Abstract. We consider the initial value problem for the fourth order nonlinear Schrödinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.
In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein-Gordon equation (NLKG) in two space dimensions:For a given asymptotic profile u ap , we construct a solution u to (NLKG) which converges to u ap as t → ∞. Here the asymptotic profile u ap is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.
Please cite this article in press as: S. Masaki, J.-i. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. I. H. Poincaré -AN (2017), http://dx.
AbstractIn this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation in the scale criticalL r space whereL r = {f ∈ S (R)| f Lr = f L r < ∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted tô L r -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.
Abstract. We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given asymptotic profile u ap , there exists a solution u to (δ-NLS) which converges to u ap in L 2 (R) as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.
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