Stability of post-fertilization traveling waves

This paper studies the stability of a family of traveling wave solutions to the system proposed by Lane et al. [D.C. Lane, J.D. Murray, V.S. Manoranjan, Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs, IMA J. Math. Appl. Med. Biol. 4 (4) (1987) 309-331], to model a pair of mechanochemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg's surface, and a free calcium concentration front. The family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these post-fertilization waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. We show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase-modulated traveling wave.     

Modelling of rotating shafts with flexible disks

This paper deals with the methodology of the modelling of rotating shafts with flexible disks. Rotating shafts are modelled as a one dimensional continuum on the basis of the Bernouli-Euler theory, which assumes that the shaft cross section remains a flat plane and is perpendicular to the centerline during vibration. Disks are modelled as a three dimensional continuum by means of the finite element method. The presented approach allows to introduce centrifugal and gyroscopic effects. The special coupling matrix is used for the connecting a rotating shaft and a mounted flexible disk. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bistable traveling waves around an obstacle

We consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity. The goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space ?^N. As a first step, we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle. This causes disturbance on the shape of the front, but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction, leaving the obstacle behind. Whether the recovery is uniform in space is shown to depend on the shape of the obstacle. © 2008 Wiley Periodicals, Inc.     

On rotating circular disks with varying material properties

Taking Young’s modulus, thermal expansion coefficient and density to be the functions of the radial coordinate, a closed form solution of rotating circular disks made of functionally graded materials subjected to a constant angular velocity and a uniform temperature change is proposed in this paper. Excellent agreement with the solution from Mathematica 5.0 indicates the correctness of the proposed closed form solution. Distributions of the radial displacement and stresses in the disks are determined with the proposed approach and how material properties, temperature change, geometric size and different material coefficients affect deformations and stresses is investigated.     

On rotating circular disks with varying material properties

Taking Young’s modulus, thermal expansion coefficient and density to be the functions of the radial coordinate, a closed form solution of rotating circular disks made of functionally graded materials subjected to a constant angular velocity and a uniform temperature change is proposed in this paper. Excellent agreement with the solution from Mathematica 5.0 indicates the correctness of the proposed closed form solution. Distributions of the radial displacement and stresses in the disks are determined with the proposed approach and how material properties, temperature change, geometric size and different material coefficients affect deformations and stresses is investigated.     

Spreading of thin liquid droplets on rotating disks

In many industrial processes solids are coated to obtain specific surface properties, as e.g. corrosion resistance, mechanical, optical, or electrical properties. Even today many of such coating processes are not fully understood and the choice of parameters is mainly based on experience. Hence, a prediction of the complete hydrodynamic process in its dependency on the parameters appears highly desirable. This would e.g. allow for a precise prediction of the (liquid) layer thickness and shape and help to optimize the quality of the coating. A common coating technique is the so–called spin coating. The coating agent is dissolved or suspended in a liquid, brought onto the solid, spread by rotation, and the carrier liquid is finally removed by evaporation or by chemical reactions. In this article an evolution equation is derived from lubrication theory, valid for thin liquid layers. The model involves a dynamic contact angle, centrifugal, capillary, gravitational, and molecular (London–van–der–Waals) forces. The evolution equation without molecular forces can even be solved analytically, provided the capillary number is small. Otherwise a numerical integration of the governing equations is engaged. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear waves in ferromagnets

Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 80, No. 3, pp. 461–469, September, 1989.     

Bounded traveling waves of the Burgers-Huxley equation

In order to investigate bounded traveling waves of the Burgers-Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle-focus.     

Elliptic traveling waves of the Olver equation

Nonlinear waves on water are studied. The method recently developed by Demina and Kudryashov is applied to the Olver water wave equation. New solutions of this equation are found. These solutions are expressed in terms of the Weierstrass elliptic function.     

