We study the homogeneous wave equation with radially symmetric data in four or higher space dimensions. In addition, to being a natural choice due to the symmetry of laplaces equation, radial solutions are natural to. Using some new integral representations for the riemann operator, we establish weighted decay estimates for the solution. Stability of radially symmetric travelling waves in. Our approach is based on the construction of suitable trace formulas which relate the impedance of the total eld at multiple frequencies to derivatives of the potential. An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i. Existence of multiple periodic solutions to asymptotically. Another, more customary derivation, writes the general solution to 87 as. The wave equation appears in a number of important applications, such as sound waves. The wave equation the heat equation the onedimensional wave equation separation of variables the twodimensional wave equation solution by separation of variables we look for a solution ux,tintheformux,tfxgt. We give explicit examples of focusing nonlinear waves that blow up in amplitude.
A standard method is used to solve a nonhomogenous system of. Pdf exact solutions of semilinear radial wave equations in n. This exact solution describes the evolution in space and time of an initial distribution of a diffusing substance. Examplesincludewaterwaves,soundwaves,electromagneticwavesradiowaves. Multiplicity of radially symmetric solutions for a p. Given the symmetric nature of laplaces equation, we look for a radial solution. Radially symmetric stationary wave for twodimensional burgers equation 3 when n 3, 1. An earthflattening transformation for waves from a point source l 197 r o c o fig. It is assumed here that the localized initial conditions are given on the ray, and the velocity on \\mathbbr3\ is radially symmetric. The expansion rate of such solutions can be either self.
In this case, it is proved that 0 is not in the spectral set of the wave operator, which is a. It corresponds to the linear partial differential equation. An interesting feature is that the solvable of the problem depends on the space dimension n and the arithmetical properties of r and t. Nonlinear stability of expanding star solutions of the. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as x our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curlcurl operator. Particle in a spherically symmetric potential wikipedia. According to 12, ux,t depends on the data g and honly on the surface. Thick clusters for the radially symmetric nonlinear. On the inverse scattering problem for radiallysymmetric. Schrodinger equation for spherically symmetric potential without making any approximation. In contrast to the heat equation we have 2 initial conditions. Abstract we discuss solutions of the spherically symmetric wave equation and klein. Radially symmetrical definition of radially symmetrical.
Numerical blowup for the radially symmetric nls equation 3 in the twodimensional case, still for radially symmetric solutions, earlier conclusions in the literature on the blowup rate of the amplitude, based on numerical and asymptotic computations, varied substantially. A point source at the origin should produce a solution with radial symmetry, i. Lecture 4 wave equations invariance, explicit solutions radial way. Weighted hls inequalities for radial functions and. Pdf existence of infinitely many periodic solutions for. Weighted decay estimates for the wave equation with radially symmetric data.
Existence of infinitely many periodic solutions for the radially symmetric wave equation with resonance article pdf available in journal of differential equations 2607 december 2015 with 43. The resulting algorithm can be used to solve for sound intensities in complex models that may include high material contrasts and arbitrary bathymetry. Coordinate system for a spherically symmetric medium with a boundary at r a between a homogeneous region co and a radially heterogeneous region cr. In the present paper, we obtain a complete asymptotic series for a solution of the cauchy problem for a wave equation with variable velocity on the simplest decorated graph obtained by gluing a ray to the euclidean space \\mathbbr3\. We obtain the sharp lower bound for the lifespan of radially symmetric solutions to a class of these systems. This paper is concerned with derivation of the global or local in time strichartz estimates for radially symmetric solutions of the free wave equation from some morawetztype estimates via weighted hardylittlewoodsobolev hls inequalities. For the radially symmetric function k, laplace equation. The general solution of steadystate on onedimensional axisymmetric mechanical and thermal stresses for a hollow thick made of cylinder functionally graded porous material is developed. Now equation 12 can be reduced to layer in the casing.
