Meromorphic solutions of nonlinear ordinary differential equations

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figure

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes,via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis

Travelling waves in two-dimensional plane Poiseuille flow

The asymptotic structure of laminar modulated travelling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper-energy branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier–Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes describes an attracting manifold of maximum-entropy configurations. The complementary modes, which have been neglected in this parameterization, are strongly damped. In order to seek a closure, a countably infinite number of modulation equations are derived on the long viscous time scale: a single equation for averaged kinetic energy and momentum; and the remaining equations for averaged powers of vorticity. Only a finite number of these vorticity modulation equations are required to determine the finite number of unknowns. The new results show that the evolution of the slowly varying amplitude parameters is determined by the vorticity field and that the phase velocity responds to these changes in the amplitude in accordance with the kinetic energy and momentum. The new results also show that the most crucial physical mechanism in the production of vorticity is the interaction between vorticity and kinetic energy, this interaction being responsible for the existence of the attractor

Seven common errors in finding exact solutions of nonlinear differential equations

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.Comment: 42 page