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Mar 2013

Volume 54, Issue 3, Articles (03xxxx)

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J. Math. Phys. 54, 033512 (2013); http://dx.doi.org/10.1063/1.4794514 (20 pages)

Andrew Neate and Aubrey Truman
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back to top Partial Differential Equations

Ground state solutions for nonlinear fractional Schrƶdinger equations in mathN

Simone Secchi

J. Math. Phys. 54, 031501 (2013); http://dx.doi.org/10.1063/1.4793990 (17 pages)

Online Publication Date: 6 March 2013

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We construct solutions to a class of Schrƶdinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.
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03.65.Ge Solutions of wave equations: bound states
02.30.Hq Ordinary differential equations
02.30.Xx Calculus of variations

Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation

JosƩ Antonio AlcƔntara FƩlix and Simone Calogero

J. Math. Phys. 54, 031502 (2013); http://dx.doi.org/10.1063/1.4793991 (9 pages)

Online Publication Date: 8 March 2013

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The relativistic Fokker-Planck equation, in which the speed of light c appears as a parameter, is considered. It is shown that in the limit c ā†’ āˆž its solutions converge in L1 to solutions of the non-relativistic Fokker-Planck equation, uniformly in compact intervals of time. Moreover in the case of spatially homogeneous solutions, and provided the temperature of the thermal bath is sufficiently small, exponential trend to equilibrium in L1 is established. The dependence of the rate of convergence on the speed of light is estimated. Finally, it is proved that exponential convergence to equilibrium for all temperatures holds in a weighted L2 norm.
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05.60.-k Transport processes
02.30.-f Function theory, analysis
02.50.-r Probability theory, stochastic processes, and statistics

Low-regularity solutions of the periodic modified two-component Camassa-Holm equation

Li-meng Xia, Lixin Tian, and Caixia Shen

J. Math. Phys. 54, 031503 (2013); http://dx.doi.org/10.1063/1.4794284 (10 pages)

Online Publication Date: 8 March 2013

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This paper studies low-regularity periodic solutions of the modified two-component Camassa-Holm equation with initial value. We prove the existence and C0-well posedness of solutions.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems

Note on intrinsic decay rates for abstract wave equations with memory

Irena Lasiecka, Salim A. Messaoudi, and Muhammad I. Mustafa

J. Math. Phys. 54, 031504 (2013); http://dx.doi.org/10.1063/1.4793988 (18 pages)

Online Publication Date: 13 March 2013

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In this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality gā€² + H(g) ā©½ 0, s ā©¾ 0 where H(s) is a given continuous, positive, increasing, and convex function such that H(0) = 0. We shall develop an intrinsic method, based on the main idea introduced by Lasiecka and Tataru [ā€œUniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation,ā€ Differential and Integral Equations 6, 507ā€“533 (1993)], for determining decay rates of the energy given in terms of the function H(s). This will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ODE. We shall show that the obtained result, while generalizing previous results obtained in the literature, is also capable of proving optimal decay rates for polynomially decaying memory kernels (H(s) āˆ¼ sp) and for the full range of admissible parameters p āˆˆ [1, 2). While such result has been known for certain restrictive ranges of the parameters p āˆˆ [1, 3/2), the methods introduced previously break down when p ā©¾ 3/2. The present paper develops a new and general tool that is applicable to all admissible parameters.
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02.30.Hq Ordinary differential equations
02.10.De Algebraic structures and number theory

Fractional wave equation and damped waves

Yuri Luchko

J. Math. Phys. 54, 031505 (2013); http://dx.doi.org/10.1063/1.4794076 (16 pages)

Online Publication Date: 13 March 2013

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In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order Ī±,ā€‰1 ā©½ Ī± ā©½ 2, both in space and in time. We show that this feature is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and ā€œmassā€ centers. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order Ī±. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time all whose moments of order less than Ī± are finite. To illustrate analytical findings, results of numerical calculations and plots are presented.
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05.60.-k Transport processes
02.50.Cw Probability theory
02.60.Lj Ordinary and partial differential equations; boundary value problems
03.65.Ge Solutions of wave equations: bound states

Global existence and non-relativistic global limits of entropy solutions to the 1D piston problem for the isentropic relativistic Euler equations

Min Ding and Yachun Li

J. Math. Phys. 54, 031506 (2013); http://dx.doi.org/10.1063/1.4792474 (28 pages)

Online Publication Date: 14 March 2013

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We study the 1D piston problem for the isentropic relativistic Euler equations when the total variations of the initial data and the speed of the piston are sufficiently small. Employing a modified Glimm scheme, we establish the global existence of shock front solutions including a strong shock without restriction on the strength. In particular, we give some uniform estimates on the perturbation waves, the reflections of the perturbation waves on the piston and the strong shock. Meanwhile, we consider the convergence of the entropy solutions as the light speed c ā†’ +āˆž to the corresponding entropy solutions of the classical non-relativistic isentropic Euler equations.
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47.75.+f Relativistic fluid dynamics
05.70.Ce Thermodynamic functions and equations of state
47.40.Nm Shock wave interactions and shock effects

Expansion of the energy of the ground state of the Grossā€“Pitaevskii equation in the Thomasā€“Fermi limit

ClƩment Gallo

J. Math. Phys. 54, 031507 (2013); http://dx.doi.org/10.1063/1.4795245 (13 pages)

Online Publication Date: 18 March 2013

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From the asymptotic expansion of the ground state of the Grossā€“Pitaevskii equation in the Thomasā€“Fermi limit given by Gallo and Pelinovsky [ā€œOn the Thomas-Fermi ground state in a harmonic potential,ā€ Asymptot. Anal. 73(1ā€“2), 53ā€“96 (2011)]10.3233/ASY-2011-1034, we infer an asymptotic expansion of the kinetic, potential, and total energy of the ground state. In particular, we give a rigorous proof of the expansion of the kinetic energy calculated by Dalfovo, Pitaevskii, and Stringari [ā€œOrder parameter at the boundary of a trapped Bose gas,ā€ Phys. Rev. A 54, 4213ā€“4217 (1996)]10.1103/PhysRevA.54.4213 in the case where the space dimension is 3. Moreover, we calculate one more term in this expansion, and we generalize the result to space dimensions 1 and 2.
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03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow
21.60.-n Nuclear structure models and methods

Jet theoretical Yang-Mills energy in the geometric dynamics of two-dimensional monolayer

M. Neagu, N. G. Krylova, and H. V. Grushevskaya

J. Math. Phys. 54, 031508 (2013); http://dx.doi.org/10.1063/1.4795715 (14 pages)

Online Publication Date: 22 March 2013

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Langmuir-Blodgett (LB)-films consist of few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe a structurization into LB-monolayers. Consequently, we develop on the 1-jet space J1([0,āˆž),math2) the single-time Lagrange geometry (in the sense of distinguished (d-) connection, d-torsions, and an abstract anisotropic electromagnetic-like d-field) for the Lagrangian governing the 2D-motion of a particle of monolayer. One assumed that an expansion near singular points for the constructed geometrical Lagrangian theory describes phase transitions to LB-monolayer. Trajectories of particles in a field of the electrocapillarity forces of monolayer have been calculated in a resonant approximation utilizing a Jacobi equation. A jet geometrical Yang-Mills energy is introduced and some computer graphic simulations are exposed.
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68.47.Pe Langmuir-Blodgett films on solids; polymers on surfaces; biological molecules on surfaces
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