arXiv subjects

Jared C. Bronski

Publications and source records attributed to Jared C. Bronski.

27 records · Page 2Linked to original sources

arXiv2010

On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions

In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multi-periodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.

math.AP↗

arXiv2009

An index theorem for the stability of periodic traveling waves of KdV type

We consider periodic solutions to equations of Korteweg-Devries type. While the stability theory for periodic waves has received much some attention the theory is much less developed than the analogous theory for solitary wave stability, and appears to be mathematically richer. We prove an index theorem giving an exact count of the number of unstable eigenvalues of the linearized operator in terms of the number of zeros of the derivative of the traveling wave profile together with geometric information about a certain map between the constants of integration of the ordinary differential equation and the conserved quantities of the partial differential equation. This index can be regarded as a generalization of both the Sturm oscillation theorem and the classical stability theory for solitary wave solutions for equations of Korteweg-de Vries type. In the case of a polynomial nonlinearity this index, together with a related one introduced earlier by Bronski and Johnson, can be expressed in terms of derivatives of period integrals on a Riemann surface. Since these period integrals satisfy a Picard-Fuchs equation these derivatives can be expressed in terms of the integrals themselves, leading to an expression in terms of various moments of the solution. We conclude with some illustrative examples.

math.AP↗

arXiv2009

The Modultional Instability for a Generalized KdV Equation

We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evan's function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first index counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second index provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Haragus and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator - in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests geometric interpretations attached to the vanishing of the modulational instability index previously mentioned.

math.AP↗

arXiv2009

Dislocation Defects and Diophantine Approximation

In this paper we consider a Schrodinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential spectrum, and in particular in counting the number of such eigenvalues. We use a homotopy argument in the width of the potential to count the eigenvalues as they are created. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in the essential spectrum is exactly one for sufficiently large gap number unless a certain Diophantine approximation problem has solutions, in which case there exists a subsequence of gaps containing 0,1 or 2 eigenvalues. We state some conditions under which the solvability of the Diophantine approximation problem can be established.

math-ph↗

arXiv2007

Eigenvalue estimates for the scattering problem associated to the sine-Gordon equation

One of the difficulties associated with the scattering problems arising in connection with integrable systems is that they are frequently non-self-adjoint, making it difficult to determine where the spectrum lies. In this paper, we consider the problem of locating and counting the discrete eigenvalues associated with the scattering problem for which the sine-Gordon equation is the isospectral flow. In particular, suppose that we take an initially stationary pulse for the sine-Gordon equation, with a profile that has either one extremum point of height less than pi and topological charge 0, or is monotone with topological charge +-1. Then we show that the point spectrum lies on the unit circle and is simple. Furthermore, we give a count of the number of eigenvalues. This result is an analog of that of Klaus and Shaw for the Zakharov-Shabat scattering problem. We also relate our results, as well as those of Klaus and Shaw, to the Krein stability theory for symplectic matrices. In particular we show that the scattering problem associated to the sine-Gordon equation has a symplectic structure, and under the above conditions the point eigenvalues have a definite Krein signature, and are thus simple and lie on the unit circle.

math.SP↗

arXiv2005

Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation

We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions ϕthe exponent 1.5 is the best possible from this line of argument. Further, this result together with a result of Molinet gives an improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.

math.AP↗

arXiv2005

Modulational instability for nonlinear Schrodinger equations with a periodic potential

We study the stability properties of periodic solutions to the Nonlinear Schrödinger (NLS) equation with a periodic potential. We exploit the symmetries of the problem, in particular the Hamiltonian structure and the $\U(1)$ symmetry. We develop a simple sufficient criterion that guarantees the existence of a modulational instability spectrum along the imaginary axis. In the case of small amplitude solutions that bifurcate from the band edges of the linear problem this criterion becomes especially simple. We find that the small amplitude solutions corresponding to the band edges alternate stability, with the first band edge being modulationally unstable in the focusing case, the second band edge being modulationally unstable in the defocusing case, and so on. This small amplitude result has a nice physical interpretation in terms of the effective mass of a particle in the periodic potential. We also consider, in somewhat less detail, some sideband instabilities in the small amplitude limit. We find that, depending on the Krein signature of the collision, these can be of one of two types. Finally we illustrate this with some exact calculations in the case where $V(x)$ is an elliptic function, where all of the relevant calculations can be done explicitly.

nlin.PS↗

arXiv2005

Instabilities in the Bogoliubov Spectrum of a condensate in a 1-D periodic potential

We study the stability of standing wave solutions to a one-dimensional Gross-Pitaevsky equation with a periodic potential. We use some simple complex analysis and the Hamiltonian structure of the problem to give a simple rigorous criterion which guarantees the existence of non-real spectrum, which corresponds to exponential instability of the standing wave solution. This criterion can be stated simply in terms of the spectrum of one of these self-adjoint operators. When the standing wave has small amplitude this criterion simplifies further, and agrees with arguments based on the effective mass in the periodic potential.

cond-mat.other↗

arXiv2000

Bose-Einstein condensates in standing waves: The cubic nonlinear Schroedinger equation with a periodic potential

We present a new family of stationary solutions to the cubic nonlinear Schroedinger equation with a Jacobian elliptic function potential. In the limit of a sinusoidal potential our solutions model a dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided the ratio of the height of the variations of the condensate to its DC offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Numerical simulations suggest such stationary states are experimentally observable.

cond-mat↗