Tube wave to p and s conversions clearly show up in figure 3a. We consider the cauchy problem for a system of semilinear wave equations with multiple propagation speeds in three space dimensions. Radially symmetric weak solutions for a quasilinear wave. Temperature, as functions of the radial direction with general thermal and mechanical boundaryconditions on the inside and outside surfaces. The fundamental solution for the axially symmetric wave. In particular, if the particle in question is an electron and the potential is derived from coulombs law, then the problem can be used to describe a hydrogenlike oneelectron atom or ion. That is, we look for a harmonic function u on rn such that ux vjxj. Construction of twobubble solutions for energycritical. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that such a stationary wave satisfies nice decay estimates and is timeasymptotically nonlinear stable under radially symmetric perturbation. Radially symmetric singular solutions of the wave equation in halfspace jarmo malinen abstract. Pdf exact solutions are derived for an ndimensional radial wave. We are concerned with the radially symmetric stationary wave for the exterior problem of twodimensional burgers equation. The lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations. Radially symmetric patterns of reactiondi usion systems.
At that time, as an outgrowth to work simulating a cylindrically symmetric millimeter wave transit time oscillator, arman 4 noted the advantages of a radially propagating planar beam and developed a. This paper is concerned with the multiplicity of radially symmetric positive solutions of the dirichlet boundary value problem for the following ndimensional pharmonic equation of the form where is a unit ball in. This is now referred to as the radial wave equation, and would be identical to the onedimensional schr odinger equation were it not for the term r 2 added to v, which. The asymptotic behaviour as t goes to infinity of solutions ux,t of the multidimensional parabolic equation u t. In mathematics, the eigenvalue problem for the laplace operator is called helmholtz equation. Based on the spectral properties of the radially symmetric wave operator, we use the saddle point reduction and variational methods to. Localized asymptotic solution of the wave equation with a. We also show global existence of radially symmetric solutions to another class of. A nonlinear twisted multicore fiber is constructed with alternating amplifying and absorbing cores, which meet the requirements of the pt symmetry.
More precisely, we consider the stability of spherically symmetric travelling waves with respect to small perturbations. Comparison of theory and simulation for a radially. Substitution into the onedimensional wave equation gives 1 c2 gt d2g dt2 1 f d2f dx2. A breather construction for a semilinear curlcurl wave. An exact solution for a nonlinear diffusion equation in a. An important outcome of our stability results is the existence of a new class of global. We present an exact solution for a nonlinear diffusion equation by considering the radially symmetric. New singular standing wave solutions of the nonlinear.
Consequently, the semilinear wave equation is reduced to an ode with r x as a parameter. The lifespan of radially symmetric solutions to nonlinear. To find the energy and the wave function of the ground state, there is no need for the calculation. Using some new integral representations for the riemann operator, we establish weighted. Behavior of solutions for radially symmetric solutions for burgers equation with a boundary corresponding to the rarefaction wave. Equations and boundary conditions consider the equation 1. Chapter 4 derivation and analysis of some wave equations wavephenomenaareubiquitousinnature.
The multidimensional wave equation n 1 special solutions. Mechanical and thermal stresses in a fgpm hollow cylinder. It is an easy exercise to verify that if is a radially symmetric weak solution of 1. In fact, some books prefer 5, rather than 3a as the standard form of the wave equation. Pdf weighted decay estimates for the wave equation with. Radially symmetric solutions for burgers equation with a boundary corresponding to the rarefaction wave itsuko hashimoto received november 10, 2014, revised august 6, 2015 abstract we investigate the largetime behavior of the radially symmetric solution for burgers equation on the exterior of a small ball in multidimensional space, where. Different interpretations of the solutions found are examined. This was proved for the radial energycritical wave equation in dimension n3 by duyckaerts, kenig and merle 19, following the earlier work of the same authors 18. A finite difference fd method is developed and analyzed for the helmholtz equation in a radially symmetric waveguide.
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