## THERMAL SCIENCE

International Scientific Journal

### INFINITE MANY CONSERVATION LAWS OF DISCRETE SYSTEM ASSOCIATED WITH A 3×3 MATRIX SPECTRAL PROBLEM

**ABSTRACT**

Differential-difference equations are often considered as an alternative approach to describing some phenomena arising in heat/electron conduction and flow in carbon nanotubes and nanoporous materials. Infinite many conservation laws play important role in discussing the integrability of non-linear differential equations. In this paper, infinite many conservation laws of the non-linear differential-difference equations associated with a 3×3 matrix spectral problem are obtained.

**KEYWORDS**

PAPER SUBMITTED: 2016-07-02

PAPER REVISED: 2016-10-15

PAPER ACCEPTED: 2016-10-25

PUBLISHED ONLINE: 2017-09-09

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**Issue 4**, PAGES [1613 - 1619]

- Zhang, D. J., Chen, D. Y., The Conservation Laws of Some Discrete Soliton Systems, Chaos, Solitons and Fractals, 14 (2002), 4, pp. 573-579
- Miura, R. M., et al., KdV Equation and Generalizations, II. Existence of Conservation Laws and Con-stants of Motion, Journal of Mathematical Physics, 9 (1968), 8, pp. 1204-1209
- Ablowitz, M. J., et al., A Note on Miura's Transformation, Journal of Mathematical Physics, 20 (1974), 6, pp. 999-1003
- Kruskal, M. D., et al. Korteweg-de Vries Equation and Generalizations: V. Uniqueness and Non-Existence of Polynomial Conservation Laws, Journal of Mathematical Physics, 11 (1970), 3, pp. 952-960
- Tu, G. Z., Qin, M. Z., The Invariant Groups and Conservation Laws of Non-linear Evolution Equations -An Approach of Symmetric Function, Science China Mathematics, 24 (1981), 1, pp. 13-26
- Wen, X. Y., A New Integrable Lattice Hierarchy Associated with a Discrete 3×3 Matrix Spectral Prob-lem: N-Fold Darboux Transformation and Explicit Solutions, Reports on Mathematical Physics, 71 (2013), 1, pp. 15-32
- Zhang, S., et al., Differential-Difference Equation Arising in Nanotechnology and it's Exact Solutions, Journal of Nano Research, 23 (2013), 1, pp. 113-116
- He, J.-H., Zhu, S. D., Differential-Difference Model for Nanotechnology, Journal of Physics: Confer-ence Series, 96 (2008), 1, doi.org/10.1088/1742-6596/96/1/01218
- Hu, Y., He, J.-H., On Fractal Space-Time and Fractional Calculus, Thermal Science, 20 (2016), 3, pp. 773-777
- Zhang, S., et al., Variable Separation for Time Fractional Advection-Dispersion Equation with Initial and Boundary Conditions, Thermal Science, 20 (2016), 3, pp. 789-792
- Zhang, S., et al., Exact Solutions of Time Fractional Heat-Like and Wave-Like Equations with Variable Coefficients, Thermal Science, 20 (2016), Suppl. 3, pp. S689-S693
- Zhang, S., Zhang, H. Q., Fractional Sub-Equation Method and its Applications to Non-Linear Fractional PDEs, Physics Letters A, 375 (2011), 7, pp. 1069-1073
- Wang, K. L., Liu, S. Y., He's Fractional Derivative for Non-Linear Fractional Heat Transfer Equation, Thermal Science, 20 (2016), 3, pp. 793-796
- Zhang, S., et al., Exact Solutions of a KdV Equation Hierarchy with Variable Coefficients, International Journal of Computer Mathematics, 91 (2014), 7, pp. 1601-1616
- Zhang, S., Wang, D., A Toda Lattice Hierarchy with Variable Coefficients and its Multi-Wave Solu-tions, Thermal Science, 18 (2014), 5, pp. 1555-1558
- Zhang, S., Cai, B., Multi-Soliton Solutions of a Variable-Coefficient KdV Hierarchy, Nonlinear Dynam-ics, 78 (2014), 3, pp. 1593-1600
- Ning, T. K., et al., The Exact Solutions for the Non-Isospectral AKNS Hierarchy through the Inverse Scattering Transform, Physica A Statistical Mechanics & Its Applications, 339 (2007), 3-4, pp. 248-266
- Zhang, S., Wang, D., Variable-Coefficient Non-Isospectral Toda Lattice Hierarchy and its Exact Solu-tions, Pramana-Journal of Physics, 86 (2016), 6, pp. 1259-1267
- Zhang, S. Zhang, L. Y., Bilinearization and New Multi-Soliton Solutions of MKdV Hierarchy with Time-Dependent Coefficients, Open Physics, 14 (2016), 1, pp. 69-75
- Wang, J., Hu, Y., On Chain Rule in Fractional Calculus, Thermal Science, 20 (2016), 3, pp. 803-806