Existence of multi-bump solutions for a system with critical exponent in RN

We consider the following system with critical exponent in ${\mathbb{R}}^N$:$\{\begin{array}{c}?\mathrm{\Delta }u=K_1(y)u^{2^{?}?1}+\frac{p}{2^{?}}V(y)u^{p?1}{v}^q\text{in}{\mathbb{R}}^N,\hfill \\ ?\mathrm{\Delta }v=K_2(y)v^{2^{?}?1}+\frac{q}{2^{?}}V(y)u^p{v}^{q?1}\text{in}{\mathbb{R}}^N,\hfill \\ u,v>0,y\in {\mathbb{R}}^N,\hfill \end{array}$ where $N\ge 5$, $p,q>1$ and $p+q=2^{?}=\frac{2N}{N?2}$. Using finite dimensional reduction method, we prove the existence of multi-bump solutions. Their bumps can be placed on arbitrarily many or even infinitely many lattice points in ${\mathbb{R}}^N$. Since $p<2$ or $q<2$, we introduce two new norms to avoid singularity.     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.     

Gamow Vectors in Exactly Solvable Models

Through two examples: the Friedrichs model and a particular case of central potential scattering, we illustrate the way of constructing Gamow vectors.     

Fundamental vibrational frequencies and spectroscopic constants for the methylperoxyl radical,CH3O2, and related isotopologues 13CH3OO,CH3 18O18O,and CD3OO

Accurate spectroscopic and geometric constants for CH₃O₂, and its isotopologues ¹³CH₃OO, CH₃ ¹⁸O¹⁸O and CD₃OO, are predicted. Employing coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)], we obtain optimized equilibrium geometries using Dunning's cc-pVTZ basis set. A Taylor expansion of the potential energy surface, including all third-order and semidiagonal fourth-order terms in a basis of normal coordinates, yields anharmonic vibrational frequencies and vibrationally-averaged properties including the effects of anharmonicity. We detail the strong influence of Fermi resonances on the problematic ν₆ vibrational mode of CD₃OO, arriving at a value of 993?cm^?1; two previous experimental measurements of this mode appear to have been incorrectly assigned. Our computed energies for the low intensity ν₁₁ transition are in excellent agreement with experimental measurements performed for CH₃ ¹⁸O¹⁸O and CD₃OO, inspiring confidence that our results will serve as a guide for experimental measurement of this yet-unobserved quantity for the CH₃OO and ¹³CH₃OO isotopologues. Given the reliability of our force field, and considering the results of other experiments, we make a number of reassignments to previously recorded spectra, which eliminate large disagreements between experimental observations. The vibrational averaging of the rotational constants and geometries are also discussed for each isotopologue.     

High spin states and evidence for octupole correlations in 117Xe

High-spin states of ¹¹⁷Xe were investigated by means of in-beam γ-ray spectroscopic techniques via the reaction of ²⁸Si bombarding a ⁹²Mo target at beam energies of 100-120 MeV. A positive-parity rotational band decaying into the yrast negative-parity band by a series of enhanced E1 transitions was observed for the first time, implying the existence of octupole correlations in ¹¹⁷Xe. The B(E1) values increase with spin. The νd_5/2 band was firmly established up to 27/2⁺ and the B(M1)/B(E2) ratios were extracted from the relative intensities of γ-rays in this band. The previously known νh_11/2 and νg_7/2 [413]5/2⁺α=−1/2 bands were confirmed and extended up to high spins and two bandcrossings are observed in the latter at ħω=0.33 and 0.44 MeV, respectively. The bandcrossings and configurations of these bands are discussed by TRS and CSM calculations. In a γ–γ–t measurement, the 11/2⁻ and 7/2⁻ levels were identified as two isomers with half-lives of 59.4±20 ns and 16.5±8.0 ns, respectively. Received: 17 June 1997 / Revised version: 29 September 1997     

Formation of intermediate coupling optical polarons and bipolarons in two-dimensional systems

The formation of the optical polaron and bipolaron in two-dimensional (2D) systems is studied in the intermediate electron–phonon coupling regime. The total energies of the 2D polaron and bipolaron are calculated by using the Buimistrov–Pekar method of canonical transformations. The obtained results are compared with other existing results obtained by using the Feynman path integral method and the modified Lee–Low–Pines unitary transformation method. It is shown that the electron–phonon correlation significantly reduces the total energy of the 2D polaron in comparison with the energy of the strong coupling (adiabatic) polaron. It is found that the polaron formation in 2D systems is possible when the electron–phonon coupling constant α is greater than the critical value ${\alpha }_c?2.94$, which is much lower than a critical value of the electron–phonon coupling constant α in three-dimensional (3D) systems. The critical values of the Fröhlich coupling constant α and the ratio $\eta ={\epsilon }_{\infty }/{\epsilon }_0$ (where ${\epsilon }_{\infty }$ and ${\epsilon }_0$ are the high frequency and static dielectric constants, respectively), which determine the bipolaron stability region in 2D systems, are calculated numerically. It is interesting for application to the layered cuprate superconductors that the (bi)polarons are formed more easily in quasi-2D regions than in the bulk. It is argued that the high-$T_c$ cuprate superconductivity can exist above the bulk superconducting transition temperature $T_c$ as the persisting superfluidity of polaronic (bosonic) Cooper pairs and large bipolarons at quasi-2D grain boundaries or in the CuO₂ layers above $T_c$.     

High-spin nuclear states and persistent currents in mesoscopic rings

An analogy is presented between periodic persistent currents in mesoscopic rings and staggerings of gamma energy transitions from some nuclear high-spin states. Various sources of damping of the expected periodic structures in both physical systems are compared. This discussion provides, in the nuclear case, a tentative explanation of the scarcity of such staggerings, their appearance near ¹⁵⁰Gd and the existence of a spin-window for their observation. Received: 21 July 1997 / Revised version: 13 October 1997     

Nonlinear resonances and self-excited oscillations of a rotor caused by radial clearance and collision

This paper investigates oscillations in a flexible rotor system with radial clearance between an outer ring of the bearing and a casing by experiments and numerical simulations. The mathematical model considers the collisions of the bearing with the casing. The following phenomena are found: (1) Nonlinear resonances of subharmonic, super-subharmonic and combination oscillation occur. (2) Self-excited oscillation of a forward whirling mode occurs in a wide range above the major critical speed. (3) Entrainment phenomena from self-excited oscillation to nonlinear forced oscillation occur at these nonlinear resonance ranges. Moreover, this study analyzes periodic solutions of the mathematical model by the Harmonic Balance Method (HBM). As the results, the nonlinear resonances of subharmonic oscillation and its entrainment phenomenon can be explained theoretically by investigating the stability of the periodic solutions. The influence of the static force and the bearing damping on these oscillation are also clarified.