Skip to main content
SpringerLink
Log in

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

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The third-order nonlinear dispersion PDE, as the key model,

$$ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + $$
((0.1))

is studied. Two Riemann’s problems for (0.1) with the initial data S ∓(x) = ∓ sgn.x create shock (u(x, t) ≡ S −(x)) and smooth rarefaction (for the data S +) waves (see [16]). The concept of “δ-entropy” solutions and others are developed for establishing the existence and uniqueness for (0.1) by using stable smooth δ-deformations of shock-type solutions. These are analogous to entropy theory for scalar conservation laws such as u t + uu x = 0, which were developed by Oleinik and Kruzhkov (in x ∊ ℝN) in the 1950s–1960s. The Rosenau-Hyman K(2, 2) (compacton) equation

$$ u_t = (uu_x )_{xx} + 4uu_x , $$

which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions are shown to be δ-entropy. Shock and rarefaction waves are discussed for other NDEs such as

$$ u_t = (u^2 u_x )_{xx} ,u_{tt} = (uu_x )_{xx} ,u_{tt} = uu_x ,u_{ttt} = (uu_x )_{xx} ,u_t = (uu_x )_{xxxxx} ,etc. $$

.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem (Oxford Univ. Press, Oxford, 2000).

    MATH  Google Scholar 

  2. W. Craig and J. Goodman, “Linear Dispersive Equations of Airy Type,” J. Differ. Equations 87, 38–61 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  3. W. Craig, T. Kappeler, and W. Strauss, “Gain of Regularity for Equations of KdV Type,” Ann. Inst. H. Poincare 9, 147–186 (1992).

    MATH  MathSciNet  Google Scholar 

  4. G. Da Prato and P. Grisvard, “Equation d’évolutions abstraites de type parabolique,” Ann. Mat. Pura Appl., IV(120), 329–396 (1979).

    Google Scholar 

  5. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer-Verlag, Berlin, 1999).

    Google Scholar 

  6. L. L. Dawson, “Uniqueness Properties of Higher Order Dispersive Equations,” J. Differ. Equations 236, 199–236(2007).

    Article  MATH  Google Scholar 

  7. Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, and S. I. Pohozaev, “Asymptotic Behavior of Global Solutions to Higher-Order Semilinear Parabolic Equations in the Supercritical Range,” Adv. Differ. Equations 9, 1009–1038(2004).

    MATH  MathSciNet  Google Scholar 

  8. J. Escher and G. Prokert, “Analyticity of Solutions to Nonlinear Parabolic Equations on Manifolds and an Application to Stokes Flow,” J. Math. Fluid Mech. 8, 1–35 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  9. J. D. Evans, V. A. Galaktionov, and J. R. King, “Source-Type Solutions of the Fourth-Order Unstable Thin Film Equation,” Eur. J. Appl. Math. 18, 273–321 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. D. Evans, V. A. Galaktionov, and J. R. King, “Unstable Sixth-Order Thin Film Equation: I. Blow-up Similarity Solutions; II. Global Similarity Patterns,” Nonlinearity 20, 1799–1841, 1843–1881 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. V. Faminskii, “On the Mixed Problem for Quasilinear Equations of the Third Order,” J. Math. Sci. 110, 2476–2507 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. V. A. Galaktionov, “Sturmian Nodal Set Analysis for Higher-Order Parabolic Equations and Applications,” Adv. Differ. Equations 12, 669–720 (2007).

    MATH  MathSciNet  Google Scholar 

  13. V. A. Galaktionov, “On Higher-Order Viscosity Approximations of Odd-Order Nonlinear PDEs,” J. Eng. Math. 60,173–208 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  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) [Comp. Math. Math. Phys. 48 (10), (2008)].

  17. 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).

    MATH  Google Scholar 

  18. I. M. Gel’fand, “Some Problems in the Theory of Quasilinear Equations,” Usp. Mat. Nauk. 14, 87–158 (1959) [Am. Math. Soc. Transl. 29, 295–381 (1963)].

    MATH  Google Scholar 

  19. T. Hoshiro, “Mouree’s Method and Smoothing Properties of Dispersive Equations,” Commun. Math. Phys. 202,255–265 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  20. M. A. Krasnosel’skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis (Fizmatgiz, Moscow, 1975; Springer-Verlag, Berlin, 1984).

    Google Scholar 

  21. S. N. Kruzhkov, “First-Order Quasilinear Equations in Several Independent Variables,” Mat. Sb. 10, 217–243(1970).

    Article  MATH  Google Scholar 

  22. N. A. Larkin, “Modified KdV Equation with a Source Term in a Bounded Domain,” Math. Methods Appl. Sci. 29,751–765 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  23. J. L. Levandosky, “Smoothing Properties of Nonlinear Dispersive Equations in Two Spatial Dimensions,” J. Differ. Equations 175, 275–372 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  24. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires (Dunod, Paris, 1969).

    MATH  Google Scholar 

  25. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems (Birkhäuser, Basel, 1995).

    MATH  Google Scholar 

  26. L. Ljusternik and V. Sobolev, Elements of Functional Analysis (Ungar, New York, 1961).

    Google Scholar 

  27. R. Mizuhara, “The Initial Value Problem for Third-and Fourth-Order Dispersive Equations in One Space Dimension,” Funk. Ekvacioj 49, 1–38 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  28. M. A. Naimark, Linear Differential Operators, Part I (Ungar, New York, 1968).

    Google Scholar 

  29. O. A. Oleinik, “Discontinuous Solutions of Nonlinear Differential Equations,” Usp. Mat. Nauk. 12, 3–73 (1957) [Am. Math. Soc. Transl. 26, 95–172 (1963)].

    MathSciNet  Google Scholar 

  30. O. A. Oleinik, “Uniqueness and Stability of the Generalized Solution of the Cauchy Problem for a Quasilinear Equation,” Usp Mat. Nauk. 14, 165–170 (1959) [Am. Math. Soc. Transl. 33, 285–290 (1963)].

    MathSciNet  Google Scholar 

  31. P. Rosenau and J. M. Hyman, “Compactons: Solitons with Finite Wavelength,” Phys. Rev. Lett. 70, 564–567(1993).

    Article  MATH  Google Scholar 

  32. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York, 1983).

    MATH  Google Scholar 

  33. H. Takuwa, “Microlocal Analytic Smoothing Effects for Operators of Real Principal Type,” Osaka J. Math. 43,13–62 (2006).

    MATH  MathSciNet  Google Scholar 

  34. T. Tao, “Multilinear Weighted Convolution of L 2 Functions, and Applications to Nonlinear Dispersive Equations,” Am. J. Math. 6, 839–908 (2000).

    Google Scholar 

  35. M. E. Taylor, Partial Differential Equations, Vol. III: Nonlinear Equations (Springer-Verlag, Now York, 1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Math, BA2 7AY, UK

    V. A. Galaktionov

Authors
  1. V. A. Galaktionov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Galaktionov.

Additional information

Dedicated to the memory of Professors O.A. Oleinik and S.N. Kruzhkov

This article was submitted by the author in English.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Galaktionov, V.A. Nonlinear dispersion equations: Smooth deformations, compactions, and extensions to higher orders. Comput. Math. and Math. Phys. 48, 1823–1856 (2008). https://doi.org/10.1134/S0965542508100084

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542508100084

Keywords

Search

Navigation