We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)| 2 for the standard and nonlocal NLS equations.

We study the nonlocal modified Korteweg-de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz-Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz-Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of local and nonlocal complex mKdV and mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two-and three-soliton solutions. We illustrate our soliton solutions by plotting their graphs for particular values of the parameters.

We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains. © Springer Science+Business Media, Inc. 2007

We study differential-difference equation of the formwith unknown t(n, x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of, such that D x F = 0 and DI = I, where D x is the operator of total differentiation with respect to x, and D is the shift operator: Dp(n) = p(n + 1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f (u, v, w) = w + g(u, v).

We study a differential-difference equation of the form t x ͑n +1͒ = f͑t͑n͒ , t͑n +1͒ , t x ͑n͒͒ with unknown t = t͑n , x͒ depending on x and n. The equation is called a Darboux integrable if there exist functions F ͑called an x-integral͒ and I ͑called an n-integral͒, both of a finite number of variables x , t͑n͒ , t͑n Ϯ 1͒ , t͑n Ϯ 2͒ , ... , t x ͑n͒ , t xx ͑n͒ ,..., such that D x F = 0 and DI = I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp͑n͒ = p͑n +1͒. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f͑x , y , z͒ = z + d͑x , y͒.