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We prove that standing-waves solutions to the non-linear Schrodinger equation in dimension one whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term $ G $ satisfies a Euler differential inequality. When the non-linear term $ G $ is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.
For both the cubic Nonlinear Schrodinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${bf M}_N$ of pure $N$-soliton states, and their associated multisoliton solutions. We prov
The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case,
We consider the Cauchy problems associated with semirelativistc NLS (sNLS) and half wave (HW). In particular we focus on the following two main questions: local/global Cauchy theory; existence and stability/instability of ground states. In between ot
We consider the Schrodinger--Poisson--Newton equations for finite crystals under periodic boundary conditions with one ion per cell of a lattice. The electron field is described by the $N$-particle Schrodinger equation with antisymmetric wave functio
We investigate a PDE-constrained optimization problem, with an intuitive interpretation in terms of the design of robust membranes made out of an arbitrary number of different materials. We prove existence and uniqueness of solutions for general smoo