Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y^4 = x^5 + λ[4]x^4 + λ[3]x^3 + λ[2]x^2 + λ[1]x + λ[0]. We construct Abelian functions using the multivariate sigma-function associated with the curve, generalizing the theory of theWeierstrass℘-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula

New Kinds of Acoustic Solitons

We find that the modified sine-Gordon equation belonging to the class of the soliton equations describes the propagation of extremely short transverse acoustic pulses through the low-temperature crystal containing paramagnetic impurities with effective spin S=1/2 in the Voigt geometry case. The features of nonlinear dynamics of strain field and effective spins, which correspond to the different kinds of acoustic solitons, are studied.Comment: 9 pages, 1 figur

A nonlocal eigenvalue problem and the stability of spikes for reaction-diffusion systems with fractional reaction rates

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction-diffusion systems with fractional reaction rates such as the Sel'kov model, the Gray-Scott system, the hypercycle Eigen and Schuster, angiogenesis, and the generalized Gierer-Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters

Biphonons in the Klein-Gordon lattice

A numerical approach is proposed for studying the quantum optical modes in the Klein-Gordon lattices where the energy contribution of the atomic displacements is non-quadratic. The features of the biphonon excitations are investigated in detail for different non-quadratic contributions to the Hamiltonian. The results are extended to multi-phonon bound states.Comment: Comments and suggestions are welcom

Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity

We address the issue of mobility of localized modes in two-dimensional nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes e.g. discrete spatial solitons in a tight-binding approximation of two-dimensional optical waveguide arrays made from photorefractive crystals. We discuss numerically obtained exact stationary solutions and their stability, focussing on three different solution families with peaks at one, two, and four neighboring sites, respectively. When varying the power, there is a repeated exchange of stability between these three solutions, with symmetry-broken families of connecting intermediate stationary solutions appearing at the bifurcation points. When the nonlinearity parameter is not too large, we observe good mobility, and a well defined Peierls-Nabarro barrier measuring the minimum energy necessary for rendering a stable stationary solution mobile.Comment: 19 pages, 4 figure

Identities for hyperelliptic P-functions of genus one, two and three in covariant form

We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3

Solitons in one-dimensional nonlinear Schr\"{o}dinger lattices with a local inhomogeneity

In this paper we analyze the existence, stability, dynamical formation and mobility properties of localized solutions in a one-dimensional system described by the discrete nonlinear Schr\"{o}dinger equation with a linear point defect. We consider both attractive and repulsive defects in a focusing lattice. Among our main findings are: a) the destabilization of the on--site mode centered at the defect in the repulsive case; b) the disappearance of localized modes in the vicinity of the defect due to saddle-node bifurcations for sufficiently strong defects of either type; c) the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects; and d) the detailed elucidation as a function of initial speed and defect strength of the different regimes (trapping, trapping and reflection, pure reflection and pure transmission) of interaction of a moving localized mode with the defect.Comment: 12 pages, 10 figure

Dynamical structure factor of a nonlinear Klein-Gordon lattice

The quantum modes of a nonlinear Klein-Gordon lattice have been computed numerically [L. Proville, Phys. Rev. B 71, 104306 (2005)]. The on-site nonlinearity has been found to lead to phonon bound states. In the present paper, we compute numerically the dynamical structure factor so as to simulate the coherent scattering cross section at low temperature. The inelastic contribution is studied as a function of the on-site anharmonicity. Interestingly, our numerical method is not limited to the weak anharmonicity and permits one to study thoroughly the spectra of nonlinear phonons

Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition

Oscillatory instabilities in Hamiltonian anharmonic lattices are known to appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions of multibreather type. Here, we analyze the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear: the three-site discrete nonlinear Schr\"odinger model with periodic boundary conditions. The stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes. We numerically analyze the nature of the two bifurcations leading to this instability and find them to be of two different types. Close to the lower-amplitude threshold stable two-frequency quasiperiodic solutions exist surrounding the unstable stationary solution, and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small. By contrast, close to the higher-amplitude threshold all two-frequency quasiperiodic solutions are detached from the unstable stationary solution, and the resulting dynamics is of 'population-inversion' type involving also the originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen. Revised and shortened version with few clarifying remarks adde

Homoclinic standing waves in focussing DNLS equations --Variational approach via constrained optimization

We study focussing discrete nonlinear Schr\"{o}dinger equations and present a new variational existence proof for homoclinic standing waves (bright solitons). Our approach relies on the constrained maximization of an energy functional and provides the existence of two one-parameter families of waves with unimodal and even profile function for a wide class of nonlinearities. Finally, we illustrate our results by numerical simulations.Comment: new version with revised introduction and improved condition (A3); 16 pages, several figure