ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## The Integration of Harry Dym and Korteweg-de Vries Equations in Parametric Form

### Author(s):

Natalya Konstantinovna Volosova

4th year student
Dept. of applied mathematics 1
Moscow State University of Railway
Engineering, 9, of. 9, St. Obraztsova, Moscow, 127994, Russia

konstantinvolosov@yandex.ru

Aleksandra Konstantinovna Volosova

konstantinvolosov@yandex.ru

Konstantin Aleksandrovich Volosov

DSc in phisics and mathematics
Professor
Dept. of applied mathematics 1
Moscow State University of Railway
Engineering, 9, of. 9, St. Obraztsova, Moscow, 127994, Russia

konstantinvolosov@yandex.ru

### Abstract:

In the present work we for the first time apply a relatively new method of constructive unfixed change of variables to the Harry Dym (HD) and the Korteweg--de Vries (KdV) equations. We construct two dynamical systems and formulate necessary conditions for the stability of phase trajectories. A system of functional algebraic equations is constructed and it is proved that two formal solvability conditions for a system of first order partial differential equations have one non--trivial common factor. An important feature of the HD and KdV equations was found: after an unfixed constructive change of variables, a new "hidden" key equation for the function of the partial first derivative can be separated from the other equations. The exact solutions constructed with the help of a non-autonomous dynamical system coincide with global solutions. That is not the case for equations with dissipation. Two classes of exact solutions are found for the HD and for the KdV equation. A possibility arises to construct new asymptotic solutions.

### Keywords

• accurately
• function for the first derivative in new variables is exact
• Harry Dym and Korteweg - de Vries equations
• method of
• not fixed constructive change of variables
• two new wide classes of exact solutions

