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Products of distributions and singular travelling waves as solutions of advection-reaction equations. (English) Zbl 1284.35112
Summary: We restrict our attention to the advection-reaction equation \(u_t+[\phi (u)]_x=\psi (u)\), where \(\phi\) and \(\psi\) are entire functions. Conditions for the propagation of a distributional wave profile are presented and the wave speed is evaluated. As an example, we prove that, under certain conditions, the propagation of delta-waves in models ruled by the diffusionless Burgers-Fisher equation is possible and compute the speeds of propagation of these waves. In the same setting, the propagation of travelling waves with the shape of a \(C^1\)-function with one jump discontinuity is also studied. These results will be easily explained by our theory of distributional products and are based on a rigorous and consistent concept of a solution that we have already introduced in previous works.

MSC:
35C07 Traveling wave solutions
35F25 Initial value problems for nonlinear first-order PDEs
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[1] A. Bressan and F. Rampazzo, ”On Differential Systems with Vector-Valued Impulsive Controls,” Boll. Unione Mat. Ital. 2B(7), 641–656 (1988). · Zbl 0653.49002
[2] J. F. Colombeau, New Generalized Functions and Multiplication of Distributions (North Holland, Amsterdam, 1985). · Zbl 0761.46021
[3] J. F. Colombeau, Elementary Introduction to New Generalized Functions (North Holland, Amsterdam, 1985). · Zbl 0584.46024
[4] J. F. Colombeau and A. Le Roux, ”Multiplication of Distributions in Elasticity and Hydrodynamics,” J. Math. Phys. 29, 315–319 (1988). · Zbl 0646.76007 · doi:10.1063/1.528069
[5] G. Dal Maso, P. LeFloch, and F. Murat, ”Definitions andWeak Stability of Nonconservative Products,” J. Math. Pures Appl. 74, 483–548 (1995). · Zbl 0853.35068
[6] V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, ”Algebras of Singularities of Singular Solutions to First-Order Quasi-Linear Strictly Hyperbolic Systems,” Theoret. Mat. Fiz. 114(1), 3–55 (1998) [English translation: Theoret. and Math. Phys. 114 (1), 1–42 (1998)]. · Zbl 0946.35049 · doi:10.4213/tmf827
[7] V. G. Danilov and V. M. Shelkovich, ”Generalized Solutions of Nonlinear Differential Equations and the Maslov Algebras of Distributions,” Integral Transforms Spec. Funct. 6(1–4), 171–180 (1998). · Zbl 0934.35089 · doi:10.1080/10652469808819161
[8] Yu. V. Egorov, ”On the Theory of Generalized Functions,” Uspekhi Mat. Nauk 45(5), 3–40 (1990) [English translation: Russian Math. Surveys 45 (5), 1–49 (1990)].
[9] A. L. Kay, D. J. Needham, and J. A. Leach, ”Travelling Waves for a Coupled Singular Reaction-Diffusion Systems Arising from a Model of Fractional Order Autocatalysis with Decay: I. Permanent Form Travelling Waves,” Nonlinearity 16(2), 735–770 (2003). · Zbl 1029.35133 · doi:10.1088/0951-7715/16/2/322
[10] R. J. LeVeque and H. C. Yee, ”A Study of Numerical Methods for Hyperbolic Conservation Laws with Stiff Source Terms,” J. Comp. Phys. 86, 187–210 (1990). · Zbl 0682.76053 · doi:10.1016/0021-9991(90)90097-K
[11] K. Lika and T. G. Hallan, ”Travelling Wave Solutions of a Nonlinear Reaction-Advection Equation,” J. Math. Biol. 38, 346–358 (1999). · Zbl 0921.92024 · doi:10.1007/s002850050152
[12] V. P. Maslov, ”Nonstandard Characteristics in Asymptotical Problems,” Uspekhi Mat. Nauk 38(6), 3–36 (1983) [English translation: Russian Math. Surveys 38 (6), 1–42 (1983)].
