Transverse instability and its long-term development for solitary waves of the (2+1)-Boussinesq equation

The stability properties of line solitary wave solutions of the (2+1)-dimensional Boussinesq equation with respect to transverse perturbations and their consequences are considered. A geometric condition arising from a multi-symplectic formulation of this equation gives an explicit relation between the parameters for transverse instability when the transverse wavenumber is small. The Evans function is then computed explicitly, giving the eigenvalues for transverse instability for all transverse wavenumbers. To determine the nonlinear and long time implications of transverse instability, numerical simulations are performed using pseudospectral discretization. The numerics confirm the analytic results, and in all cases studied, transverse instability leads to collapse.Comment: 16 pages, 8 figures; submitted to Phys. Rev.

Solitons in Triangular and Honeycomb Dynamical Lattices with the Cubic Nonlinearity

We study the existence and stability of localized states in the discrete nonlinear Schr{\"o}dinger equation (DNLS) on two-dimensional non-square lattices. The model includes both the nearest-neighbor and long-range interactions. For the fundamental strongly localized soliton, the results depend on the coordination number, i.e., on the particular type of the lattice. The long-range interactions additionally destabilize the discrete soliton, or make it more stable, if the sign of the interaction is, respectively, the same as or opposite to the sign of the short-range interaction. We also explore more complicated solutions, such as twisted localized modes (TLM's) and solutions carrying multiple topological charge (vortices) that are specific to the triangular and honeycomb lattices. In the cases when such vortices are unstable, direct simulations demonstrate that they turn into zero-vorticity fundamental solitons.Comment: 17 pages, 13 figures, Phys. Rev.

Effects of nonlocal dispersive interactions on self-trapping excitations

A one-dimensional discrete nonlinear Schrodinger (NLS) model with the power dependence r(-s) on the distance r of the dispersive interactions is proposed. The stationary states psi(n) of the system are studied both analytically and numerically. Two types of stationary states are investigated: on-site and intersite states. It is shown that for s sufficiently large all features of the model are qualitatively the same as in the NLS model with a nearest-neighbor interaction. For s less than some critical value s(cr), there is an interval of bistability where two stable stationary states exist at each excitation number N = Sigma(n)\psi(n)\(2). For cubic nonlinearity the bistability of on-site solitons may occur for dipole-dipole dispersive interaction (s = 3), while s(cr) for intersite solitons is close to 2.1. For increasing degree of nonlinearity sigma, s(cr) increases. The long-distance behavior of the intrinsically localized states depends on s. For s > 3 their tails are exponential, while for 2 <s <3 they are algebraic. In the continuum limit the model is described by a nonlocal MLS equation for which the stability criterion for the ground state is shown to be s <sigma + 1

Asymptotic stability of solitary waves

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, In particular, we study the case where f(u)=u p+1 / (p+1) , p =1, 2, 3 (and 30, with f ∈ C 4 ). The same asymptotic stability result for KdV is also proved for the case p =2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values of p between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameter p increases beyond the critical value p =4.) The solution is decomposed into a modulating solitary wave, with time-varying speed c(t) and phase γ( t ) ( bound state part ), and an infinite dimensional perturbation ( radiating part ). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. As p →4 − , the local decay or radiation rate decreases due to the presence of a resonance pole associated with the linearized evolution equation for solitary wave perturbations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46489/1/220_2005_Article_BF02101705.pd