Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves
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J. L. Bona and F. B. Weissler, “Blowup of Spatially Periodic Complex-Valued Solutions of Nonlinear Dispersive Equations,” Indiana Univ. Math. J. 50, 759–782 (2001).
A. Bressan, Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem (Oxford Univ. Press, Oxford, 2000).
P. A. Clarkson, A. S. Fokas, and M. Ablowitz, “Hodograph Transformations of Linearizable Partial Differential Equations,” SIAM J. Appl. Math. 49, 1188–1209 (1989).
G. M. Coclite and K. H. Karlsen, “On the Well-Posedness of the Degasperis-Procesi Equation,” J. Funct. Anal. 233, 60–91 (2006).
W. Craig, T. Kappeler, and W. Strauss, “Gain of Regularity for Equations of KdV Type,” Ann. Inst. H. Poincaré 9, 147–186 (1992).
C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer-Verlag, Berlin, 1999).
B. Dey, “Compacton Solutions for a Class of Two Parameter Generalized Odd-Order Korteweg-De Vries Equations,” Phys. Rev. E 57, 4733–4738 (1998).
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications (Chapman and Hall/CRC, Boca Raton, FL, 2004).
V. A. Galaktionov, “Sturmian Nodal Set Analysis for Higher-Order Parabolic Equations and Applications,” Adv. Differ. Equations 12, 669–720 (2007).
V. A. Galaktionov, “On Higher-Order Viscosity Approximations of Odd-Order Nonlinear PDEs,” J. Eng. Math. 60, 173–208 (2008).
V. A. Galaktionov, “Shock Waves and Compactons for Fifth-Order Nonlinear Dispersion Equations,” Eur. J. Appl. Math. (submitted).
V. A. Galaktionov and S. I. Pohozaev, “Existence and Blowup for Higher-Order Semilinear Parabolic Equations: Majorizing Order-Preserving Operators,” Indiana Univ. Math. J. 51, 1321–1338 (2002).
V. A. Galaktionov and S. I. Pohozaev, “Blowup for Nonlinear Initial-Boundary Value Problems,” Dokl. Akad. Nauk 412, 444–447 (2007) [Dokl. Math. 75, 76–79 (2007)].
V. A. Galaktionov, “Nonlinear Dispersion Equations: Smooth Deformations, Compactons, and Generalizations to High Orders,” Zh. Vychisl. Mat. Mat. Fiz. 48, (2008) [Comput. Math. Math. Phys. 48, (2008)].
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, Boca Raton, FL, 2007).
V. A. Galaktionov and J. L. Vazquez, A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach (Birkhäuser, Boston, 2004).
I. M. Gel’fand, “Some Problems in the Theory of Quasilinear Equations,” Usp. Mat. Nauk 14, 87–158 (1959).
J. M. Hyman and P. Rosenau, “Pulsating Multiplet Solutions of Quintic Wave Equations,” Phys. D 123, 502–512 (1998).
M. Inc, “New Compacton and Solitary Pattern Solutions of the Nonlinear Modified Dispersive Klein-Gordon Equations,” Chaos Solitons Fractals 33, 1275–1284 (2007).
S. Kawamoto, “An Exact Transformation from the Harry Dym Equation to the Modified KdV Equation,” J. Phys. Soc. Jpn. 54, 2055–2056 (1985).
S. N. Kruzhkov, “First-Order Quasilinear Equations in Several Independent Variables,” Mat. Sb. 10, 217–243 (1970).
J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires (Dunod, Paris, 1969).
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems (Birkhäuser, Basel, 1995).
O. A. Oleinik, “Discontinuous Solutions of Nonlinear Differential Equations,” Usp. Mat. Nauk 12, 3–73 (1957).
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).
A. V. Porubov and M. G. Velarde, “Strain Kinks in an Elastic Rod Embedded in a Viscoelastic Medium,” Wave Motion 35, 189–204 (2002).
P. Rosenau, “On a Class of Nonlinear Dispersive-Dissipative Interactions,” Phys. D 123, 525–546 (1998).
P. Rosenau and J. M. Hyman, “Compactons: Solitons with Finite Wavelength,” Phys. Rev. Lett. 70, 564–567 (1993).
P. Rosenau and S. Kamin, “Thermal Waves in an Absorbing and Convecting Medium,” Phys. D 8, 273–283 (1983).
P. Rosenau and D. Levy, “Compactons in a Class of Nonlinearly Quintic Equations,” Phys. Lett. A 252, 297–306 (1999).
J. Shen, W. Xu, and W. Li, “Bifurcation of Traveling Wave Solutions in a New Integrable Equation with Peakon and Compactons,” Chaos Solitons Fractals 27, 413–425 (2006).
H. Takuwa, “Microlocal Analytic Smoothing Effects for Operators of Real Principal Type,” Osaka J. Math. 43, 13–62 (2006).
Z. Yan, “Constructing Exact Solutions for Two-Dimensional Nonlinear Dispersion Boussinesq Equation: II. Solitary Pattern Solutions,” Chaos Solitons Fractals 18, 869–880 (2003).