We consider the master equation of quantum Brownian motion, and with the application of the group... more We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation.
ABSTRACT We examine the elements of the Riccati Differential Sequence and its integrals in terms ... more ABSTRACT We examine the elements of the Riccati Differential Sequence and its integrals in terms of their structure and their symmetry properties. We find that there is a marked change from the fourth member of the sequence (the well-known Painlevé–Ince Equation) to subsequent members of the sequence both in terms of the number of Lie point symmetries of the differential equation and of the integrals.
We introduce an inhomogeneous term, f (t, x), into the right-hand side of the usual Burgers equat... more We introduce an inhomogeneous term, f (t, x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f (t, x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra sl(2, R). When f is a function of only t, there are five symmetries with the algebra sl(2, R) ⊕ s 2A 1. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.
Abstract. We study the generalised Chazy equation, "x'-bxqx"l-kxq-lx 2 ~- O, which... more Abstract. We study the generalised Chazy equation, "x'-bxqx"l-kxq-lx 2 ~- O, which is characterised by the symmetries of time translation and rescaling. For a large class of initial conditions numerical computations reveal the asymptotic appearence of periodic solutions for k = q + 1. These ...
We consider the master equation of quantum Brownian motion, and with the application of the group... more We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation.
ABSTRACT We examine the elements of the Riccati Differential Sequence and its integrals in terms ... more ABSTRACT We examine the elements of the Riccati Differential Sequence and its integrals in terms of their structure and their symmetry properties. We find that there is a marked change from the fourth member of the sequence (the well-known Painlevé–Ince Equation) to subsequent members of the sequence both in terms of the number of Lie point symmetries of the differential equation and of the integrals.
We introduce an inhomogeneous term, f (t, x), into the right-hand side of the usual Burgers equat... more We introduce an inhomogeneous term, f (t, x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f (t, x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra sl(2, R). When f is a function of only t, there are five symmetries with the algebra sl(2, R) ⊕ s 2A 1. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.
Abstract. We study the generalised Chazy equation, "x'-bxqx"l-kxq-lx 2 ~- O, which... more Abstract. We study the generalised Chazy equation, "x'-bxqx"l-kxq-lx 2 ~- O, which is characterised by the symmetries of time translation and rescaling. For a large class of initial conditions numerical computations reveal the asymptotic appearence of periodic solutions for k = q + 1. These ...
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Papers by Peter Leach