Narrow Results

In this paper, we mainly study the resonant interactions of lump chains of (2+1)-dimensional generalized Korteweg–de Vries (gKdV) equation, which can be used to describe the nonlinear wave phenomena in plasma physics, weakly dispersive media and fluid dynamics. We construct the lump chain solutions by adding some constraints to the soliton solutions of the gKdV equation. Then through the asymptotic analysis, the analytical solution of each branch of the lump chains after interaction is obtained. Through the phase shift analysis of the analytical solution, we get the condition of resonance, and then the resonant interactions of the two- and three-lump chains are analyzed in detail. In particular, we explore two cases including the cross-resonant interaction and parallel-resonant interaction of the lump chains. In addition, 2D and 3D images are used to describe the spatial evolution of resonant interactions. The results can further help us study the resonant interactions between the lump chains and other nonlinear waves.

The relaxation mechanisms of local vibrational modes in solids are of central importance in understanding how such modes decay into phonons, the diffusion of impurities, and the loss of hydrogen passivation in technologically important materials. Interstitial oxygen in silicon is a model system for studying the relaxation of local vibrational modes into extended vibrational modes by measuring the interactions between the two. We have used hydrostatic pressure to bring the antisymmetric stretch mode of Si:¹⁸O_i into resonance with the second harmonic of the ¹⁸O_i resonant mode and observed an anticrossing between the two vibrational modes near pressures of 4 GPa. A theoretical model of this interaction produced excellent agreement with the experimentally observed frequencies and linewidths.

It has previously been established that in a prestressed incompressible elastic plate, a near-critical instability mode can interact resonantly with two non-critical vibration modes and that the evolution equations for the three modes take the form d² A₁/dτ² = -c₀A₁ - c₁|A₁|²A₁ - γ₁Ā₂Ā₃, dA₂/dτ = γ₂Ā₁Ā₃ and dA₃/dτ = γ₃Ā₁Ā₂, where a bar denotes complex conjugation, τ is a slow time variable and c₀, c₁, γ₁, γ₂, γ₃ are real constants. In this paper, we analyze these amplitude equations using dynamical systems theory. Regions of the parametric space that correspond to bounded solutions are determined and some explicit representations of the bounded solutions are obtained. It is shown that the resonant-triad interaction can lead to chaotic motions.

In this paper, we presented a physical scheme to generate the multi-cavity maximally entangled W state via cavity QED. All the operations needed in this scheme are to modulate the interaction time only once.