Dmitry E. Pelinovsky

Bifurcations and stability of gap solitons in periodic potentials

We analyze the existence, stability, and internal modes of gap solitons in nonlinear periodic systems described by the nonlinear Schrödinger equation with a sinusoidal potential, such as photonic crystals, waveguide arrays, optically-induced photonic lattices, and Bose-Einstein condensates loaded onto an optical lattice. We study bifurcations of gap solitons from the band edges of the Floquet-Bloch spectrum, and show that gap solitons can appear near all lower or upper band edges of the spectrum, for focusing or defocusing nonlinearity, respectively. We show that, in general, two types of gap solitons can bifurcate from each band edge, and one of those two is always unstable. A gap soliton corresponding to a given band edge is shown to possess a number of internal modes that bifurcate from all band edges of the same polarity. We demonstrate that stability of gap solitons is determined by location of the internal modes with respect to the spectral bands of the inverted spectrum and, when they overlap, complex eigenvalues give rise to oscillatory instabilities of gap solitons.

Wave group dynamics in weakly nonlinear long-wave models

Abstract The dynamics of wave groups is studied for long waves, using the framework of the extended Korteweg–de Vries equation. It is shown that the dynamics is much richer than the corresponding results obtained just from the Korteweg–de Vries equation. First, a reduction to a nonlinear Schrodinger equation is obtained for weakly nonlinear wave packets, and it is demonstrated that either the focussing or the defocussing case can be obtained. This is in contrast to the corresponding reduction for the Korteweg–de Vries equation, where only the defocussing case is obtained. Next, the condition for modulational instability is obtained. It is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term in the extended Korteweg–de Vries equation, and for a high carrier frequency. At the boundary of this parameter space, a modified nonlinear Schrodinger equation is derived, and its steady-state solutions, including an algebraic soliton, are found. The exact breather solution of the extended Korteweg–de Vries equation is analysed. It is shown that in the limit of weak nonlinearity it transforms to a wave group with an envelope described by soliton solutions of the nonlinear Schrodinger equation and its modification as described above. Numerical simulations demonstrate the main features of wave group evolution and show some differences in the behaviour of the solutions of the extended Korteweg–de Vries equation, compared with those of the nonlinear Schrodinger equation.

Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations

Spectral stability analysis for solitary waves is developed in the context of the Hamiltonian system of coupled nonlinear Schrödinger equations. The linear eigenvalue problem for a non–self–adjoint operator is studied with two self–adjoint matrix Schrödinger operators. Sharp bounds on the number and type of unstable eigenvalues in the spectral problem are found from the inertia law for quadratic forms, associated with the two self–adjoint operators. Symmetry–breaking stability analysis is also developed with the same method.

Averaging for Solitons with Nonlinearity Management

We develop an averaging method for solitons of the nonlinear Schrödinger equation with a periodically varying nonlinearity coefficient, which is used to effectively describe solitons in Bose-Einstein condensates, in the context of the recently proposed technique of Feshbach resonance management. Using the derived local averaged equation, we study matter-wave bright and dark solitons and demonstrate a very good agreement between solutions of the averaged and full equations.

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

Count of eigenvalues in the generalized eigenvalue problem

We study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite-dimensional Hamiltonian systems. The theory is based on Pontryagin’s invariant subspace theorem and extends beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others. Our main results are (i) the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self-adjoint operators, (ii) the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self-adjoint operators, and (iii) the quadratic forms defined by the two self-adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem. Applicatio...

Stable all-optical limiting in nonlinear periodic structures. I. Analysis

We consider propagation of coherent light through a nonlinear periodic optical structure consisting of two alternating layers with different linear and nonlinear refractive indices. A coupled-mode system is derived from the Maxwell equations and analyzed for the stationary-transmission regimes and linear time-dependent dynamics. We find the domain for existence of true all-optical limiting when the input–output transmission characteristic is monotonic and clamped below a limiting value for output intensity. True all-optical limiting can be managed by compensating the Kerr nonlinearities in the alternating layers, when the net-average nonlinearity is much smaller than the nonlinearity variance. The periodic optical structures can be used as uniform switches between lower-transmissive and higher-transmissive states if the structures are sufficiently long and out-of-phase, i.e., when the linear grating compensates the nonlinearity variations at each optical layer. We prove analytically that true all-optical limiting for zero net-average nonlinearity is asymptotically stable in time-dependent dynamics. We also show that weakly unbalanced out-of-phase gratings with small net-average nonlinearity exhibit local multistability, whereas strongly unbalanced gratings with large net-average nonlinearity display global multistability.

Matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates.

We study matter-wave bright solitons in spin-orbit coupled Bose-Einstein condensates with attractive interactions. We use a multiscale expansion method to identify solution families for chemical potentials in the semi-infinite gap of the linear energy spectrum. Depending on the linear and spin-orbit coupling strengths, the solitons may present either a sech2-shaped or a modulated density profile reminiscent of the stripe phase of spin-orbit coupled repulsive Bose-Einstein condensates. Our numerical results are in excellent agreement with our analytical findings and demonstrate the potential robustness of solitons for experimentally relevant conditions.

Global Well-Posedness of the Short-Pulse and Sine–Gordon Equations in Energy Space

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space H 2. Our analysis relies on local well-posedness results of Schäfer and Wayne [15], the correspondence of the short-pulse equation to the sine–Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and global well-posedness of the sine–Gordon equation in an appropriate function space.

On the exchange of energy in coupled Klein-Gordon equations

Abstract We consider a system of coupled Klein–Gordon equations, which models one-dimensional nonlinear wave processes in two-component media. We find both linear and nonlinear solutions involving the exchange of energy between the different components of the system. The solutions are a continuum generalization of the classical example of energy exchange in Mandelshtam’s system of coupled pendulums.