Nonlinear differential-difference and difference equations: integrability and exact solvability

Difference integrability conditions for parameterized linear difference and differential equations

This paper is devoted to integrability conditions for systems of linear difference and differential equations with difference parameters. It is shown that such a system is difference isomonodromic if and only if it is difference isomonodromic with respect to each parameter separately. Due to this result, it is no longer necessary to solve non-linear difference equations to verify isomonodromici...

Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

and Applied Analysis 3 Step 6. Using the results obtained in above steps, we can finally obtain exact solutions of (5). 3. Applications of the Modified Truncated Expansion Method In this section, we apply the proposed modified truncated expansion method to construct the exact solutions for the nonlinear DDEs via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable ...

Solvability of Nonlinear Difference Equations of Fourth Order

In this article we show the existence of solutions to the nonlinear difference equation xn = xn−3xn−4 xn−1(an + bnxn−2xn−3xn−4) , n ∈ N0, where the sequences (an)n∈N0 and (bn)n∈N0 , and initial the values x−j , j = 1, 4, are real numbers. Also we find the set of initial values for which solutions are undefinable when an 6= 0 and bn 6= 0 for every n ∈ N0. When these two sequences are constant, w...

Nonstandard finite difference schemes for differential equations

In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs). Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with ...

Preface: Symmetries and Integrability of Difference Equations