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New classes of exact solutions to general nonlinear equations and systems of equations in mathematical physics. (English. Russian original) Zbl 1217.35166
Dokl. Math. 78, No. 1, 607-611 (2008); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 421, No. 6, 744-748 (2008).
From the text: New classes of exact solutions to some general nonlinear equations and systems of equations of mathematical physics are described that involve arbitrary functions. Special attention is given to equations and systems of equations encountered in the theory of mass and heat transfer, mathematical biology, and wave theory.
In this paper, an exact solution is regarded in the sense of the definition given in [A. D. Polyanin, V. F. Zaitsev and A. I. Zhurov, Methods for solving nonlinear equations in mathematical physics and mechanics. Fizmatlit, Mowcow, (2005)], page 10.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35C05 Solutions to PDEs in closed form
80A20 Heat and mass transfer, heat flow (MSC2010)
35A24 Methods of ordinary differential equations applied to PDEs
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References:
[1] A. D. Polyanin, V. F. Zaitsev, and A. I. Zhurov, Methods for Solving Nonlinear Equations in Mathematical Physics and Mechanics (Fizmatlit, Moscow, 2005) [in Russian].
[2] R. Cherniha and J. R. King, J. Phys. A: Math. Gen. 33, 267–282; 7839–7841 (2000). · Zbl 0947.35026 · doi:10.1088/0305-4470/33/2/304
[3] A. G. Nikitin and R. J. Wiltshire, J. Math. Phys. 42, 1667–1688 (2001). · Zbl 1053.35067 · doi:10.1063/1.1331318
[4] T. Barannyk, Proc. Inst. Math. Natl. Acad. Sci. Ukr. 43,Part 1, 80–85 (2002).
[5] R. Cherniha and J. R. King, J. Phys. A: Math. Gen. 36, 405–425 (2003). · Zbl 1059.35058 · doi:10.1088/0305-4470/36/2/309
[6] T. A. Barannyk and A. G. Nikitin, Proc. Inst. Math. Natl. Acad. Sci. Ukr. 50,Part 1, 34–39 (2004).
[7] A. D. Polyanin, Dokl. Math. 71, 148–153 (2005) [Dokl. Akad. Nauk 400, 606–611 (2005)].
[8] A. D. Polyanin and E. A. Vyaz’mina, Dokl. Math. 74, 597–602 (2006) [Dokl. Akad. Nauk 409, 455–460 (2006)]. · doi:10.1134/S1064562406040338
[9] A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists (Chapman & Hall; CRC, Boca Raton, FL, 2007). · Zbl 1267.00006
[10] A. G. Nikitin, J. Math. Anal. Appl. 332, 666–690 (2007). · Zbl 1128.35054 · doi:10.1016/j.jmaa.2006.10.032
[11] A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (Chapman and Hall; CRC, Boca Raton, FL, 2004). · Zbl 1053.35001
[12] CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1: Symmetries, Exact Solutions and Conservation Laws, Ed. by N. H. Ibragimov (CRC, Boca Raton, FL, 1994).
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