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Register a new userOn the Dynamics of solitons in the nonlinear Schroedinger equation
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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).
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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 try to obtain meromorphic solution of Time dependent Schroedinger equation which partially satisfies Painleve Integrable property. Our study and analysis exhibits meromorphic behavior of classical particle trajectory. In other words, particle is confined in punctured complex domain in singular fundamental domain. . We have explicitly developed solution from Jacobi Elliptic function and pole expansion approach in which solution remains meromorphic . Branch point analysis also shows solution branches out near such singular point. Meromorphic behavior is significant for a classical particle within quantum limit.
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We study large time behavior of quantum walks (QW) with self-dependent coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QW such as Strichartz estimate. Such argument is standard in the study of nonlinear Schrodinger equations but it seems to be the first time to be applied to QW. We also numerically study the dynamics of QW and observe soliton like solutions.
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.
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