The algebraic structure behind the derivative nonlinear Schroedinger equation

The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schr\" odinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of a $\hat{s\ell}_2$ Kac-Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows, respectively. The equivalence between the latter and the massive Thirring model is explicitly demonstrated also. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.Comment: references adde

Nonlinear Supersymmetry as a Hidden Symmetry

Ver abstrac

Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations

A reduced-order model algorithm, called ALP, is proposed to solve nonlinear evolution partial differential equations. It is based on approximations of generalized Lax pairs. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the basis on which the solution is searched for evolves in time according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive front or wave propagation. Another difference with other reduced-order methods is that it is not based on an off-line / on-line strategy. Numerical examples are shown for the linear advection, KdV and FKPP equations, in one and two dimensions

Global Existence Without Decay

We treat some problems related to the small data global existence theory of some dispersive PDEs. That is, we try to abstractly construct solutions which exist globally in time and have “good” properties, under a smallness condition on the initial data, typically given at time t = 0. What is special here is that we do not impose strong decay on this data, that is, we only assume that they are L2 based Sobolev functions. More precisely, we treat 1. the Klein-Gordon equation with a quadratic nonlinearity Q, positive mass and initial data from appropriate standard Sobolev spaces. We can also treat systems under a condition on the masses involved in the nonlinear interactions. 2. a special quadratic nonlinear Schrödinger equation with initial data in the scaling critical Sobolev space and 3. the modified Novikov-Veselov equation in two space dimensions, which has a cubic nonlinearity containing roughly one derivative. 3. the modified Novikov-Veselov equation in two space dimensions, which has a cubic nonlinearity containing roughly one derivative. For each of the above equations and initial data from a sufficiently small ball around the origin, we construct global solutions which scatter and depend smoothly on the initial data, using a fixed point argument. In the second part of this work, we turn towards negative results and start with the observation that a solution operator constructed by the techniques used in the proofs of the statements above imply that there is a smooth scattering operator, which in turn shows that a trilinear spacetime interaction of free waves can be bounded by their inital data. Such an estimate is very close to so-called convolution estimates in Fourier spacetime, for which the behavior is known, and we can use this to derive contradictions in some cases. This is related to the concept of time resonance, and we can show that the results above for the Klein-Gordon and Schrödinger equations are sharp in some sense. Regarding the modified Novikov-Veselov equation, we show a negative result for the related Novikov-Veselov equation, for which the nonlinearity is replaced by a quadratic nonlinearity containing roughly one derivative