Radial Wave Equation Research Articles

Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation

We establish Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a (time-dependent) self-similar potential. As an application we consider the critical wave equation and prove the asymptotic stability of the ODE blowup profile in the energy space.

Scattering for radial energy-subcritical wave equations in dimensions 4 and 5

ABSTRACTIn this paper, we consider the focusing and defocusing energy-subcritical, nonlinear wave equation in ℝ1+d with radial initial data for d = 4,5. We prove that if a solution remains bounded in the critical space on its interval of existence, then the solution exists globally and scatters at ±∞. The proof follows the concentration compactness/rigidity method initiated by Kenig and Merle, and the main obstacle is to show the nonexistence of nonzero solutions with a certain compactness property. A main novelty of this work is the use of a simple virial argument to rule out the existence of nonzero solutions with this compactness property rather than channels of energy arguments that have been proven to be most useful in odd dimensions.

A Green's function approach to the nonrelativistic radial wave equation of hydrogen atom

We report an efficient and consistent method for the solution of Schrödinger wave equation using the Green's function technique and its successful application to the nonrelativistic radial wave equation of hydrogen atom. For the radial wave equation, the Green's function is worked out analytically by means of Laplace transform method and the wave function is proposed under the boundary conditions. Computationally the product of potential term, the proposed wave function and the Green's function are integrated iteratively to get the nonrelativistic radial wave function. The resultant wave after each iteration is normalized and plotted against the standard nonrelativistic radial wave function. The solution converges to the standard wave with the increasing number of iterations. Results are verified for the first 15 states of hydrogen atom. The method adopted here can be extended to many‐body problem and hope that it can enhance our knowledge about complex systems.

Investigation of Bohr–Mottelson Hamiltonian in γ-rigid version with position dependent mass

In this paper, we consider the Bohr–Mottelson Hamiltonian in γ-rigid version with position dependent mass. The separation of variables has been done for the related wave equation. The obtained radial wave equation is solved for Kratzer potential. Then, the corresponding wave function, energy spectra and transition rates have been obtained for some nuclei. In addition, our results have been compared with experimental data.

A GDQ approach to thermally nonlinear generalized thermoelasticity of disks

ABSTRACTGeneralized thermoelasticity response of an annular disk subjected to thermal shock on its inner surface is analyzed in this research. The Lord–Shulman theory, which accounts for one relaxation time in the conventional Fourier law, is used to avoid the infinite speed of thermal wave propagation. Unlike the other available works in which the first law of thermodynamics is linearized, the nonlinearity arising from the temperature change is taken into consideration. The first law of thermodynamics in this case becomes nonlinear and the analysis under such formulation is called thermally nonlinear. Two coupled equations, i.e., the radial displacement wave equation and temperature wave propagation equation are obtained. These equations and the associated boundary conditions are discreted through the generalized differential quadrature method. Solution of the time-dependent system of equations is obtained using the Newmark time marching scheme and the successive Picard method. Numerical results are provided for both thermally linear and thermally nonlinear temperature and radial displacement wave propagations. Parametric studies reveal that at higher temperature levels, thermally nonlinear first law of thermodynamics should be considered instead of thermally linear one. Furthermore, the higher the coupling parameter and/or relaxation time, the higher the divergence of thermally nonlinear-/linear-based results.

Stable blowup for wave equations in odd space dimensions

We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions d≥5. We prove that for every p>d+3d−1 there exists an open set of radial initial data in Hd+12×Hd−12 such that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions.

Quaternionic spherical wave functions

The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu‐type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of ‐hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.

Profiles for the radial focusing energy-critical wave equation in odd dimensions

In this paper, we consider global and non-global radial solutions of the focusing energy--critical wave equation on $\mathbb{R} \times \mathbb{R}^N$ where $N \geq 5$ is odd. We prove that if the solution remains bounded in the energy space as you approach the maximal forward time of existence, then along a sequence of times converging to the maximal forward time of existence, the solution decouples into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume a bound on the evolution that rules out the formation of multiple solitons, then this decoupling holds for all times approaching the maximal forward time of existence.

Viscosity for isotopes of helium with HCB model

The calculation of viscosities for the fluids of Helium isotopes, He3 and He4, has been made by using HCB models. The phase shifts required for its calculation of viscosities have been obtained by numerical integration of the radial wave equation expressed by taking a simple form of Leenard-Jones intermolecular potential specified in terms of the support function (hx) and surface to surface coordinate representation K. For both He3 and He4 the phase shifts resemble the rigid sphere behavior for high collision energies of the two colliding helium molecules of the same species. The cross-sections required for the calculation of viscosities below temperature 5K for the fluids of helium isotopes has also been discussed.

