We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi‐linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from Stability of Travelling wave solutions of Nonlinear Dispersive equations of NLS type, where Leisman et al. compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of Leismen et al.​to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave.

We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi-linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from [LBJM2019], where Leisman et al. compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of [LBJM2019] to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave. This paper deals with modulational stability of equation ( 1), that is, the spectral stability subject to long-wavelength perturbations. Rigorously speaking, this amounts to expanding the spectrum of the linearized operator in a neighbourhood of the origin in the spectral plane. Whitham modulation theory [

We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill's equation. Applying the Evans-Krein function theory of [KM2014] to our Evans function, we provide a new method for computing the Krein signatures of simple characteristic values of the linearised sine-Gordon equation. By varying the Floquet exponent parametrising the quasi-periodic solutions, we compute the linearised spectra of periodic travelling wave solutions of the sine-Gordon equation and the locations of Hamiltonian-Hopf bifurcations therein. Finally, we show that our new Evans function can be readily applied to the general case of the nonlinear Klein-Gordon equation with a non-periodic potential.

The LM libel case in which ITN successfully sued the left-wing magazine ended in March, with LM ordered to pay £375,000 in damages. The result has been to farce the magazine into closure. The case raised some fundamental issues about press freedom, reporting methods, the role of a magazine like LM, F.nglish libel law and, indeed, the nature of modern journalism.British Journalism Review belie11es the case is of such importance that we in11ited the two main contestants to put their case. MICK HUME, &Jitor of LM, and RICHARD TAIT, &Jitor-in-Chief of ITN present their arguments:Was LM bullied into oblivion ... LM magazine closed in April, after the High Court awarded ITN and two reporters a total of £375,000 in libel damages over an article we published in 1997 about their famous pictures of an emaciated Bosnian Muslim and a barbed wire fence at the Bosnian Serb-run Trnopolje camp. The verdict was bad news for me, as LM editor, for the magazine's co-publishers Helene Guldberg and Claire Fox, for all of LM's friends -and for everybody else

We prove that the modulational instability criterion of the formal Whitham modulation theory agrees with the spectral stability of long-wavelength perturbations of periodic traveling wave solutions to the generalized Whitham equation. We use the standard WKB procedure to derive a quasilinear system of three Whitham modulation equations, written in terms of the mass, momentum, and wave number of a periodic traveling wave solution. We use the same quantities as parameters in a rigorous spectral perturbation of the linearized operator, which allows us to track the bifurcation of the zero eigenvalue as the Floquet parameter varies. We show that the hyperbolicity of the Whitham system is a necessary condition for the existence of purely imaginary eigenvalues in the linearized system, and hence also a prerequisite for modulational stability of the underlying wave. Since the generalized Whitham equation has a Hamiltonian structure, we conclude that strict hyperbolicity is a sufficient condition for modulational stability.