Nonlinear Schr�dinger equations and sharp interpolation estimates

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

“…In particular, from [8] we know that e iωt ϕ ω,0 (x) is stable in H 1 (R) for any ω > 0 if 1 < p < 5. On the other hand, it was shown that e iωt ϕ ω,0 (x) is unstable in H 1 (R) for any ω > 0 if p 5 (see [3] for p > 5 and [45] for p = 5).…”

“…With this definition and Remark 2, it is clear that stability in H 1 (R) implies stability in H 1 rad (R) and conversely that instability in H 1 rad (R) implies instability in H 1 (R). For γ = 0, the orbital stability for (2) has been extensively studied (see [3,7,8,44,45] and the references therein). In particular, from [8] we know that e iωt ϕ ω,0 (x) is stable in H 1 (R) for any ω > 0 if 1 < p < 5.…”

Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities

“…It is well known that solutions of (1.1) can become singular in finite time if |ψ 0 | 2 2 ≥ N c , where |ψ 0 | 2 2 = |ψ 0 | 2 2 dx is the input beam power, and N c , the critical power for singularity formation, is a constant which depends only on d. The critical power N c , thus, sets an upper limit on the amount of power (|ψ| 2 2 ) that can be propagated with a single beam. The critical NLS (1.1) admits solitary waves ψ = e iωt R ω (x) whose power is exactly equal to the critical power, i.e., |R ω | 2 2 ≡ N c [17]. These solitary waves are, however, strongly unstable.…”

Chebfun: A New Kind of Numerical Computing

“…12 This equation has an infinite number of solutions in H 1 (R 3 ). The solution of minimal mass is positive, radial, and exponentially decaying and is called the ground state [19]. We shall seek a positive radial solution to (3.5) with exponential decay.…”