Absorption cross-section and decay rate of rotating linear dilaton black holes

We analytically study the scalar perturbation of non-asymptotically flat (NAF) rotating linear dilaton black holes (RLDBHs) in 4-dimensions. We show that both radial and angular wave equations can be solved in terms of the hypergeometric functions. The exact greybody factor (GF), the absorption cross-section (ACS), and the decay rate (DR) for the massless scalar waves are computed for these black holes (BHs). The results obtained for ACS and DR are discussed through graphs.

Channels of energy for the linear radial wave equation

Exterior channel of energy estimates for the radial wave equation were first considered in three dimensions in [6], and for the 5-dimensional case in [12]. In this paper we find the general form of the channel of energy estimate in all odd dimensions for the radial free wave equation. This will be used in the companion paper [11] to establish the soliton resolution for equivariant wave maps in R3 exterior to the ball B(0,1) and in all equivariance classes.

Stable soliton resolution for exterior wave maps in all equivariance classes

In this paper we consider finite energy ℓ-equivariant wave maps from Rt,x1+3\\(R×B(0,1))→S3 with a Dirichlet boundary condition at r=1, and for all ℓ∈N. Each such ℓ-equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary ℓ-equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d=2ℓ+3, which are established in the companion paper [13].

Scattering for the radial 3D cubic wave equation

Consider the Cauchy problem for the radial cubic wave equation in 1+3 dimensions with either the focusing or defocusing sign. This problem is critical in $\dot{H}^{\frac{1}{2}} \times \dot{H}^{-\frac{1}{2}}$ and subcritical with respect to the conserved energy. Here we prove that if the critical norm of a solution remains bounded on the maximal time-interval of existence, then the solution must in fact be global-in-time and scatter to free waves as $t \to \pm \infty$.

The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method

The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method

On Type I Blow-Up Formation for the Critical NLW

We introduce a suitable concept of weak evolution in the context of the radial quintic focusing semilinear wave equation on ℝ3+1, that is adapted to continuation past type II singularities. We show that the weak extension leads to type I singularity formation for initial data corresponding to: (i) the Kenig-Merle blow-up solutions with initial energy below the ground state and (ii) the Krieger-Nakanishi-Schlag blow-up solutions sitting initially near and “above” the ground state static solution.

The Equation of State for Non-ideal Quark Gluon Plasma

The mass spectra of quarkonium systems at T= 0 are analyzed by solving the nonrelativistic radial wave equation using the internal energy potential. The QGP matter is studied through the dissociations of quarkonium states. A modified form of the internal energy potential function is used to determine the EoS at different number of quark flavors by using Mayer's cluster expansion theory and phenomenological thermodynamic model. The thermodynamic model gives a good agreement with the lattice results rather than Mayer’s cluster expansion theory. One can conclude that, the Mayer’s cluster expansion theory may be more suitable to study a weakly coupled plasma while, the QGP may be considered as a strongly interacting plasma.

Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime

In this paper, we study the exact boundarycontrollability for the cubic focusing semilinear wave equation onSchwarzschild black hole background in radially symmetrical case.When the initial data and the finaldata are in the so called potential well, we find that thesufficient condition for the global existence is also sufficient toensure the exact boundary controllability of the problem. Moreover,under the assumption of radial symmetry, our problem is changed toone space dimension case, and then the control time can be that of the linearwave equation.

Energy partition for the linear radial wave equation

We consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy, as time goes to infinity. We show that the exterior energy estimate, which Duyckaerts et al. obtained in odd dimensions (Duyckaerts et al., J Eur Math Soc 13:533–599, 2011; J Eur Math Soc, 2013) fails in even dimensions. Positive results for restricted classes of data are obtained.

The Dirac equation in Kerr-Taub-NUT spacetime

We study the Dirac equation in the Kerr–Taub–NUT spacetime. We use Boyer–Lindquist coordinates and separate the resulting equations into radial and angular parts. We obtain some exact analytical solutions of the angular equations for some special cases. We also obtain the radial wave equations with an effective potential. Finally, we discuss the potentials by plotting them as a function of radial distance in a physically acceptable region.

Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in R3 via two distinct approaches. The first approach invokes the method established in the works Bourgain (1994, 1996) [3–5] based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schrödinger equation on the unit ball in subsequent works Bourgain and Bulut (2013) [10,11].