The observation that the soliton-like solutions of a given second-order nonlinear differential equation define the separatrix of the equivalent autonomous system is used to obtain the one-soliton solutions for theφ⁴ theories (the usual and the one with the wrong sign of the mass term), theφ⁶, theφ⁸, the sine-Gordon theories and the KdV equation. Transformations are given which transform the sine-Gordon equation into an equation belonging to theφ²ⁿ class of theories. A procedure is evolved for obtaining the two-soliton solutions for the sine-Gordon theory without the use of Backlund transformations; it is suggested that this procedure may be useful for investigating the existence of similar solutions for theories of the polynomial type.

An attempt is made to study the interaction Hamiltonian,H_int =Gψ²(x)U(φ(x)) in the Bethe-Salpeter framework for the confined states of theψ particles interactingvia the exchange of theU field, whereU(φ) = cos (gφ). An approximate solution of the eigenvalue problem is obtained in the instantaneous approximation by projecting the Wick-rotated Bethe-Salpeter equation onto the surface of a four-dimensional sphere and employing Hecke’s theorem in the weak-binding limit. We find that the spectrum of energies for the confined states,E =2m+B (B is the binding energy), is characterized byE ∼n⁶, wheren is the principal quantum number.

A virial theorem for solitons derived by Friedberg, Lee and Sirlin is used to reduce a system of second order equations to an equivalent first order set. It is shown that this theorem, when used in conjunction with our earlier observation that soliton-like solutions lie on the separatrix, helps in obtaining soliton-like solutions of theories involving coupled fields. The method is applied to a system of equations studied extensively by Rajaraman. The ’t-Hooft-Polyakov monopole equations are then studied and we obtain the well-known monopole solutions in the Prasad-Sommerfeld limit (λ=0); for the case λ≠0, we succeed in obtaining a non-trivial algebraic constraint between the fields of the theory.

The one-soliton solutions found earlier through the inverse scattering method for the complex sine-Gordon theory by Lund (m² < 0) and by Vega and Maillet (m² > 0) are reobtained by using the virial theorem for solitons. An attempt is made to understand the physics of the virial approach.

By reexamining the analysis of Basu and Biswas, based on the stereographic projection method of Fock and Levy, it is shown that the general solution of the Wick-Cutkosky model in the instantaneous approximation, hitherto unreported, involves only one quantum number; this is contrasted with the well-known solution which involves two quantum numbers, but for which the spectrum is degenerate with respect to one of them. The latter situation is shown to hold under a rather special circumstance.

The finite-temperature Schrödinger equation, derived recently from the Bethe-Salpeter equation for the bound states of an electron and a proton interacting via the instantaneous Coulomb interaction, is studied in the coordinate space. An expression for the temperature-modified Coulomb potential is obtained and briefly discussed.

Following Salpeter, the Bethe-Salpeter equation for the bound system of two oppositely charged particles is reduced to a Schrödinger equation for each of the following cases: (a) both particles are spin 1/2 particles, (b) one particle is a spinor while the other is spinless, and (c) both particles are spinless. It is shown that ife is the magnitude of charge carried by each of the particles whose masses are set equal to the electron and proton masses then, strictly speaking, only in case (a) do we obtain the familiar Schrödinger equation for the hydrogen atom. The latter equation is recovered in the other two cases only if relativistic remnants—terms of the order of 10⁻⁵ and smaller—are neglected in comparison with unity. Attention is drawn to a situation where such remnants may not be negligibly small, viz. the problem of confinement of quarks.

We show that the temperature-generalization of a popular model of quark-confinement seems to provide a rather interesting insight into the origin of mass of elementary particles: as the universe cooled, there was an era when particles did not have an identity since their masses were variable; the temperature at which the conversion of these ‘nomadic’ particles into ‘elementary’ particles took place seems to have been governed by the value of a dimension-less coupling constantC_c. ForC_c=0.001(0.1) this temperature is of the order of 10⁹ K (10¹¹ K), below which the particle masses do not change.