Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-13T13:30:27.114Z Has data issue: false hasContentIssue false
  • English
  • Français

The initial-value problem for the forced Korteweg-de Vries equation*

Published online by Cambridge University Press:  14 November 2011

Jerry L. Bona  and
Bing-Yu Zhang
Jerry L. Bona
Affiliation:
Department of Mathematics and Computational & Applied Mathematics Program, University of Texas, Austin, TX 78712, U.S.A.
Bing-Yu Zhang
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gained prominence as a model for a number of interesting physical situations. At the same time, the modern theory for the initial-value problem for the unforced Korteweg-de Vries equation has taken great strides forward. The mathematical theory pertaining to the forced equation is currently set in narrow function classes and has not kept up with recent advances for the homogeneous equation. This aspect is rectified here with the development of a theory for the initial-value problem for the forced Korteweg-de Vries equation that entails weak assumptions on both the initial wave configuration and the forcing. The results obtained include analytic dependence of solutions on the auxiliary data and allow the external forcing to lie in function classes sufficiently large that a Dirac δ-function or its derivative is included. Analyticity is proved by an infinite-dimensional analogue of Picard iteration. A consequence is that solutions may be approximated arbitrarily well on any bounded time interval by solving a finite number of linear initial-value problems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 571 - 598
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Akylas, T. R.. On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141 (1984), 455–66.CrossRefGoogle Scholar
2Benjamin, T. B., Bona, J. L. and Mahony, J. J.. Model equations for long waves in nonlinear, dispersive media. Philos. Trans. Roy. Soc. London. Ser. A 272 (1972), 47–78.Google Scholar
3Bona, J. L.. Convergence of periodic wavetrains in the limit of large wavelength. Appl. Sci. Res. 37 (1981), 21–30.CrossRefGoogle Scholar
4Bona, J. L. and Smith, R.. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555–601.Google Scholar
5Bona, J. L. and Scott, L. R.. Solutions of the Korteweg-de Vries equations in fractional order Sobolev spaces. Duke Math. J. 43 (1976), 87–99.CrossRefGoogle Scholar
6Bourgain, J.. Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part I: Schrödinger equations. Geom. Fund. Anal. 3 (1993), 107–56.CrossRefGoogle Scholar
7Bourgain, J.. Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear equations, part II; the KdV equation. Geom. Fund. Anal. 3 (1993), 209–62.CrossRefGoogle Scholar
8Cohen, A.. Solutions of the Korteweg-de Vries equation from irregular data. Duke Math. J. 45 (1978), 149–81.Google Scholar
9Cole, S. L.. Transient waves produced by flow past a bump. Wave Motion 7 (1986), 579–87.CrossRefGoogle Scholar
10Constantin, P. and Saut, J.-C.. Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1 (1988), 413–46.CrossRefGoogle Scholar
11Craig, W., Kappeler, T. and Strauss, W. A.. Gain of regularity for equations of the Korteweg-de Vries type. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 147–86.CrossRefGoogle Scholar
12Ginibre, J. and Tsutsumi, Y.. Uniqueness for the generalized Korteweg-de Vries equations. SIAM J. Math. Anal. 20 (1989), 1388–425.CrossRefGoogle Scholar
13Ginibre, J. and Velo, G.. Smoothing properties and retarded estimates for some dispersive evolution equations (Preprint).Google Scholar
14Grimshaw, R. H. J. and Smyth, N.. Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169(1986), 429–64.CrossRefGoogle Scholar
15Kato, T.. Quasilinear equations of evolutions, with applications to partial differential equation. Lecture Notes in Math. 449 (1975), 27–50.Google Scholar
16Kato, T.. On the Korteweg-de Vries equation. Manuscripta Math. 29 (1979), 89–99.CrossRefGoogle Scholar
17Kato, T.. On the Cauchy problem for the (generalised) Korteweg–de Vries equations. Adv. Math. Suppl. Stud. Stud. Appl. Math. 8 (1983), 93–128.Google Scholar
18Kenig, C. E., Ponce, G. and Vega, L.. Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40 (1991), 33–69.CrossRefGoogle Scholar
19Kenig, C. E., Ponce, G. and Vega, L.. Well-posedness of the initial value problem for the KdV equation. J. Amer. Math. Soc. 4 (1991), 323–47.CrossRefGoogle Scholar
20Kenig, C. E., Ponce, G. and Vega, L.. Well-posedness and scattering results for the generalized Korteweg-de Vries equations via the concentration principle. Comm. Pure Appl. Math. 46 (1993), 527–620.Google Scholar
21Kenig, C. E., Ponce, G. and Vega, L.. The Cauchy problem for the Korteweg-de Vries equation in Sobolev of negative indices. Duke Math. J. 71 (1993), 1–21.CrossRefGoogle Scholar
22Kruzhkov, S. N. and Faminskii, A. V.. Generalized solutions of the Cauchy problem for the Kortewegde Vries equation. Math. U.S.S.R. Sb. 48 (1984), 93–138Google Scholar
23Lee, S.-J.. Generation of long water waves by moving disturbances (Ph.D. thesis, Carlifornia Institute of Technology, 1985).Google Scholar
24Miura, R. M.. The Korteweg-de Vries equation: A survey of results. SIAM Rev. 18 (1967), 412–59.Google Scholar
25Sachs, R. L.. Classical solutions of the Korteweg-de Vries equation for non-smooth initial data via inverse scattering. Comm. Partial Differential Equations 10 (1985), 29–89.CrossRefGoogle Scholar
26Saut, J.-C. and Temam, R.. Remarks on the Korteweg–de Vries equation. Israel J. Math. 24 (1976), 78–87.CrossRefGoogle Scholar
27Tsutsumi, Y.. The Cauchy problem for the Korteweg-de Vries equation with measures as initial data. SIAM J. Math. Anal. 20 (1989), 582–88.CrossRefGoogle Scholar
28Wu, T. Y.. Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184 (1987), 75–99.CrossRefGoogle Scholar
29Zhang, B.-Y.. Some results for the nonlinear dispersive wave equations with applications to control (Ph.D. thesis, University of Wisconsin-Madison, 1990).Google Scholar
30Zhang, B.-Y.. Taylor series expansion for solutions of the KdV equation with respect to their initial values. J. Fund. Anal. 129 (1995), 293–324.CrossRefGoogle Scholar
31Zhang, B.-Y.. A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain. Differential and Integral Equations 8 (1995), 1191–204.CrossRefGoogle Scholar