Darboux transformation for the Zn-Hirota systems

Hirota equation is a modified nonlinear Schrödinger (NLS) equation, which takes into account higher order dispersion and delay correction of cubic nonlinearity. The propagation of the waves in the ocean is described, and the optical fiber can be regarded as a more accurate approximation than the NLS equation. Using the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-Hirota systems. Considering the potential applications of two-mode nonlinear waves in nonlinear optical fibers, including its Lax pairs, we use the algebraic reductions of the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text]. Then, we construct Darboux transformation of the strongly coupled Hirota equation, which implies the new solutions of [Formula: see text] generated from the known solution [Formula: see text]. The new solutions [Formula: see text] furnish soliton solutions and breather solutions of the strongly coupled Hirota equation. Furthermore, using Taylor series expansion of the breather solutions, the rogue waves of the strongly coupled Hirota equation can be given demonstrably. It is obvious that different images can be obtained by choosing different parameters.

Bäcklund transformations of Zn-sine-Gordon systems

In this paper, from the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon systems which contain many multi-component sine-Gordon type and sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon equations which can generate new solutions from seed solutions. To see the [Formula: see text]-systems clearly, we consider the [Formula: see text]-sine-Gordon and [Formula: see text]-sine-Gordon equations explicitly including their Bäcklund transformations, the nonlinear superposition formula and Lax pairs.

Excitation of forbidden lines in gaseous nebulae I. Formulation and calculations for 2pq ions

A formulation is given for electron collisions with ions in configurations s22s22pq and The main approximation is neglect of coupling to other configurations. Hartree-Fock functions are used for the ion states and the complete wave functions are expressed as sums of vector-coupled anti-symmetrized products of ion functions and orbitals for the colliding electron. Variational principles are used to obtain coupled integro-differential equations for the radial functions for the colliding electron, and to correct results obtained from approximate solutions of these equations. All algebraic reductions are carried out without the introduction of subsidiary approximations, and conservation and reciprocity theorems are therefore satisfied exactly. Expressions are tabulated for all algebraic coefficients. Numerical calculations are made in two approximations: in the exact resonance approximation, used only for />-waves, the wave functions are calculated with quadrupole interactions neglected; and in the distorted wave approximation the wave functions are calculated from static central potentials. Variational corrections are calculated and are found to be reasonably small. It is concluded that the final corrected results should agree closely with results which would be obtained from exact solutions of the coupled equations. Collision strengths are calculated for all inelastic collisions in configurations 1 to 5, for at least three different energies, and for values of the residual charge z = 1, 2, 3, 4, 5 and the limit of z-> oo. Results may be interpolated for all members of the iso-electronic sequences. Results for energies such that some channels are open and others closed are obtained by means of extrapolation techniques.