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Exact solutions of the Burgers-Huxley equation. (Russian, English) Zbl 1092.35084

Prikl. Mat. Mekh. 68, No. 3, 462-469 (2004); translation in J. Appl. Math. Mech. 68, No. 3, 413-420 (2004).
The authors study the Burgers-Huxley equation \[ U_t+\alpha UU_x = DU_{xx}+\beta U+\gamma U2 - \delta U3,\quad d\neq 0\tag{1} \] which describes some nonlinear wave effects. By means of the Cole-Hopf substitution [J. D. Cole, Q. Appl. Math. 9, No. 3, 225–236 (1951; Zbl 0043.09902)] an exact solution to equation (1) is obtained. The types of exact solution are analysed depending on the parameter values of the equation.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q51 Soliton equations

Citations:

Zbl 0043.09902
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References:

[1] Osipov, V.V., The simplest autowaves, Sorosovskii obrazovatel’nyi zhurnal, 7, 115-121, (1999)
[2] Loskutov, A.Yu.; Mikhailov, A.S., Introduction to synergetics, (1990), Nauka Oxford
[3] Kolmogorov, A.N.; Petrovskii, I.G.; Piskunov, N.S., Investigation of the diffusion equation coupled with an increase in the quantity of matter and its application to a biological problem, Byul. mosk. GoS. univ., ser. A, 6, 1-26, (1937)
[4] Kudrayashov, N.A., Analytical theory of non-linear differential equations, (2002), Mosk. Inzk-Firz. Inst. Moscow
[5] Kudryashov, N.A., Partial differential equations with solutions having movable first-order singularities, Phys. lett. A., 169, 4, 237-242, (1992)
[6] Kudryashov, N.A., Exact soliton solutions of the generalized evolution equation of wave dynamics, Prikl. mat. mekh., 51, 3, 465-470, (1988)
[7] Kudryashov, N.A.; Sukhare, M.B., Exact solutions of a non-linear fifth-order equation for describing waves on water, Prikl. mat. mekh., 65, 5, 884-894, (2001) · Zbl 1040.35072
[8] Cole, J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Q. appl. math., 9, 3, 225-236, (1951) · Zbl 0043.09902
[9] Hope, E., The partial differential equation u_{t} + uux = uuxx, Comm. pure appl. math., 3, 3, 201-230, (1950)
[10] Burgers, J.M., A mathematical model illustrating the theory of turbulance, (), 171-199
[11] Rozhdestvenskii, B.L.; Yanenko, N.N., Systems of quasilinear equations and their application to gas dynamics, (1978), Nauka New York · Zbl 0177.14001
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