Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation

Abstract

A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.

Rational solutions of the Toda lattice equation in Casoratian form

“…A Jordan block of the first type in (2.19) has the real eigenvalue k j with algebraic multiplicity k j , and a Jordan block of the second type in (2.20) has the pair of complex eigenvalues k j;AE ¼ a j AE b j i with algebraic multiplicity l j . The case of real eigenvalues greater than 2 and less than 2 corresponds to positons and negatons [22], respectively. The case of complex eigenvalues corresponds to complexitons [16].…”

Generalized Matrix Exponential Solutions to the AKNS Hierarchy

“…The MIST can determine solutions more directly and generate more diverse solutions than the traditional IST. Actually, such solutions can be also obtained by applying the Wronskian technique [19][20][21][22][23]. The arbitrariness of the matrices involved leads to the diversity of exact solutions.…”

“…Successful examples are the Wronskian technique and the IST. Ma et al had introduced the matrix element in Wronskian determinants when they used the Wronskian technique to solve soliton equations [19][20][21][22][23]. They obtained various kinds of solutions such as soliton, rational, Matveev, and complexiton solutions.…”

Complexiton solutions to integrable equations

“…These three types of eigenfunctions lead to rational solutions (see, for example, [ 18]), negatons and positons (see, for example, [ 2]), respectively. Case l i = 1 of Type 2: To construct complexitons, we solve…”