We obtain exact self-similar solutions to an inhomogeneous nonlinear Schrödinger equation, describing propagation of optical pulses in fiber amplifiers with distributed dispersion and gain. We show that there exists a one-to-one correspondence between such self-similar waves and solitons of the standard, homogeneous, nonlinear Schrödinger equation if a certain compatibility condition is satisfi...

The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining noninvariant solutions of partial differential equations with infinite dimensional symmetry groups. ...

A new approach is proposed to the description of thin viscous flows near the edges of a solid surface. For a steady flow, the lubrication approximation and the no-slip condition are assumed to be valid on most of the surface, except for relatively small neighborhoods of the edges, where an universality principle is postulated: the behavior of the liquid in these regions is universally determine...

Using a method of expansion similar to Chapman Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an error O(ǫ2) where ǫ is the small parameter appearing in the high frequency approximatio...

Difficulties associated with free-surface finite element flow simulations are related to a) nonlinear and advective nature of most hydrodynamic flows, b) requirements for compatibility between velocity and pressure interpolation, c) maintaining a valid computational mesh in the presence of moving boundaries, and d) enforcement of the kinematic conditions at the free surface. Focusing on the las...

A parsimonious and compatible system of taper, total, and merchantable volume equations sharing a common vector of three parameter estimates was derived from Clark and Saucier’s (1990) volume ratio model. Clark and Saucier’s model expresses volume inside bark as a function of outside bark diameter and is itself not compatible with the derived taper equation. To ensure compatibility, models were...