Stability theory of solitary waves in the presence of symmetry, II☆

We study a type of ‘eigenvalue’ problem for systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function D (λ) introduced by J. W. Evans (1975) in his study of the stability of nerve impulses. We develop a general theory of D (λ) that clarifies the role of the essential spectrum in applications. New formulae for derivatives of D (λ) are used to study linear exponential instabilities of solitary waves for generalizations of: (1) the Korteweg-de Vries equation (KdV); (2) the Benjamin-Bona-Mahoney equation (BBM); and (3) the regularized Boussinesq equation. A pair of real eigenvalues exists, indicating a non-oscillatory instability, when the ‘momentum’ of the wave (a time-invariant functional associated with the hamiltonian structure of the equation) is a decreasing function of wave speed. Also we explain the mechanism of the transition to instability. Unexpectedly, these transitions are unlike typical transitions to instability in finite-dimensional hamiltonian systems. Instead they can be understood in terms of the motion of poles of the resolvent formula extended to a multi-sheeted Riemann surface. Finally, for a generalization of the KdV-Burgers equation (a model for bores), we show that a conjectured transition to instability does not involve real eigenvalues emerging from the origin, suggesting an oscillatory type of instability.

Eigenvalues, and Instabilities of Solitary Waves

An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green's functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of well-separated multi-bump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.

Stability of Travelling Waves

In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrodinger equation
 for sufficiently small ℏ>0, in which sense we call them “semiclassical bound states.” We assume that 1≦p<∞ forn=1,2 and 1≦p<1+4/(n−2) forn≧3, and thatV is in the class(V)
a in the sense of Kato for somea. For any finite collection {x
1,...,x
N} of nondegenerate critical points ofV, we construct a solution of the forme
−iEt/ℏv(x) forE<a, wherev is real and it is a small perturbation of a sum of one-lump solutions concentrated nearx
1,...,x
N respectively. The concentration gets stronger as ℏ→0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller ℏ>0. Indeed, for each such collection of critical points we construct 2
N−1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.

/pdf/on-positive-multi-lump-bound-states-of-nonlinear-schrodinger-120gehyrfd.pdf

On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential

In this paper we review a number of recent developments in the study of the Discrete Nonlinear Schrodinger (DNLS) equation. Results concerning ground and excited states, their construction, stability and bifurcations are presented in one and two spatial dimensions. Combinations of such steady states lead to the study of coherent structure bound states. A special case of such structures are the so-called twisted modes and their two-dimensional discrete vortex generalization. The ideas on such multi-coherent structures and their interactions are also useful in treating finite system effects through the image method. The statistical mechanics of the system is also analyzed and the partition function calculated in one spatial dimension using the transfer integral method. Finally, a number of open problems and future directions in the field are proposed.

The discrete nonlinear schrödinger equation: a survey of recent results

In this paper I am introducing a new technique for proving instability of bound states for Hamiltonian systems. There are already two disparate types of instability results in the literature. The approach developed by Strauss-Shatah [20] gave an instability criterion coming from the variational structure of the problem; on the other hand, Jones' approach [11] produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques. It turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results. Finally in order to demonstrate how this method works I apply the instability criterion in some specific examples.

Linearized instability for nonlinear schrödinger and klein‐gordon equations

Partial differential equations that conserve energy can often be written as infinite-dimensional Hamiltonian systems of the following general form: du/dt=JE'(u), where J:X * →X is a symplectic matrix (i.e., JJ * =−1) and E: X→R is a C 2 functional defined on some Hilbert space X

Manoussos Grillakis

Papers

Linearized instability for nonlinear schrödinger and klein‐gordon equations

Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system