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On the formulation of mass, momentum and energy conservation in the KdV equation. (English) Zbl 1310.35206

Summary: The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable \(\eta\) representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities \(Q\), \(R\) and \(S\) used in the work of T. B. Benjamin and M. J. Lighthill [Proc. R. Soc. Lond., Ser. A 224, 448–460 (1954; Zbl 0055.45605)] on cnoidal waves and undular bores.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs

Citations:

Zbl 0055.45605
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Full Text: DOI

References:

[1] Ablowitz, M., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, vol. 4. SIAM, Philadelphia (1981) · Zbl 0472.35002 · doi:10.1137/1.9781611970883
[2] Ali, A., Kalisch, H.: Mechanical balance laws for Boussinesq models of surface water waves. J. Nonlinear Sci. 22, 371-398 (2012) · Zbl 1253.35113 · doi:10.1007/s00332-011-9121-2
[3] Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485-541 (2008) · Zbl 1131.76012 · doi:10.1007/s00222-007-0088-4
[4] Benjamin, T.B., Lighthill, M.J.: On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448-460 (1954) · Zbl 0055.45605 · doi:10.1098/rspa.1954.0172
[5] Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12, 283-318 (2002) · Zbl 1022.35044 · doi:10.1007/s00332-002-0466-4
[6] Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373-410 (2005) · Zbl 1108.76012 · doi:10.1007/s00205-005-0378-1
[7] Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55-108 (1872)
[8] Bridges, T.J.: Spatial Hamiltonian structure, energy flux and the water-wave problem. Proc. R. Soc. Lond. A 439, 297-315 (1992) · Zbl 0777.76012 · doi:10.1098/rspa.1992.0151
[9] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. Partial Differ. Equ. 10, 787-1003 (1985) · Zbl 0577.76030 · doi:10.1080/03605308508820396
[10] Craig, W., Groves, M.D.: Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367-389 (1994) · Zbl 0929.76015 · doi:10.1016/0165-2125(94)90003-5
[11] Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comp. Physiol. 108, 73-83 (1993) · Zbl 0778.76072 · doi:10.1006/jcph.1993.1164
[12] Dutykh, D., Dias, F.: Energy of tsunami waves generated by bottom motion. Proc. R. Soc. Lond. A 465, 725-744 (2009) · Zbl 1186.86004 · doi:10.1098/rspa.2008.0332
[13] Gardner, C.S., Green, J.M., Kruskal, M.D., Miura, R.M.: A method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095-1097 (1967) · Zbl 1061.35520 · doi:10.1103/PhysRevLett.19.1095
[14] Grimshaw, R.: Wave action and wave-mean-flow interaction with application to stratified shear flows. Annu. Rev. Fluid Mech. 16, 11-44 (1984) · Zbl 0599.76024 · doi:10.1146/annurev.fl.16.010184.000303
[15] Grimshaw, R., Joshi, N.: Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation. SIAM J. Appl. Math. 55, 124-135 (1995) · Zbl 0814.34043 · doi:10.1137/S0036139993243825
[16] Hayes, W.D.: Conservation of action and modal wave action. Proc. R. Soc. Lond. A 320, 187-208 (1970) · doi:10.1098/rspa.1970.0205
[17] Keulegan, G.H., Patterson, G.W.: Mathematical theory of irrotational translation waves. Nat. Bur. Standards J. Res. 24, 47-101 (1940) · Zbl 0061.45911 · doi:10.6028/jres.024.027
[18] Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave. Philos. Mag. 39, 422-443 (1895) · doi:10.1080/14786449508620739
[19] Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605-654 (2005) · Zbl 1069.35056 · doi:10.1090/S0894-0347-05-00484-4
[20] Lannes, D.: The Water Waves Problem. Mathematical Surveys and Monographs, vol. 188. Am. Math. Soc., Providence (2013) · Zbl 1410.35003
[21] Lannes, D., Bonneton, P.: Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21, 016601 (2009) · Zbl 1183.76294 · doi:10.1063/1.3053183
[22] Longuet-Higgins, M.S., Fenton, J.D.: On the mass, momentum, energy and circulation of a solitary wave. II. Proc. R. Soc. Lond. A 340, 471-493 (1974) · Zbl 0306.76025 · doi:10.1098/rspa.1974.0166
[23] Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9, 1204-1209 (1968) · Zbl 0283.35019 · doi:10.1063/1.1664701
[24] Schneider, G., Wayne, C.E.: The long-wave limit for the water wave problem. I. The case of zero surface tension. Commun. Pure Appl. Math. 53, 1475-1535 (2000) · Zbl 1034.76011 · doi:10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V
[25] Stoker, J.J.: Water Waves: The Mathematical Theory with Applications. Pure and Applied Mathematics, vol. 4. Interscience Publishers, New York (1957) · Zbl 0078.40805
[26] Whitham, G.B.: A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273-283 (1965) · Zbl 1467.35244 · doi:10.1017/S0022112065000745
[27] Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974) · Zbl 0373.76001
[28] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39-72 (1997) · Zbl 0892.76009 · doi:10.1007/s002220050177
[29] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445-495 (1999) · Zbl 0921.76017 · doi:10.1090/S0894-0347-99-00290-8
[30] Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190-194 (1968) · doi:10.1007/BF00913182
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