Nonlinear waves in ferromagnets

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

Nonlinear waves in optical media

In nonlinear optical systems quasi-monochromatic waves satisfy the classical, constant dispersion and dispersion managed nonlocal, nonlinear Schrödinger equations, both of which exhibit localized pulse solutions. Current research has shown that mode-locked lasers are also described by dispersion managed equations. In spatial systems recent developments have attracted considerable interest in 2D photonic lattices. A computational method is introduced to find these and other localized waves in nonlinear optical media with vanishing and non-vanishing boundary conditions.     

Nonlinear longitudinal waves in inhomogeneous rods

The work contains two basic results. The first consists in the derivation of an equation for the longitudinal vibrations of a rod which accounts for both nonlinear and dispersion effects as well as effects caused by the inhomogeneity of the material of the rod. This equation is found to be a perturbed Korteweg-de Vries equation. The second result consists in the development of a perturbation method for solving the Cauchy problem for this equation. The solution found describes the deformation of a soliton under the influence of the inhomogeneity of the rod.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 64–73, 1980.     

Nonlinear void-fraction waves in two-phase bubbly flow

It is shown that nonlinear void-fraction waves constitute exact solutions of the two-fluid equations for bubbly flow, if and only if the virtual-mass coefficientm() is a definite function of the void fraction. In the case of spherical bubbles the functional dependence is given bym()=(1/2)(1–a)(1–3). A similar expression was obtained recently by means of an elaborate linear stability analysis of a uniform two-phase flow. It is remarkable that the greater part of the physical quantities of a nonlinear void-fraction wave depends linearly on. The virtual-mass coefficient may be derived directly from a special form of Hamilton's variational principle.

---

Zusammenfassung Es wird gezeigt daß nichtlineare Konzentrationswellen exakte Lösungen der Zweifluiden Gleichungen einer Blasenströmung darstellen dann und nur dann, wenn der Koeffizientm() der virtuellen Masse eine bestimmte Funktion der Volumenkonzentration der Blasen ist. Die funktionelle Abhängigkeit ist im Falle kugelförmiger Blasen gegeben durchm()= (1/2)(1–) (1–3). Ein gleicher Ausdruck wurde neulich mittels einer umständlichen linearen Stabilitätsanalysis einer uniformen Zweiphasen-Strömung erhalten. Es ist bemerkenswert daß der größere Teil der physikalischen Größen einer nichtlinearen Konzentrationswelle in linearer Weise von abhängt. Der Koeffizient der virtuellen Masse kann in direkter Weise von einer speziellen Form des Hamiltonschen Variationsprinzips hergeleitet werden.

     

Nonlinear resonant traveling waves in rotating disks

The resonant conditions for traveling waves in rotating disks are derived. The nonlinear resonant spectrum of a rotating disk is computed from the resonant conditions. Such a resonant spectrum is useful for the disk drive industry to determine the range of operational rotation speed. The resonant wave motions for linear and nonlinear, rotating disks are simulated numerically for a 3.5-inch diameter computer memory disk.     

Nonlinear waves in a general magnetic fluid

Summary A nonlinear acceleration wave analysis is developed for a general magnetohydrodynamic theory studied by Roberts [6]. The sixth order wavespeed equation is solved exactly and then Roberts' linearized stability restriction on the free energy is confirmed from a nonlinear analysis.

---

Sommario Un analisi nonlineare delle onde di discontinuità è svillupato per una teoria generale di magnetoidrodinamica che era studiato da Roberts [6]. L'equazione per la velocità dell'onda è risolto esattamente e la condizione sull'energia libera, che era derivato da Roberts dagli analisi lineare, è riderivato dal punto di vista nonlineare.

     

Nonlinear amplification of ocean waves in straits

Theoretical and Mathematical Physics - Using the exact Hasselmann equation, we study wind-driven deep-water ocean waves in a strait with the wind directed orthogonally to the shore. The strait has...     

Nonlinear differential identities for cnoidal waves

This article presents a family of nonlinear differential identities for the spatially periodic function , which is essentially the Jacobian elliptic function with one non‐trivial parameter . More precisely, we show that this function fulfills equations of the form for all . We give explicit expressions for the coefficients and for given s. Moreover, we show that for any s the set of functions constitutes a basis for . By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well‐known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.     

