On the ground states for the X‐ray free electron lasers Schrödinger equation

Abstract

We consider the following X‐ray free electron lasers Schrödinger equation false(i∇−Afalse)2u+Vfalse(xfalse)u−μfalse|xfalse|u=()1false|xfalse|∗false|ufalse|2u−Kfalse(xfalse)false|ufalse|q−2u,0.1em0.1emx∈ℝ3,$$ {\left(i\nabla -A\right)}&amp;#x0005E;2u&amp;#x0002B;V(x)u-\frac{\mu }{\mid x\mid }u&amp;#x0003D;\left(\frac{1}{\mid x\mid}\ast {\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2\right)u-K(x){\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{q-2}u,x\in {\mathbb{R}}&amp;#x0005E;3, $$ where A∈Lloc2false(ℝ3,ℝ3false)$$ A\in {L}_{loc}&amp;#x0005E;2\left({\mathbb{R}}&amp;#x0005E;3,{\mathbb{R}}&amp;#x0005E;3\right) $$ denotes the magnetic potential such that the magnetic field B=curl0.4emA$$ B&amp;#x0003D;\operatorname{curl}\kern0.4em A $$ is ℤ3$$ {\mathbb{Z}}&amp;#x0005E;3 $$‐periodic, μ∈ℝ,0.1emK∈L∞()ℝ3$$ \mu \in \mathbb{R},K\in {L}&amp;#x0005E;{\infty}\left({\mathbb{R}}&amp;#x0005E;3\right) $$ is ℤ3$$ {\mathbb{Z}}&amp;#x0005E;3 $$ periodic and non‐negative, q∈false(2,4false)$$ q\in \left(2,4\right) $$. Using the variational method, based on a profile decomposition of the Cerami sequence in HA1()ℝ3$$ {H}_A&amp;#x0005E;1\left({\mathbb{R}}&amp;#x0005E;3\right) $$, we obtain the existence of the ground state solution for suitable μ≥0$$ \mu \ge 0 $$. When μ<0$$ \mu &amp;lt;0 $$ is small, we also obtain the non‐existence. Furthermore, we give a description of the asymptotic behavior of the ground states as μ→0+$$ \mu \to {0}&amp;#x0005E;{&amp;#x0002B;} $$.

Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

“…The nonlinear Schrödinger equation (NSE) has an important role in many fields of science and technology, such as nonlinear optics, electromagnetic theory, water wave, superconductivity, super fluids, chemical and biological dynamics, lasers and plasmas [16][17][18][19][20][21][22][23].…”