Nonlinear dispersion equations: Smooth deformations, compactions, and extensions to higher orders

Computational Mathematics and Mathematical Physics

Volumn 48, Issue 10, 2008, Pages 1823-1856

  Galaktionov, V A ^a  

^a University of Bath   (United Kingdom)

Author keywords

Entropy solutions; Odd order quasi linear PDE; Self similar patterns; Shock and rarefaction waves

Indexed keywords

EID: 54249107908     PISSN: 09655425     EISSN: 15556662     Source Type: Journal    
DOI: 10.1134/S0965542508100084     Document Type: Article

References (36)

1. A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem (Oxford Univ. Press, Oxford, 2000).
2. W. Craig J. Goodman 1990 Linear Dispersive Equations of Airy Type J. Differ. Equations 87 38 61
3. W. Craig T. Kappeler W. Strauss 1992 Gain of Regularity for Equations of KdV Type Ann. Inst. H. Poincare 9 147 186
4. G. Da Prato P. Grisvard 1979 Equation d'évolutions abstraites de type parabolique Ann. Mat. Pura Appl. IV 120 329 396
5. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer-Verlag, Berlin, 1999).
6. L. L. Dawson 2007 Uniqueness Properties of Higher Order Dispersive Equations J. Differ. Equations 236 199 236
7. Yu. V. Egorov V. A. Galaktionov V. A. Kondratiev S. I. Pohozaev 2004 Asymptotic Behavior of Global Solutions to Higher-Order Semilinear Parabolic Equations in the Supercritical Range Adv. Differ. Equations 9 1009 1038
8. J. Escher G. Prokert 2006 Analyticity of Solutions to Nonlinear Parabolic Equations on Manifolds and an Application to Stokes Flow J. Math. Fluid Mech. 8 1 35
9. J. D. Evans V. A. Galaktionov J. R. King 2007 Source-Type Solutions of the Fourth-Order Unstable Thin Film Equation Eur. J. Appl. Math. 18 273 321
10. J. D. Evans V. A. Galaktionov J. R. King 2007 Unstable Sixth-Order Thin Film Equation: I. Blow-up Similarity Solutions; II. Global Similarity Patterns Nonlinearity 20 1799 1841
11. A. V. Faminskii 2002 On the Mixed Problem for Quasilinear Equations of the Third Order J. Math. Sci. 110 2476 2507
12. V. A. Galaktionov 2007 Sturmian Nodal Set Analysis for Higher-Order Parabolic Equations and Applications Adv. Differ. Equations 12 669 720
13. V. A. Galaktionov 2008 On Higher-Order Viscosity Approximations of Odd-Order Nonlinear PDEs J. Eng. Math. 60 173 208
14. V. A. Galaktionov, "Shock Waves and Compactions for Fifth-Order Nonlinear Dispersion Equations," Eur. J. Appl. Math. (submitted).
15. V. A. Galaktionov and S. I. Pohozaev, "Refined Complicated Structure of Similarity Solutions of Higher-Order Parabolic, Hyperbolic, and Nonlinear Dispersion PDEs: An Analytic-Numerical Approach," Izv. Russ. Acad. Sci. (submitted).
16. V. A. Galaktionov and S. I. Pohozaev, "Third-Order Nonlinear Dispersion Equations: Shocks, Rarefaction, and Blowup Waves," Vychisl. Mat. Mat. Fiz. 48(10), (2008)
17. [Comp. Math. Math. Phys. 48 (10), (2008)].
18. V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman and Hall/CRC, Florida, Boca Raton, FL, 2007).
19. I. M. Gel'fand 1959 Some Problems in the Theory of Quasilinear Equations Usp. Mat. Nauk. 14 87 158 [Am. Math. Soc. Transl. 29, 295-381 (1963)]
20. T. Hoshiro 1999 Mouree's Method and Smoothing Properties of Dispersive Equations Commun. Math. Phys. 202 255 265
21. M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis (Fizmatgiz, Moscow, 1975; Springer-Verlag, Berlin, 1984).
22. S. N. Kruzhkov 1970 First-Order Quasilinear Equations in Several Independent Variables Mat. Sb. 10 217 243
23. N. A. Larkin 2006 Modified KdV Equation with a Source Term in a Bounded Domain Math. Methods Appl. Sci. 29 751 765
24. J. L. Levandosky 2001 Smoothing Properties of Nonlinear Dispersive Equations in Two Spatial Dimensions J. Differ. Equations 175 275 372
25. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires (Dunod, Paris, 1969).
26. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems (Birkhäuser, Basel, 1995).
27. L. Ljusternik and V. Sobolev, Elements of Functional Analysis (Ungar, New York, 1961).
28. R. Mizuhara 2006 The Initial Value Problem for Third-and Fourth-Order Dispersive Equations in One Space Dimension Funk. Ekvacioj 49 1 38
29. M. A. Naimark, Linear Differential Operators, Part I (Ungar, New York, 1968).
30. O. A. Oleinik 1957 Discontinuous Solutions of Nonlinear Differential Equations Usp. Mat. Nauk. 12 3 73 [Am. Math. Soc. Transl. 26, 95-172 (1963)]
31. O. A. Oleinik 1959 Uniqueness and Stability of the Generalized Solution of the Cauchy Problem for a Quasilinear Equation Usp Mat. Nauk. 14 165 170 [Am. Math. Soc. Transl. 33, 285-290 (1963)]
32. P. Rosenau J. M. Hyman 1993 Compactons: Solitons with Finite Wavelength Phys. Rev. Lett. 70 564 567
33. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983).
34. H. Takuwa 2006 Microlocal Analytic Smoothing Effects for Operators of Real Principal Type Osaka J. Math. 43 13 62
35. 2 Functions, and Applications to Nonlinear Dispersive Equations Am. J. Math. 6 839 908
36. M. E. Taylor, Partial Differential Equations, Vol. III: Nonlinear Equations (Springer-Verlag, Now York, 1996).