New Sets of Kink Bearing Hamiltonians

Given a kink bearing Hamiltonian, Isospectral Hamiltonian approach is used in generating new sets of Hamiltonains which also admit kink solutions. We use Sine-Gordon model as a example and explicitly work out the new sets of potentials and the solutions. * E-mail address: [email protected] 1 Nonlinear field theory models in (1+1) dimensions, with Lagrangian density L = 12∂μφ∂φ − V (φ) , for which the equations of motion admit finite energy, finite width solutions have been well studied . Sine-Gordon model and φ models are two popular models used for modelling some physical systems. While the solutions of φ are called kinks/lumps, the solutions of Sine-Gordon model have the extra property that they retain their identity after collisions, and they are called solitons. In order to model a variety of physical systems, various attempts are made to enlarge the class of nonlinear field theoretic models. Parametrically modified Sine-Gordon model was one of such earlier attempts made by Remoissenet and Peyrard . In this model the potential is V (φ, r) whose shape can be varied continuosly as a function of r, −1 < r < 1 and has the Sine-Gordon (SG) potential for r = 0 as a special case. The implicit kink solutions for this model and their rest masses are calculated. For r 6= 0 the model is not completely integrable. Kink-antikink interactions are studied for this model and was shown that the structure is similar to that observed in KK̄ interactions for the φ model for a range of r . In this article we present a prescription to construct a family of potentials which admit kink solutions. The method is based on Isospectral Hamiltonian approach which enables us to generate two sets of potentials from a given potential. In one case we can give explicit kink solution whereas implicit solutions are obtained in the other case. A partial result of this approach using φ model was presented earlier by one of us . Once the field theoretic model admits kink type solutions, the stability of the kink is ensured by the occurence of the zero energy ground state of the stability equation when small oscillations around the kink are considered . Considering the stability equation as a one dimensional Schrodinger-like equation for a particle in a potential V (x) we can construct a isospectral partner for it. Then, as we explain below, following the work of Christ and Lee 5 we shall construct the kink solution and the potential which admits the solution from the partner stability equation. Let the Lagrangian density of a single hermitian scalar field φ in 1+1 dimensions be 2 given by L = 1 2 ∂μφ∂ φ − V (φ) (1) In order to have kink solutions the potential V (φ) is assumed to have atleast two degenerate absolute minima. The time independent field equations reads dφ dx2 − dV dφ = 0 (2) which can also be written as 1 2 ( dφ dx ) = V (φ) (3) on taking squareroot of equn (3) and integrating