The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data

Venakides, Stephanos

doi:10.1090/S0002-9947-1987-0879569-7

The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data
HTML articles powered by AMS MathViewer

by Stephanos Venakides PDF

Trans. Amer. Math. Soc. 301 (1987), 189-226 Request permission

Abstract:

We study the initial value problem for the Korteweg-de Vries equation \[ ({\text {i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0\] in the limit of small dispersion, i.e., $\varepsilon \to 0$. When the unperturbed equation \[ ({\text {ii}})\quad {u_t} - 6u{u_x} = 0\] develops a shock, rapid oscillations arise in the solution of the perturbed equation (i) In our study: a. We compute the weak limit of the solution of (i) for periodic initial data as $\varepsilon \to 0$. b. We show that in the neighborhood of a point $(x, t)$ the solution $u(x, t, \varepsilon )$ can be approximated either by a constant or by a periodic or by a quasiperiodic solution of equation (i). In the latter case the associated wavenumbers and frequencies are of order $O(1/\varepsilon )$. c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of $x$ and $t$. d. We explain the mechanism of the generation of oscillatory phases. Our computations in a and c are subject to the solution of the Lax-Levermore evolution equations (7.7). Our results in b-d rest on a plausible averaging assumption.

References

Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 538168

Periodic multisolitons of the Korteweg-de Vries equation and Toda lattice

B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Uspehi Mat. Nauk 31 (1976), no. 1(187), 55–136 (Russian). MR 0427869
H. Flaschka, M. G. Forest, and D. W. McLaughlin, Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 33 (1980), no. 6, 739–784. MR 596433, DOI 10.1002/cpa.3160330605
Joseph B. Keller, Discriminant, transmission, coefficient, and stability bands of Hill’s equation, J. Math. Phys. 25 (1984), no. 10, 2903–2904. MR 760562, DOI 10.1063/1.526036
Peter D. Lax and C. David Levermore, The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), no. 3, 253–290. MR 697466, DOI 10.1002/cpa.3160360302
H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143–226. MR 427731, DOI 10.1002/cpa.3160290203
H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321–337. MR 430403, DOI 10.1002/cpa.3160300305

Asymptotic behavior of stability regions for Hill’s equation

Stephanos Venakides, The zero dispersion limit of the Korteweg-de Vries equation for initial potentials with nontrivial reflection coefficient, Comm. Pure Appl. Math. 38 (1985), no. 2, 125–155. MR 780069, DOI 10.1002/cpa.3160380202
Stephanos Venakides, The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 883–909. MR 812354, DOI 10.1002/cpa.3160380616
G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954

On the construction of modulating multiphase wavetrains

M. Gregory Forest and David W. McLaughlin, Modulations of sinh-Gordon and sine-Gordon wavetrains, Stud. Appl. Math. 68 (1983), no. 1, 11–59. MR 686249, DOI 10.1002/sapm198368111
M. Gregory Forest and David W. McLaughlin, Modulations of perturbed KdV wavetrains, SIAM J. Appl. Math. 44 (1984), no. 2, 287–300. MR 739305, DOI 10.1137/0144021
Nicholas Ercolani, M. Gregory Forest, and David W. McLaughlin, Modulational stability of two-phase sine-Gordon wavetrains, Stud. Appl. Math. 71 (1984), no. 2, 91–101. MR 760226, DOI 10.1002/sapm198471291
Maria Elena Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982), no. 8, 959–1000. MR 668586, DOI 10.1080/03605308208820242
Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI 10.1007/BF00752112
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398

The small dispersion limit of the Korteweg-de Vries equation: Multiwell initial data

Stephanos Venakides, The small dispersion limit of the Korteweg-de Vries equation, Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986) Math. Sci. Res. Inst. Publ., vol. 7, Springer, New York, 1987, pp. 295–336. MR 920841, DOI 10.1007/978-1-4613-9583-6_{1}2