Nonlinear waves in complex oscillator network with delay

We investigate the spatio-temporal patterns of Hopf bifurcating periodic solutions in a delay complex oscillator network. Firstly, we calculate the critical values of Hopf bifurcation. Secondly, the bifurcating periodic solutions can take on two cases: one is synchronization or anti-synchronization, and another is the coexistence of two phase-locked, N mirror-reflecting and N standing waves, because the system has group symmetry. Finally, the stability of these nonlinear oscillations is determined using the center manifold theorem and normal form method with the imaginary eigenvalues being simple and double.     

Lightlike periodic trajectories in space-times

Let ., _z be a Lorentz metric on a manifold such that isnot compact. We prove the existence of infinitely many lightlike periodic trajectories in by using variational methods and Ljusternik-Schnirelman theory.Sponsored by M.U.R.S.T. (fondi 60% «Problemi differenziali nonlineari e teoria dei punti critici»; fondi 40% «Equazioni differenziali e calcolo delle variazioni»).     

Geodesic completeness in static space-times

Let (H, h) be a Riemannian manifold and assume f: H(0, ) is a smooth function. The Lorentzian warped product (a, b) _f ×H,–a<b, with metric ds ²=(–f ² dt ²) h is called a standard static space-time. A study is made of geodesic completeness in standard static space-times. Sufficient conditions on the warping function f: H(0, ) are obtained for (a, b) _f ×H ×H to be timelike and null geodesically complete. In the timelike case, the sufficient condition is independent of the completeness of the Riemannian manifold (H, h).     

Nonlinear localized waves in a medium with nonlocal interaction

Soliton solutions are found for nonlinear integro-differential equations with a type λ/(τ-τ′) kernel used to describe particle tunneling and magnetic and superconducting vortices in a medium with nonlocal interaction. The Fourier transform method is applied to derive asymptotic formulas for even and odd localized solutions. Analytical solutions are found for particular parameter values. A complete pattern is constructed for the behavior of soliton solutions in an arbitrary range of the interaction parameter λ by means of numerical calculations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 3, pp. 366–379, March, 1998.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Nonlinear analysis of flow pulses and shock waves in arteries

Zusammenfassung In jeder Gruppe von Menschen variieren die Kreislaufparameterwerte von Individuum zu Individuum ziemlich stark. Um die Effekte realistischer Schwankungen dieser Parameter auf die Strom-und Druckpulse in arteriellen Leitungen zu untersuchen und um mögliche Manifestationen gewisser pathologischer Zustände wie Arteriosklerosis und die Insuffizienz der Aortenklappen zu identifizieren, werden hier einige der Resultate präsentiert, die sich auf Grund des im ersten Teil beschriebenen mathematischen Modells ergeben. Insuffizienz der Aortenklappen ist nachgebildet durch einen Ausströmungspuls mit aussergewöhnlich grossem Rückstrom. Der zugehörige Druckpuls zeigt die bekannten Eigenschaften von Stosswellen, die anscheinend verantwortlich sind für die pistol shot sounds, die man über der arteria femoralis und der arteria radialis von Patienten mit Aorteninsuffizienz hört.

---

---

For Part I see Z. ang. Math. Phys.22, 217 (1971).     

Nonlinear evolution equation for describing waves in a viscoelastic tube

Propagation of the nonlinear waves in a viscoelastic tube filled with a liquid is studied. The bending of the tube wall is taken into account. By using the reductive perturbation method the fifth order nonlinear evolution equation is derived. Exact solutions for this equation is obtained by means of the simplest equation method. Properties of nonlinear waves in a viscoelastic tube filled with a liquid are discussed.     

