## THERMAL SCIENCE

International Scientific Journal

### DARBOUX TRANSFORM AND CONSERVATION LAWS OF NEW DIFFERENTIAL-DIFFERENCE EQUATIONS

**ABSTRACT**

Darboux transforms, exact solutions and conservation laws are important topics in thermal science and other fields as well. In this paper, the new non-linear differential-difference equations associated a discrete linear spectral problem are studied analytically. Firstly, the Darboux transform of the studied equations is constructed, and exact solutions are then obtained. Finally, infinite many conservation laws are derived.

**KEYWORDS**

PAPER SUBMITTED: 2019-04-28

PAPER REVISED: 2019-08-10

PAPER ACCEPTED: 2019-09-08

PUBLISHED ONLINE: 2020-06-21

**THERMAL SCIENCE** YEAR

**2020**, VOLUME

**24**, ISSUE

**4**, PAGES [2519 - 2527]

- He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- He, J. H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
- Li, X. X., et al., A Fractal Modification of the Surface Coverage Model for an Electrochemical Arsenic Sensor, Electrochimica Acta, 296 (2019), Feb., pp. 491-493
- Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals, 26 (2018), 6, 1850086
- Wang, Y., Deng, Q., Fractal Derivative Model for Tsunami Travelling, Fractals, 27 (2019), 1, 1950017
- He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
- Ain, Q. T., He, J. H., On Two-Scale Dimension and its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
- He, J. H., Zhu, S. D., Differential-Difference Model for Nanotechnology, Journal of Physics: Conference Series, 96 (2008), 1, ID 012189
- Zhang, S., et al., Differential-Difference Equation Arising in Nanotechnology and its Exact Solutions, Journal of Nano Research, 23 (2013), 1, pp. 113-116
- Zhang, S., Liu, D. D., Infinite Many Conservation Laws of Discrete System Associated with a 3×3 Matrix Spectral Problem, Thermal Science, 21 (2017), 4, pp. 1613-1619
- He, J. H., A Modified Li-He's Variational Principle for Plasma, International Journal of Numerical Meth-ods for Heat and Fluid Flow, On-line first, doi.org/10.1108/HFF-06-2019-0523, 2019
- He, J. H., Lagrange Crisis and Generalized Variational Principle for 3D unsteady flow, International Journal of Numerical Methods for Heat and Fluid Flow, On-line first, doi.org/10.1108/HFF-07-2019-0577, 2019
- He, J. H., Wu, X. H., Exp-Function Method for Non-Linear Wave Equations, Chaos, Solitons and Fractals, 30 (2006), 3, pp. 700-708
- Zhang, S., et al., A Direct Algorithm of Exp-Function Method for Non-Linear Evolution Equations in Fluids, Thermal Science, 20 (2016), 3, pp. 881-884
- Anjum, N., He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138
- He, J. H., Some Asymptotic Methods for Strongly Nonlinear Equations, International Journal of Modern Physics B, 20 (2006), 10, pp. 1141-1199
- Wu, Y., He, J. H., Homotopy Perturbation Method for Nonlinear Oscillators with Coordinate Dependent Mass., Results in Physics, 10 (2018), Sept., pp. 270-271
- Wu, Y., He, J. H., A Remark on Samuelson's Variational Principle in Economics, Applied Mathematics Letters, 84 (2018), Oct., pp. 143-147
- He, J. H., Hamilton's Principle for Dynamical Elasticity, Applied Mathematics Letters, 72 (2017), Oct., pp. 65-69
- He, J. H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57 (2019), 8, pp. 1932-1934
- He, J. H., The Simplest Approach to Nonlinear Oscillators, Results in Physics, 15 (2019), Dec., ID 102546
- Matveev, V. B., Salle, M. A., Darboux Transformation and Solitons, Springer-Verlag: Berlin, Germany, 1991
- Tian, Y., Symmetry Reduction a Promising Method for Heat Conduction Equations, Thermal Science, 23 (2019 ), 4, pp. 2219-2227
- Tian, Y., Diffusion-Convection Equations and Classical Symmetry Classification, Thermal Science, 23 (2019), 4, pp. 2151-2156
- Bai, Y. S., Zhang, Q., Perturbed Korteweg-de Vries Equations Symmetry Analysis and Conservation Laws, Thermal Science, 23 (2019), 4, pp. 2281-2289
- Xu, X. X., A Family of Integrable Differential-Difference Equations, its Bi-Hamiltonian Structure and Binary Nonlinearization of the Lax Pairs and Adjoint Lax Pairs, Chaos, Solitons & Fractals, 45 (2012), 4, pp. 444-453