On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

Fìz.-mat. model. ìnf. tehnol. 2021, 32:7-12

Abstract

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

References

1. Olver, P. J. (1986). Applications of Lie Groups to Differential Equations. New York: Springer-Verlag.
2. Mitropol'skiy, Yu. A., Bogolyubov, N. N., Prikarpatskiy, A. K., Samoylenko, V. G. (1987). Integriruyemyye dinamicheskiye sistemy: spektral'nyye i differentsial'no-geometricheskiye aspekty. K.: Naukova dumka.
3. Sanders, J. A., Roelofs, M. (1994). An algorithmic approach to conservation laws using the 3-dimensional Heisenberg algebra. Tech. Rep. 2 RIACA.
4. Sanders, J. A., Wang, J. P. (1995). On the (non)existence of conservation laws for evolution equations. Tech. Rep. WS-445. Amsterdam : Vrije Universiteit.
5. Wolf, T., Brand, A., Mohammadzadeh, M. (1999). Computer algebra algorithms and routines for the computa¬tions of conservation laws and fixing of gauge in differential expressions. J.Symb.Comp., 27, 221-238. DOI doi.org/10.1006/jsco.1998.0250
6. Göktas, Ü., Hereman, W. (1996). Symbolic computation of conserved densities for systems of nonlinear evolution equations. Technical Report MCS-96-06. Colorado School of Mines.
   DOI doi.org/10.1006/jsco.1998.0250
7. Ruo-Xia, Yao, Zhi-Bin, Li. (2006). CONSLAW: A Maple package to construct the conservation laws for nonlinear evolution equations. Applied Mathematics and Computation, 173, 616-635.
   DOI doi.org/10.1016/j.amc.2005.04.049
8. Prykarpats'kyy, A. K., Fil', B. M. (1992). Category of topological jet manifolds and certain applications in the theory of nonlinear infinite-dimensional dynamical systems. Ukrainian Mathematical Journal, 44(9), 1136-1148.
   DOI doi.org/10.1016/j.amc.2005.04.049
9. Mitropol'skii, Yu. O., Prikarpats'kii, A. K., Fil', B. M. Some aspects of a gradient holonomic algorithm in the theory of integrability of nonlinear dynamic systems and computer algebra problems. Ukrainian Mathematical Journal, 43(1), 63-74.
   DOI doi.org/10.1007/bf01066906
10. Fil', B. N., Prikarpatskii, A. K., Pritula, N. N. 1988. Quantum lie algebra of currents - The universal algebraic structure of symmetries of completely integrable dynamical systems. Ukrainian Mathematical Journal, 40(6), 645-649.
    DOI doi.org/10.1007/bf01057184
11. Samoilenko, V. G., Fil', B. N. 1988. Algebras of symmetries of completely integrable dynamical systems. Ukrainian Mathematical Journal, 40(2), 164-168.
    DOI doi.org/10.1007/bf01056471
12. Pukach, P. Y. 2012. On the unboundedness of a solution of the mixed problem for a nonlinear evolution equation at a finite time, Nonlinear Oscillations, 14(3), 369-378.
    DOI doi.org/10.1007/s11072-012-0164-6
13. Fil', B., Pritula, M. 1991. Primeneniye metodov kompyuternoy algebry k nakhozhdeniyu invariantov nelineynykh dinamicheskikh sistem. Tezisy dokladov nauchno-tekhnicheskoy konferentsii "Primeneiye vycheslitel'noy tekhniki i matematicheskikh metodov v nauchnykh iekonomicheskikh issledovaniyakh".