Nonlinear modulation of SH waves in an incompressible hyperelastic plate

Propagation of nonlinear shear horizontal (SH) waves in a homogeneous, isotropic and incompressible elastic plate of uniform thickness is considered. The constituent material of the plate is assumed to be generalized neo-Hookean. By employing a perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the nonlinear modulation of waves is governed asymptotically by a nonlinear Schr?dinger (NLS) equation. Then the effect of nonlinearity on the propagation characteristics of asymptotic waves is discussed on the basis of this equation. It is found that, irrespective of the plate thickness, the wave number and the mode number, when the plate material is softening in shear then the nonlinear plane periodic waves are unstable under infinitesimal perturbations and therefore the bright (envelope) solitary SH waves will exist and propagate in such a plate. But if the plate material is hardening in shear in this case nonlinear plane periodic waves are stable and only the dark solitary SH waves may exist.     

Nonlinear analysis of flow pulses and shock waves in arteries

Zusammenfassung Druck- und Strompulse mit grosser Amplitude erzeugt in der Aorta und in andern Hauptarterien des Hundes werden theoretisch berechnet für vorgeschriebene Ausströmungspulse von der linken Herzkammer und für gegebene physikalische und geometrische Eigenschaften des Kreislauf-systems. Der Blutausfluss durch die Äste und Verzweigungen der uns interessierenden arteriellen Leitung ist durch ein kontinuierlich verteiltes, von Druck und Ort abhängiges Ausflussmodell nachgeahmt. Am herzfernen Ende der Leitung ist als Randbedingung entweder der periphere Widerstand oder ein konstanter druck vorgeschrieben. Die Geometrie der Leitung ist durch ihren kreisförmingen Querschnitt und den mit Herzdistanz exponentiell abnehmenden Radius definiert. Die elastischen Eigenschaften der Gefässwand sind durch die von Ort und Druck abhängige Geschwindigkeit kleiner Druckwellen gegeben. Durch Integration der Beziehung zwischen Wellen-geschwindigkeit und Querschnittsdehnung ist damit auch der Querschnitt als Funktion des Druckes und der Herzdistanz vorgeschrieben. Die nichtlinearen Gleichungen für eindimensionale Strömung einer inkompressibler Flüssigkeit werden mit Hilfe der Charakteristikenmethode integriert für Kreislaufparameterwerte die einem hypothetischen Hund von 30 kg Gewicht entsprechen. Das verwendete mathematische Modell für die arterielle Leitung wiedergibt manche der bekannten Eigenschaften des vom Herzen erzeugten Pulses, einschliesslich die Klappenincisur, das Ansteigen und Abfallen der Höhe des systolischen Druckgipfels mit wachsender Entfernung vom Herzen. Während der Fortpflanzung der Pulswelle zeichnet sich eine zunehmende Steilheit der Wellenfront ab, die jedoch nicht merkbar ist wenn man die grundlegenden Gleichungen linearisiert. Die numerischen Ergebnisse weisen darauf hin, dass die sekundäre (dicrotic) Welle durch Reflexionen erzeugt wird, und als solche von der Verjüngung des Querschnittes und vom Blutausfluss abhängt.

---

Notation A cross-sectional area of artery at the reference intraluminal pressurep ₀ - c local wave speed - c ₀,c ₁ parameters in wave speed expression - c _L local wave speed for linearized analysis - C, C designations for general curves inz, t plane - D ₀ diameter at aortic valve whenp=p ₀ - D _t diameter at distal end of artery whenp=p ₀ - E circumferential Young's modulus - f axial frictional force per unit mass of fluid - h wall thickness - L distance from aortic valve to distal end of artery - n parameter in wave speed expression - p intraluminal pressure - calculated mean pressure - p ₀ reference pressure - p _c capillary pressure - p _L pressure at distal end of artery - q S v=local volume flow rate - q ₀ q ₀(t)=volume flow rate ejected by heart - r radial coordinate - R _e Reynolds number for steady flow - R _L peripheral resistance - s curvilinear coordinate - S cross-sectional area of artery - S _L cross-sectional area for linearized analysis - t time - v axial fluid velocity averaged over cross section - z axial distance coordinate - z ^* distance from aortic valve to femoral artery - exponent in cross-sectional area expression - outflow parameter - undetermined multiplier - blood viscosity coefficient - blood density - outflow function simulating effect of side branches     

Huygen's principle on nonempty space-times

Huygen's principle is considered on some nonempty space-time manifolds. It is proved that if Huygen's principle holds in a space with spherically symmetric time-independent metric, then the space must have constant curvature.