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I.K. Mylonas A.K. Rossides V.M. Rothos 《The European physical journal. Special topics》2016,225(6-7):1187-1197
In this work, we study the stability and internal modes of one-dimensional gap solitons employing the modified nonlinear Schrödinger equation with a sinusoidal potential together with the presence of a weak nonlocality. Using an analytical theory, it is proved that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self-defocusing nonlinearity, and one of these is always unstable. Also we study the oscillatory instabilities and internal modes of the modified nonlinear Schrödinger equation. 相似文献
2.
We study the instability of algebraic solitons for integrable
nonlinear equations in one spatial dimension that include modified
KdV, focusing NLS, derivative NLS, and massive Thirring equations.
We develop the analysis of the Evans function that defines
eigenvalues in the corresponding Lax operators with algebraically
decaying potentials. The standard Evans function generically has
singularities in the essential spectrum, which may include embedded
eigenvalues with algebraically decaying eigenfunctions. We construct
a renormalized Evans function and study bifurcations of embedded
eigenvalues, when an algebraically decaying potential is perturbed
by a generic potential with a faster decay at infinity. We show that
the bifurcation problem for embedded eigenvalues can be reduced to
cubic or quadratic equations, depending on whether the algebraic
potential decays to zero or approaches a nonzero constant. Roots of
the bifurcation equations define eigenvalues which correspond to
nonlinear waves that are formed from unstable algebraic solitons. Our results provide precise information on the transformation
of
unstable algebraic solitons in the time-evolution problem associated
with the integrable nonlinear equation. Algebraic solitons of the
modified KdV equation are shown to transform to either travelling
solitons or time-periodic breathers, depending on the sign of the
perturbation. Algebraic solitons of the derivative NLS and massive
Thirring equations are shown to transform to travelling and rotating
solitons for either sign of the perturbation. Finally, algebraic
homoclinic orbits of the focusing NLS equation are destroyed by the
perturbation and evolve into time-periodic space-decaying solutions. 相似文献
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We establish the splitting of homoclinic orbits for a near-integrable lattice modified KdV (mKdV) equation with periodic boundary conditions. We use the Bäcklund transformation to construct homoclinic orbits of the lattice mKdV equation. We build the Melnikov function with the gradient of the invariant defined through the discrete Floquet discriminant evaluated at critical points. The criteria for the persistence of homoclinic solutions of the perturbed lattice mKdV equation are established. 相似文献
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M. Fečkan M. Pospíšil V. M. Rothos H. Susanto 《Journal of Dynamics and Differential Equations》2013,25(3):795-820
In this article, damped Fermi–Pasta–Ulam-type lattices driven by extended external forces are considered. The existence and uniqueness results of periodic travelling waves of the system are presented. The existence and the stability of periodic waves are also computed and discussed numerically. 相似文献
5.
V.M. Rothos 《The European physical journal. Special topics》2016,225(6-7):943-958
In this review we try to capture some of the recent excitement induced by a large volume of theoretical and computational studies addressing nonlinear Schrödinger models (discrete and continuous) and the localized structures that they support. We focus on some prototypical structures, namely the breather solutions and solitary waves. In particular, we investigate the bifurcation of travelling wave solution in Discrete NLS system applying dynamical systems methods. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. We also offer an outlook on interesting possibilities for future work on this theme. 相似文献
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