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Register a new userThe Dirac equation without spinors
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In the first part of the paper we give a tensor version of the Dirac equation. In the second part we formulate and analyse a simple model equation which for weak external fields appears to have properties similar to those of the 2--dimensional Dirac equation.
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We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrodinger equation at zero energy are also shown.
begin{abstract} We show that if the initial profile $qleft( xright) $ for the Korteweg-de Vries (KdV) equation is essentially semibounded from below and $int^{infty }x^{5/2}leftvert qleft( xright) rightvert dx<infty,$ (no decay at $-infty$ is required) then the KdV has a unique global classical solution given by a determinant formula. This result is best known to date. end{abstract}
We study the behavior of the soliton solutions of the equation i((partial{psi})/(partialt))=-(1/(2m)){Delta}{psi}+(1/2)W_{{epsilon}}({psi})+V(x){psi} where W_{{epsilon}} is a suitable nonlinear term which is singular for {epsilon}=0. We use the strong nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for {epsilon}to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x).
We consider the mass-critical focusing nonlinear Schrodinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrodinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.
We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self--similar solution $chi_T$ of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long time behaviour (in similarity coordinates) of linear perturbations around $chi_T$ is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of $chi_T$ with the sharp decay rate for the perturbations.
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