Open Access
2010 On Time-Space Dependent Conservation Laws of Nonlinear Evolution Differential Equations
Mahouton N. Hounkonnou , Pascal D. Sielenou
Commun. Math. Anal. 8(3): 102-119 (2010).
Abstract

In this paper, we propose an alternative direct algebraic method of constructing, for nonlinear evolution partial differential equations, conservation laws that depend not only on dependent variables and its derivatives but also explicitly on independent variables. As illustration, the fifth order Korteweg de Vries (fKdV) and modified $(n+1)$-dimensional Zakharov-Kuznetvov (ZK) equations are probed.

References

1.

S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation laws classifications. Europ. J. Appl. Math. 13 (2002), pp 545-566. MR1939160 1034.35070 S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation laws classifications. Europ. J. Appl. Math. 13 (2002), pp 545-566. MR1939160 1034.35070

2.

S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential equations. Part II: General treatment. Europ. J. Appl. Math. 13 (2002), pp 567-585. MR1939161 1034.35071 S. C. Anco and G. W. Bluman, Direct construction method for conservation laws of partial differential equations. Part II: General treatment. Europ. J. Appl. Math. 13 (2002), pp 567-585. MR1939161 1034.35071

3.

R. W. Antherton and G. M. Homsy, On the existence and formulation of variation principles for nonlinear differential equations. Stud. Appl. Math. 54 (1975), pp 31-60. MR458271 0322.49019 R. W. Antherton and G. M. Homsy, On the existence and formulation of variation principles for nonlinear differential equations. Stud. Appl. Math. 54 (1975), pp 31-60. MR458271 0322.49019

4.

M. N. Hounkonnou and P. A. Dkengne Sielenou, Symmetry reductions and new exact solutions of fKdV. Int. J. Contemp. Math. Sciences 4(35) (2009), pp 1719 - 1738. MR2674930 05759612 M. N. Hounkonnou and P. A. Dkengne Sielenou, Symmetry reductions and new exact solutions of fKdV. Int. J. Contemp. Math. Sciences 4(35) (2009), pp 1719 - 1738. MR2674930 05759612

5.

P. S. Laplace, Celestrial Mechanic: Paris 1 (1798). P. S. Laplace, Celestrial Mechanic: Paris 1 (1798).

6.

E. Noether, Invariant Variation problems. Nachr. v. d. Ges. d. Wiss. zu Göttingen (1918), pp 235-257. E. Noether, Invariant Variation problems. Nachr. v. d. Ges. d. Wiss. zu Göttingen (1918), pp 235-257.

7.

P. J. Olver, Applications of Lie groups to differential equations. Springer, New york (1993). MR1240056 P. J. Olver, Applications of Lie groups to differential equations. Springer, New york (1993). MR1240056

8.

L. D. Poole, Symbolic computation of conservation laws of nonlinear partial differential equations using homotopy operators. PhD Thesis, Colodaro School of Mines (2007). L. D. Poole, Symbolic computation of conservation laws of nonlinear partial differential equations using homotopy operators. PhD Thesis, Colodaro School of Mines (2007).

9.

V. Rosenhaus, On conserved densities and asymptotic behaviour for the potential Kadomtsev-Petviashili equation. J. Phys. A: Math. Gen. 39 (2006), pp 7693-7703. MR2236646 10.1088/0305-4470/39/24/006 V. Rosenhaus, On conserved densities and asymptotic behaviour for the potential Kadomtsev-Petviashili equation. J. Phys. A: Math. Gen. 39 (2006), pp 7693-7703. MR2236646 10.1088/0305-4470/39/24/006

10.

J. M. Sanz-Serna, An explicit finite-difference scheme with exact conservation properties. J. Comput. Phys. 47 (1982), pp 199-210. MR674048 0484.65062 10.1016/0021-9991(82)90074-2 J. M. Sanz-Serna, An explicit finite-difference scheme with exact conservation properties. J. Comput. Phys. 47 (1982), pp 199-210. MR674048 0484.65062 10.1016/0021-9991(82)90074-2

11.

H. Steudel, Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen. Zeit Naturforsch 17A (1962), pp 129-132. MR156591 H. Steudel, Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen. Zeit Naturforsch 17A (1962), pp 129-132. MR156591

12.

T. Wolf, Application of CRACK in the classification of integrable systems. CRM proceding and lectures notes 37 (2004), pp 283-300. MR2102988 1073.37081 T. Wolf, Application of CRACK in the classification of integrable systems. CRM proceding and lectures notes 37 (2004), pp 283-300. MR2102988 1073.37081

13.

T. Wolf, Partial and complete linearization of PDEs based on conservation laws. Trends in Mathematics: Differential Equations with Symbolic Computation (2005), pp 291-306. MR2187385 1157.70312 10.1007/3-7643-7429-2_16 T. Wolf, Partial and complete linearization of PDEs based on conservation laws. Trends in Mathematics: Differential Equations with Symbolic Computation (2005), pp 291-306. MR2187385 1157.70312 10.1007/3-7643-7429-2_16

14.

T. Wolf, A comparison of four appoaches to the calculation of conservation laws. Euro. J. Appl. Math. 13 (2002), pp 129-152.  MR1896225 1002.35008 10.1017/S0956792501004715 T. Wolf, A comparison of four appoaches to the calculation of conservation laws. Euro. J. Appl. Math. 13 (2002), pp 129-152.  MR1896225 1002.35008 10.1017/S0956792501004715

15.

R. X. Yao and Z. B. Li. Commun. Thero. Phys. 41 (2004), pp 487. MR2234687 1167.35394 R. X. Yao and Z. B. Li. Commun. Thero. Phys. 41 (2004), pp 487. MR2234687 1167.35394

16.

R. X. Yao and Z. B. Li. Chinese Journal of Physics 42 (2004), pp 315-322. R. X. Yao and Z. B. Li. Chinese Journal of Physics 42 (2004), pp 315-322.

17.

V. E. Zakharov and E. A. Kuznetsov, Three dimensional solitons. Sov. Phys. JETP 39(2) (1974), pp 285-296. V. E. Zakharov and E. A. Kuznetsov, Three dimensional solitons. Sov. Phys. JETP 39(2) (1974), pp 285-296.
Copyright © 2010 Mathematical Research Publishers
Mahouton N. Hounkonnou and Pascal D. Sielenou "On Time-Space Dependent Conservation Laws of Nonlinear Evolution Differential Equations," Communications in Mathematical Analysis 8(3), 102-119, (2010). https://doi.org/
Published: 2010
Vol.8 • No. 3 • 2010
Back to Top