fr.
23
,
1
(
1877
).
10.
D. J.
Kaup
, “
A higher-order water-wave equation and the method for solving it
,”
Prog. Theor. Phys.
54
,
396
(
1975
).
11.
V. B.
Matveev
and
M. I.
Yavor
, “
Solutions presque périodiques er a N-solitons de l'équation hydrodynamique non linéaire de Kaup
,”
Ann. Inst. Henry Poincaré
31
,
25
(
1979
).
12.
G. A.
El
,
R. H. J.
Grimshaw
, and
M. V.
Pavlov
, “
Integrable shallow-water equations and undular bores
,”
Stud. Appl. Math.
106
,
157
(
2001
).
13.
G. A.
El
,
R. H. J.
Grimshaw
, and
A. M.
Kamchatnov
, “
Wave breaking and the generation of undular bores in an integrable shallow water system
,”
Stud. Appl. Math.
114
,
395
(
2005
).
14.
G. A.
El
,
R. H. J.
Grimshaw
, and
A. M.
Kamchatnov
, “
Analytic model for a weakly dissipative shallow-water undular bore
,”
Chaos
15
,
037102
(
2005
).
15.

The negative dispersion in Eq. (1) corresponds to a “+” sign in front of the third derivative term in the first equation of the system (3).

16.
S. K.
Ivanov
 et al, “
The Riemann problem for polarization waves in a two-component Bose-Einstein condensate,”
(unpublished).
17.
A. V.
Gurevich
and
A. L.
Krylov
, “
Dissipationless shock waves in media with positive dispersion
,”
Sov. Phys. JETP
65
,
944
953
(
1987
).
18.
G. A.
El
,
V. V.
Geogjaev
,
A. V.
Gurevich
, and
A. L.
Krylov
, “
Decay of an initial discontinuity in the defocusing NLS hydrodynamics
,”
Physica D
87
,
186
192
(
1995
).
19.
s1=i=14λi,s2=i<jλiλj,s3=i<j<kλiλjλk and s4=Πi=14λi.
20.
A. M.
Kamchatnov
,
Y.-H.
Kuo
,
T.-C.
Lin
,
T.-L.
Horng
,
S.-C.
Gou
,
R.
Clift
,
G. A.
El
, and
R. H. J.
Grimshaw
, “
Undular bore theory for the Gardner equation
,”
Phys. Rev. E
86
,
036605
(
2012
).
21.
T.
Congy
,
A. M.
Kamchatnov
, and
N.
Pavloff
, “
Dispersive hydrodynamics of nonlinear polarization waves in two-component Bose-Einstein condensates
,”
SciPost Phys.
1
,
006
(
2016
).
22.
A. V.
Gurevich
and
A. P.
Meshcherkin
, “
Expanding self-similar discontinuities and shock waves in dispersive hydrodynamics
,”
Zh. Eksp. Teor. Fiz.
87
,
1277
1292
(
1984
)
A. V.
Gurevich
and
A. P.
Meshcherkin
, [
Sov. Phys. JETP
60
,
732
740
(
1984
)].
23.
G. A.
El
, “
Resolution of a shock in hyperbolic systems modified by weak dispersion
,”
Chaos
15
,
037103
(
2005
).
24.
R. F.
Bikbaev
, “
Finite-gap attractors and transition processes of the shock-wave type in integrable systems
,”
Zapadn. Nauchno. Semin. POMI
199
,
25
(
1992
)
R. F.
Bikbaev
, [
J. Math. Sci.
77
,
3033
(
1995
)].
25.
E.
Iacocca
,
Th.
Silva
, and
M. A.
Hoefer
, “
Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films
,”
Phys. Rev. Lett.
118
,
017203
(
2017
).
You do not currently have access to this content.