Dynamics of rotating thermoelastic disks with stationary heat source

Rotating disks are important components of car brakes or sawing units. In both cases, heat effects to be induced via stationary local contacts between pads and disk or workpiece and saw blade, respectively, influence the dynamic behavior and raise interesting problems in theory and practice. Therefore, the discussion of dynamic thermoelasticity in rotating disks with stationary heat sources is of basic interest to understand the interaction of temperature and displacement or stress fields in such structural elements. It will be analyzed here in detail for the case of an elastic disk for which there is a full (but weak) coupling of strain and temperature within the two applications mentioned. As a relatively general case, the combined excitation by a mechanical load and a simultaneously acting heat source will be examined.     

Dynamic programming and the optimum design of rotating disks

The minimum-weight design of elastic rotating disks under various design criteria has been studied using ideas of dynamic programming and invariant imbedding. It is shown that the criterion of minimum potential energy furnishes a remarkably simple solution in terms of a linear partial differential equation subject to initial conditions. The treatment of more general design constraints leads to the development of a number of stable, two-sweep iterative procedures. A series of numerical examples incorporating various design constraints is presented to show the accuracy and feasibility of the method. The sensitivity of the radial stresses to small changes in the thickness of the disk is noticed. On the other hand, the high stability of the volume with respect to the design criteria is stressed.This work was partially supported by a University of California grant. The author wishes to thank Mr. R. Todeschini, who carried out the numerical computations.     

Asymptotic representation of surface waves in the form of two traveling burgers waves

Chair of General Mathematics, Department of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 25–40, July–September, 1995.     

Turning points and traveling waves in FitzHugh-Nagumo type equations

Consider the following FitzHugh-Nagumo type equation

     

Stick-slip-separation waves in a rotating shaft-bush system

We consider an elastic annular bush fitted to a rigid rotating shaft with a diameter mismatch. The formation of stick-slip, slip-separation and stick-slip-separation waves due to dynamic friction acting between the bush and the shaft is studied. Numerically, we calculate the loci of non-smooth bifurcations in the parameter space for rotating mode-1 waves on the shaft-bush interface, varying the angular velocity and the static coefficient of friction. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of traveling waves in the fractional bistable equation

We construct traveling waves of the fractional bistable equation by approximating the fractional Laplacian ${(D^{2})^{\alpha}, \alpha \in (0, 1)}$ , with operators ${J \ast u - (\int_{R} J)u}$ , where J is nonsingular. Since the resulting approximating equations are known to have traveling waves, the solutions are obtained by passing to the limit. This provides an answer to the statement (about existence and properties) “This construction will be achieved in a future work” before Assumption 2 in Imbert and Souganidis [6]. With a modification of a part of the argument, we also get the existence of traveling waves for the ignition nonlinearity in the case ${\alpha \in (1/2, 1)}$ .     

Existence and modulation of traveling waves in particles chains

We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where q_n(t) is the displacement of the n^th particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C∞ function. For this system, we prove the existence of C∞, time‐periodic, traveling‐wave solutions of the form q_n(t) = q(wt kn + where q is a periodic function q(z) = q(z+1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small‐amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling‐wave solution and the non‐linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and ω. © 1999 John Wiley & Sons, Inc.     

Periodic waves in rotating plane Couette flow

A class of nonlinear equations of Navier-Stokes type of the form (**) $$\frac{{dw}}{{dt}} + L_0 w + (\lambda - \lambda _0 )L_1 w + \gamma \lambda (M_1 w + M_2 w) + \gamma ^2 L_3 (\lambda ,\gamma )w + B(w) = 0$$ is investigated, where λ is a “load” parameter (i.e., a Reynolds number), γ is a “structure” parameter,L _o L ₁,M ₁,M ₂ andL ₃(λ, γ) are linear operators, andB is a quadratic operator. An equation of the form (**) describes a variety of spiral flow problems including rotating plane Couette flow which is studied here in detail. Under suitable hypotheses on the operators in (**), it is shown that Hopf bifurcation occurs for γ sufficiently small. In the problem of rotating plane Couette flow, by determining the sign of the real part of a certain “cubic” coefficient, it is shown, in addition, that the bifurcating periodic orbits are supercritical and asymptotically stable, and correspond to periodic waves.