Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation

We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established.

Fundamentals of Metamaterials

This chapter describes the fundamentals of left-handed metamaterials. From Maxwell's equations, constant phase term is originated, and it is revealed that its negative value is chosen in a negative permittivity and negative permeability (DNG) medium whereas its positive value is designated in a DPS medium. The negative phase constant results in a negative phase velocity and negative index of refraction in the medium. Complementary split ring resonator (CSRR) as a valuable metamaterial component is illustrated. The resonant frequencies of the CSRR are associated with the features of their arrangements. CSRR is agitated with the E field of the electric and magnetic wave together with the axis of the CSRR. Consequently, the CSRR presents negative permittivity in a particular frequency band.