We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.
We prove the existence of time-periodic solutions and spatially localised solutions (breathers), in general nonlinear Klein-Gordon infinite lattices. The existence problem is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder's Fixed Point Theorem.
Klein-Gordon equations describe the dynamics of waves/particles in sub-atomic scales. For nonlinear Klein-Gordon equations, their breather solutions are usually known as time periodic solutions with the vanishing spatial-boundary condition. The existence of breather solution is known for the Sine-Gordon equations, while the Sine-Gordon equations are also known as the soliton equation. The breather solutions is a certain kind of time periodic solutions that are not only play an essential role in the bridging path to the chaotic dynamics, but provide multi-dimensional closed loops inside phase s…
In this paper we consider a discrete Klein-Gordon (dKG) equation on $\ZZ^d$ in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength $\eps$. By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG breather from existence and linear stability of the corresponding dNLS soliton. Nonlinear stability, for an exponentially long time scale of the order $\mathcal{O}(\exp(\eps^{-1}))$, is also obtained via the normal form technique, together with highe…
The generalized perturbative reduction method is used to find the two-component vector breather solution of the nonlinear Klein-Gordon equation. It is shown that the nonlinear pulse oscillates with the sum and difference of frequencies and wave numbers in the region of the carrier wave frequency and wave number. Explicit analytical expressions for the profile and parameters of the nonlinear pulse are obtained. In the particular case, the vector breather coincides with the vector $0π$ pulse of self-induced transparency.
In this report, the various 1D single soliton and multi-soliton solutions of the Sine-Gordon equation are explored. First the topological kink solitons and their properties for the Sine-Gordon, as well as the $ϕ^{4}$ model are investigated. This includes energy and momentum conservation laws for both models, as well as static kinks and antikinks. The Bogomol'nyi equation and the importance of Lorentz invariance are also discussed. Then in the second half, the focus is on Sine-Gordon multi-kink solutions, which are found by a separation of variables method rather than using inverse scattering o…
We revisit the problem of transverse instability of a 2D breather stripe of the sine-Gordon (sG) equation. A numerically computed Floquet spectrum of the stripe is compared to analytical predictions developed by means of multiple-scale perturbation theory showing good agreement in the long-wavelength limit. By means of direct simulations, it is found that the instability leads to a breakup of the quasi-1D breather in a chain of interacting 2D radial breathers that appear to be fairly robust in the dynamics. The stability and dynamics of radial breathers in a finite domain are studied in detail…
We consider the existence and spectral stability of multi-breather structures in the discrete Klein-Gordon equation, both for soft and hard symmetric potentials. To obtain analytical results, we project the system onto a finite-dimensional Hilbert space consisting of the first $M$ Fourier modes, for arbitrary $M$. On this approximate system, we then take a spatial dynamics approach and use Lin's method to construct multi-breathers from a sequence of well-separated copies of the primary, single-site breather. We then locate the eigenmodes in the Floquet spectrum associated with the interaction …
To study the propagation of nonlinear waves across Y- and T-type junctions, we consider the 2D sine--Gordon equation as a model and study the dynamics of kinks and breathers in such geometries. The comparison of the energies reveals that the angle of the fork plays no role. Motivated by this, we introduce a 1D effective equation whose solutions agree well with the 2D simulations for kink and breather solutions. For branches of equal width, breather crossing occurs approximately when $v > 1 - ω$, where $v$ is the breather celerity and $ω$ is its frequency. We then characterize the breathers in …
The dynamics of sine-Gordon breathers is studied in the presence of dissipative and stochastic perturbations. Taking a stationary breather with a random phase value as the initial state, the performed simulations demonstrate that a spatially-homogeneous noisy source can make the oscillatory excitation more stable, i.e., it enables the latter to last significantly longer than it would in a noise-free scenario. Both the frequency domain and the localization of energy are examined to document the effectiveness of the noise-enhanced stability phenomenon, which emerges as a nonmonotonic behavior of…
Evolution of the nonequilibrium inhomogeneities and topological defects is studied in terms of complex kink solutions of the sine-Gordon equation. The weakly damped oscillation of the sine-Gordon kink, named as the kink quasimode, is described explicitly. It is shown that the oscillatory kink behavior and the wave packet generation depend significantly on the initial nonequilibrium kink profile. In order to specify conditions of the generation of wobbling kinks with a multibreather structure we reformulate the direct scattering problem associated with the SG equation as the spectral problem of…
We prove that finite energy solutions to the nonlinear Schroedinger equation and nonlinear Klein--Gordon equation which have the compact time spectrum have to be one-frequency solitary waves. The argument is based on the generalization of the Titchmarsh convolution theorem to partial convolutions.
在非线性展开系数的符号条件下，我们构造了Klein-Gordon方程呼吸子的所有阶数的级数解，以振幅参数的幂为单位。由于一维双孤子势薛定谔方程的性质，所有项都可以计算。证明了该级数在复平面的单位圆内无极点的充要条件是该方程本质上等价于sine-Gordon方程。这似乎是唯一的结果，解释了是什么使sine-Gordon方程特殊的偏微分方程的观点，并在同一时间，为什么近似，长寿呼吸存在。在一篇相关的论文（Classical and Quantum Gravity，25（2008）245004）中，我们扩展了这种方法来解决广义相对论中的孤子星问题。