### References:

1. Maslov V. P. Asymptotic and perturbation Theory. Nauka, Moscow. 1988. (in Russian)
2. Maslov V. P., Tsupin V. A. Sovrem. problems of Math., 1977, № 8, p. 273
3. Maslov V. P., Omel'yanov G. A. Geometric Asymptotics for Nonlinear PDE. Amer. Math. Soc. Providence 2001, v. 202, 360 p
4. Maslov V. P. Quantum Economics, Nauka, Moscow, 2007, 80 p. (in Russian)
5. Maslov V. P., Danilov V. G., Volosov K. A. Mathematical Modeling of Technological Processes on LSI Circuil Production. (MIEM, Moscow, 1984, Moscow, MIEM, 1984. ) (in Russian)
6. Maslov V. P., Danilov V. G., Volosov K. A. Mathematical Modeling of Heat and Mass Transfer Processes. (Evolution of Dissipative Structures, addition of Kolobov N. A. ) (Nauka, Moscow, 1987. ) (in Russian)
7. Maslov V. P., Danilov V. G., Volosov K. A. Mathematical Modelling of Heat and Mass Transfer Processes. Kluver Academic Publishers. Dordrecht, Boston, London, 1995. - 316 p
8. Clarkson P. A., Kruskal M. D. New similarity reduction of the Boussinesq equation. J. Math. Phys., 1989, V. 30, № 10, pp. 2201-2213
9. Solitons. Edited by R. K. Bullough, P. J. Caudrey (with contributions by R. K. Bullought, F Calogero, P. J. Caudrey, A. Degasperis, L. D. Faddev, H. M. Gibbs, R. Hirota, G. L. Lamb, Jr, A. H. Luther, D. W. McLaughlin, A. C. Newell, S. P. Novikov, M. Toda, M. Wadati, V. E. Zakharov). New Yurk. 1980
10. Volosov K. A. Transformation of Approximate solutions of liner parabolic equations into asymptotic solutions of quasilinear parabolic equations. Mathematical Notes. - 1994. - Vol. 56. № 5-6, P. 1295-1299. English transl. in Mat. Zametki 56 (6) 122-126 (1994)
11. Volosov K. A. Differential Equations. Pleiades Publishing Ltd., ISSN 0012-2661. Vol. 43, № 4, p. 507-512, 2007
12. Volosova A. K., Volosov K. A. Construction Solutions of PDE in Parametric Form. Hindawi Publishing Corporation. International Journal of Mathematics and Mathematical Sciences. V. 2009, Article ID 319269, 17 p., http:// www. hindawi. com/journals/ijmms/2009/319269. htmt. doi:10. 1155/2009/319269
13. Volosova A. K., Volosov K. A. Stochastic systems under periodic and white noise external excitation. The 3rd International Conference on Nonlinear Dynamics Nd-KhPI 2010, Khakov, Ukraine, p. 437-442
14. Volosov K. A. Construction of solutions of PDE. Journal of Applied and Industtrial Mathematics, 2009, V. 3, № 4, pp. 519-527
15. Volosov K. A. Doctorial dissertation, MIEM, Moscow, Russia, MIIT, 2007, http://eqworld.ipmnet.ru dis volosov Doc2007. pdf
16. Volosov K. A. Implicit formulas for exaxt solutions of quasilinear partial differential equations. (Russian) Dokl. Akad. Nauk 2008, V. 418, № 1, p. 11-14. Engl. tran. in Doklady Mathematics. 2008, V. 77, № 1, pp. 1-4
17. Kudrashov N. A. From singular manifold to integrable evolution equations. J. Phys. A: Math. Gen. 27, (1994) 2457-2470. print in the UK
18. Volosov K. A., Vdovina E. K., Volosova A. K. Accompany matrix of the Korteweg - de Vries equation. International conference on the differential equations. Samara-Diff 2011, 26-30 june 2011, p. 32-33. Math-Net. Ru, Google Schoar, ZentralBlatt
19. Volosov K. A., Vdovina E. K., Volosova A. K. Accompany matrix of the Korteweg - de Vries equation. 8th International ISAAC Congress. Moscow, 22-27 august 2011, p. 278 Math-Net. Ru, Google Schoar, ZentralBlatt
20. Volosov K. A., Vdovina E. K. About expansion of number of models which have pairs of Lax's. International Journal Equation and Applications. 2012, v. 11, № 1, p. 27-30
21. Volosov K. A., Volosova N. K., Volosova A. K., Vakulenko S. P. The theory adding to equation of Korteweg - de Vries. LXX International jubilee conference. St. Peterburg. Herzen State Pedagogical University of Russia, 10-15 April 2017, p. 38-52
22. Bratus A. S., Volosov К. A. Exact solutions of the Hamilton-Jacobi-Bellman equation for problems of optimal correction with an integral constraint on the total control resources. (Russian) Dokl. Akad. Nauk 385 (2002), № 3, p. 319-322. Engl. tran. in Doklady Mathematics. V. 66, № 1, 2002, pp. 148-151
23. Bratus A. S., Volosov К. A. Exact solutions of the Hamilton-Jacobi-Bellman equation for problems of optimal correction with a constrained overall control resourse. (Russian) Prikl. Mat. Mekh. V. 68 (2004) - № 5, pp. 819-832. Engl. tran. in J. Appl. Math. and Mech. V. 68, (2004), pp. 731-742
24. Volosov K. A., Danilov V. G., Maslov V. P. Combustion wave asymptotics in nonlinear inhomogeneous media with slowly varying properties. Dokl. Akad. Nauk SSSR 290:5 (1986), 1089-1094 (Russian); English transl. in Soviet Math. Dokl
25. Volosov K. A., Danilov V. G., Maslov V. P. Weak discontinuity structure of solutions of quasilinear parabolic equations. Mat. Zametki 43:6 (1988), 829-838 (Russian); English transl. in Math. Notes
26. Maslov V. P., Tsupin V. A. !!!! ERROR!!! IMAGE IS NOT ALLOWERD! -shaped Sobolev generalized solutions of quasilinear equations. Russian mathematical Surveys. 1979. V. 34, № 1 (205), 231-236. http://dx.doi org/10. 1070/RM1979v034n01ABEH002884
27. Maslov V. P., Omel'yanov G. A., Tsupin V. A. Asymptotics of some differential and pseudodifferential equations, and dynamical systems with small dispersion. Math. USSR - Sb. 50:1 (1985), 191-212
28. Maslov V. P., Tsupin V. A. Propagation of a shock wave in an isentropic gas with small viscosity. Journal of Soviet Mathematics. 1980, V. 13, № 1, pp. 163-185
29. Maslov V. P., Omel'yanov G. A. Asymptotic soliton - form solutions of equations with small dispersion. Russian math. Surveys, V. 36, № 3, (1981), pp. 73-149
30. Dobrochotov S., Maslov V. Multyphase asymptotics of nonlinear partial equations with a small parameter. Tn. Sov. Science rev., Phys. Rev., 1981. Amsterdam: Over Publ. Ass, 1982
31. Maslov V. P., Omel'yanov G. A. Soliton - like asymptotics of internal waves in a stratified fluid with small dispersion (Russian) Differentsial'nye Uravneniya V. 21. (1985), № 10, 1766-1775, 1837
32. Maslov V. P. Three algebras corresponding to nonsmooth solutions of quasilinear hiperbolic equations, Uspekhi Math. Nauk, V. 35, (1980) (Russian); Englis transl. in Russian Math. Surveys
33. Maslov V. P. The Complex WKB Method for Nonlinear Equations. I. Linear Theory, Birkhä user, Basel - Boston - Berlin, 1994
34. Maslov V. P. Nonstandard characteristics in asymptotic problems, Uspeki Mat. Nauk V. 38:6 (1983), pp. 3-36
35. Danilov V. G., Omel'yanov G. A., Shelkovich V. M. Weak asymptotics and interaction of nonlinear waves. Translation of the American Mathematical Society. Series 2 208 (2003), 33-164