[13] V. P. Maslov, ”Nonstandard Characteristics in Asymptotical Problems,” Proc. International Congress of Mathematicians 1, 139–183 (Warsaw, 1983).
[14] V. P. Maslov, V. G. Danilov, and K. A. Volosov, Mathematical Modeling of Heat and Mass Transfer Processes (Nauka, Moscow, 1987) [English translation: Kluwer Academic Publishers Group, Dordrecht, 1995]. · Zbl 0645.73049
[15] V. P. Maslov and G. A. Omel’yanov, ”Asymptotic Soliton-Form Solutions of Equations with Small Dispersion,” Uspekhi Mat. Nauk 36(3), 63–126 (1981) [English translation: Russian Math. Surveys 36 (3), 73–149 (1981)].
[16] V. P. Maslov and V. A. Tsupin, ”Necessary Conditions for Existence of Infinitely Narrow Solitons in Gas Dynamics,” Dokl. Akad. Nauk SSSR 246(2), 298–300 (1979) [English translation: Soviet Phys. Dock. 24 (5), 354–356 (1979)].
[17] P. M. McCabe, J. A. Leach, and D. J. Needham, ”A Note on the Non-Existence of Permanent Form Travelling Wave Solutions in a Class of Singular Reaction-Diffusion Problems,” Dyn. Syst. 17(2), 131–135 (2002). · Zbl 1010.34020 · doi:10.1080/14689360110116498
[18] R. E. Mickens, ”A Nonstandard Finite Difference Scheme for the Diffusionless Burgers Equation with Logistic Reaction,” Math. Comput. Simulation 62, 117–124 (2003). · Zbl 1015.65036 · doi:10.1016/S0378-4754(02)00180-5
[19] R. Monneau and G. S. Weiss, ”Pulsating Travelling Waves in the Singular Limit of a Reaction-Diffusion System in Solid Combustion,” Ann. Inst. H. PoincarĂ©-AN 26(4), 1207–1222 (2009). · Zbl 1178.35121 · doi:10.1016/j.anihpc.2008.09.002
[20] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations (Longman Scientific & Technical, 1992). · Zbl 0818.46036
[21] E. E. Rosinger, Distributions and Nonlinear Partial Differential Equations (Lecture Notes Math. 684, Springer, Berlin, 1978). · Zbl 0469.35001
[22] E. E. Rosinger, Nonlinear Partial Differential Equations (Sequential and Week Solutions, North Holland, Amsterdam, 1980). · Zbl 0447.35001
[23] E. E. Rosinger, Generalized Solutions of Nonlinear Partial Differential Equations (North Holland, Amsterdam, 1987). · Zbl 0635.46033
[24] E. E. Rosinger, Nonlinear Partial Differential Equations. An Algebraic View of Generalized Solutions (North Holland, Amsterdam, 1990). · Zbl 0717.35001
[25] C. O. R. Sarrico, ”Distributional Products and Global Solutions for Nonconservative Inviscid Burgers Equation,” J. Math. Anal. Appl. 281, 641–656 (2003). · Zbl 1026.35078 · doi:10.1016/S0022-247X(03)00187-2
[26] C. O. R. Sarrico, ”About a Family of Distributional Products Important in the Applications,” Port. Math. 45, 295–316 (1988). · Zbl 0664.46042
[27] C. O. R. Sarrico, ”Distributional Products with Invariance for the Action of Unimodular Groups,” Riv. Math. Univ. Parma 4, 79–99 (1995). · Zbl 0888.46013
[28] C. O. R. Sarrico, ”New Solutions for the One-Dimensional Nonconservative Inviscid Burgers Equation,” J. Math. Anal. Appl. 317, 496–509 (2006). · Zbl 1099.35121 · doi:10.1016/j.jmaa.2005.06.037
[29] C. O. R. Sarrico, ”Collision of Delta-Waves in a Turbulent Model Studied via a Distribution Product,” Nonlinear Anal. 73, 2868–2875 (2010). · Zbl 1198.35051 · doi:10.1016/j.na.2010.06.036
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