# On the Construction of a Modulating Multiphase Wavetrain for a Perturbed Kdv Equation

## Abstract

This paper summarizes the status of a direct construction of an asymptotic representation of a modulating multiphase wavetrain for a class of perturbed kdV equations. This class includes the kdV-Burgers equation. The calculations apply on a “boundary” between dispersive and dissipative behavior. The construction proceeds by standard asymptotic methods. The result of the construction is an invariant representation of the reduced equations which permits their diagonalization. While mathematically the construction is incomplete, care is taken to identify the mathematical status of each step in the construction. The equivalence of this constructive approach with postulated averages of conservation laws is established for two phase waves. Finally, the Young measure for this program is constructed explicitly.

## Keywords

Weak Limit Modulation Equation Phase Wave Young Measure Periodic Travel Wave Solution## Preview

Unable to display preview. Download preview PDF.

## References

- 1.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, pp. 739–784.MathSciNetADSzbMATHCrossRefGoogle Scholar
- 2.D.W. McLaughlin, “Modulations of Kdv Wavetrains,” Physica D3, 1981, pp. 335–343.ADSGoogle Scholar
- 3.M.G. Forest and D.W. McLaughlin, “Modulations of Perturbed KdV Wavetrains,” SIAM J. Appl. Math, 44, 1984, 287–300.MathSciNetADSzbMATHCrossRefGoogle Scholar
- 4.M.G. Forest and D.W. McLaughlin, “Modulations of sinh-Gordon and sine-Gordon wavetrains,” Stud. Appl. Math. 68, 1983, pp. 11–59.MathSciNetzbMATHGoogle Scholar
- 5.N. Ercolani, M.G. Forest, and D.W. McLaughlin, “Modulational stability of two phase sine-Gordon wavetrains,” Stuc. Appl. Math. 71, 1984, 91–101.MathSciNetADSzbMATHGoogle Scholar
- 6.A.V. Gurevich and L.P. Pitaevskii, “Nonstationary Structure of a Collionsless Shock Wave,” Sov. Phys. JETP 38, 1974.Google Scholar
- 7.B. Fornberg and G.B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Phil. Trans. Roy. Soc. Lond. 289, 1978, pp. 373–404.MathSciNetADSzbMATHCrossRefGoogle Scholar
- 8.P.D. Lax and C.D. Levermore, “Zero dispersion limit for the KdV equation,” Proc. Nat. Acad. Science (U.S.A.), 1979. Also, Comm. Pure. Appl. Math. 36, 1983, 253–290. Comm. Pure Appl. Math. 36, 1983, 571–594. Also Comm. Pure Appl. Math. 36, 1983, 809–829.Google Scholar
- 9.S. Venekides, Thesis, New York University, 1982.Google Scholar
- 10.M.E. Schonbek, “Convergence of solutions to nonlinear dispersive equations,” preprint, U. Rhode Island, 1981.Google Scholar
- 11.G.B. Whithan, “Nonlinear dispersive waves,” Proc. Roy. Soc. A 283, 1965, pp. 238–261.ADSCrossRefGoogle Scholar
- 12.G.B. Whithan, Linear and Nonlinear Dispersive Waves, Wiley-Interscience, New York, 1974.Google Scholar
- 13.R. Miuraadn M. Kruskal, “Application of a nonlinear WKB method to the Korteweg de Vries equation,” SIAM J. Appl. Math. 26, 1974, pp. 376–395.MathSciNetCrossRefGoogle Scholar
- 14.J.C. Luke, “A perturbation method for nonlinear dispersive wave problems,” Proc. Roy. Soc. A 292, 1966 (403–412).MathSciNetADSGoogle Scholar
- 15.M.J. Albowitz and D.J. Benney, “The evolution of multi-phase modes for nonlinear dispersive waves,” Stud. Appl. Math. 49, 1970, pp. 225–238.Google Scholar
- 16.Jiminez, Thesis, Cal Tech, 1973.Google Scholar
- 17.E.N. Pelinovsky and S. Kh. Sharvatsky, “Breaking of stationary waves in nonlinear dispersive media,” Physica D 3, 1980, pp. 317–328.ADSCrossRefGoogle Scholar
- 18.R. Rosales, private notes.Google Scholar
- 19.L. Tartar, “Compensated compactness and applications to partial differential equations,” Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol IV, Research Notes in Mathematics 39, R.S. Knops, Ed., Pitman Publishing, 1979.Google Scholar
- 20.R. Diperna, “Measure valued solutions to conservation laws,” Duke Univ. Preprint (1984).Google Scholar
- 21.H.P. McKean and P. van Moerbeke, “The spectrum of Hill’s equation,” Invent. Math. 30, 1975, 11, 217–274.MathSciNetADSzbMATHCrossRefGoogle Scholar
- 22.I.M. Gel’fand and L.A. Dikii, “Integrable nonlinear equations and the Liouville theorem,” Funkt. Analiz. Egr. Prilozheniya 13, 1979, pp. 8–20.MathSciNetzbMATHGoogle Scholar
- 23.S. Venekides, these proceedings.Google Scholar
- 24.N. Ercolani, M. Forest, and D.W. McLaughlin, “Oscillations and Instabilities in Near Integrable PDE’s” proc. of Sante Fe Conference on Evolution Equation, 1985 (to appear).Google Scholar