Further observations on the anharmonic motion equation

Peters [1] discussed some theoretical aspects of the ‘ringing’ sound associated with barbershop quartet singing. These combination tones were first investigated theoretically by Helmholtz [2] who showed that the transient displacement of the ear membrane is governed by the anharmonic motion equation. Rayleigh [3] modelled the situation as an initial value problem and this is now referred to as the Rayleigh problem. Following [1] there has been a renewed interest in the problem (see [4‐8]). The intention of the present paper is to:

• illustrate the qualitative behaviour of the solutions of the Rayleigh problem by a phase plane analysis;

• describe how the method of multiple scales provides a framework within which previous analyses can be seen;

• describe an improvement to the method of multiple scales, which may also prove useful in other applications.

     

Series solutions of some anharmonic motion equations

Consider the class of nonlinear oscillators of the form

(1)     

Power series solutions for the KPP equation

In this paper we solve the KPP equation by a non numerical method. To this end we find power series solutions where the coefficients are computed recursively. We also prove convergence of the series and illustrate the method by few examples.      

Some special solutions of the Helmholtz equation

Functions are constructed which describe a cylindrical wave diverging from the origin and tangent to a semi-infinite plane wave. The use of such functions is illustrated by the example of the problem of diffraction of a plane wave by a semi-infinite screen.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 197–202, 1975.     

Some solutions of discrete sine-Gordon equation

In this paper, a series of exact solutions of discrete sine-Gordon equation are obtained by the different transformations and symbolic computation.     

Some properties of the solutions of a quasipotential equation

Scientific Council for the Complex Problem Kibernetika, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 1, pp. 155–160, October, 1990.     

Some exact solutions of the variable coefficient Schrodinger equation

By constructing appropriate transformations, we find some exact solutions of variable coefficient Schrodinger equation, which include elliptic function and trigonometric solutions and so on.     

A series of the solutions for the Heisenberg ferromagnetic spin chain equation

The Heisenberg ferromagnetic spin chain equation is investigated. By applying the improved F‐expansion method (Exp‐function method) and the Jacobi elliptic method, respectively, a series of exact solutions is constructed. The parametric conditions of the existence for the solutions are presented. These solutions comprise periodic wave solutions, doubly periodic wave solutions, and dark and bright soliton solutions, which are expressed in several different function forms, namely, Jacobi elliptic function, trigonometric function, hyperbolic function, and exponential function. The results illustrate that the Exp‐function method is a powerful symbolic algorithm to look for new solutions for the nonlinear evolution systems.     

Some properties of solutions to the weakly dissipative Degasperis-Procesi equation

In this paper, we consider the weakly dissipative Degasperis-Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis-Procesi equation are also investigated.     

Some isoperimetric norm bounds for solutions of the helmholtz equation

Summary Previous results of the authors [2] relating theL ₁ andL ₂ integrals of the first eigenfunction in the two dimensional fixed membrane problem are here extended to the analogous problem in higher dimensions.

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Zusammenfassung Frühere Resultate der Verfasser [2], welche eine Beziehung zwischen demL ₁- und demL ₂-Integral der ersten Eigenfunktion einer zweidimensionalen eingespannten Membran ergaben, werden hier auf das analoge Problem in höheren Dimensionen erweitert.

     

Some special solutions for divergence structure quasilinear equation

We consider the divergence structure quasilinear equation−div⃗a(∇u) = f(x, u,∇u),which is not a variational equation. By applying the method of Galerkinapproximation, we give some special solutions of the above equation.     

On the solutions of the translation equation in rings of formal power series

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г¹∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L¹∞),Ф = (f_n ⁿ≤1,forwhich f₁ = l, f₂ = ... = f_p+l =0,f_p+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w _n ^p )_n≥p+2 of universal polynomials in f_p+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w _n ^p )_n≥p+2 we may also use another sequence (L _n ^p )_n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions.     

Some pathological examples of solutions to a Beltrami equation

We construct an example of a bounded solution to a uniformly elliptic Beltrami equation that has no nontangential limit values almost everywhere on the boundary of the unit disk and also an example of a solution to such an equation that is not identically zero and has zero nontangential limit values almost everywhere on the boundary of the unit disk. These examples show that, in the general case of the Hardy spaces of solutions to a uniformly elliptic Beltrami equation (and to more general noncanonical first-order elliptic systems), the usual statement of boundary value problems used for holomorphic and generalized analytic functions is ill-posed.     

Some exact solutions of KdV equation with variable coefficients

In this paper, the extended mapping transformation method is used to obtain some new exact solutions of a variable-coefficient KdV equation arising in arterial mechanics. The obtained solutions include soliton solutions, periodic solutions and rational solutions.