Characterization of the Ground State to the Intercritical NLS with a Linear Potential by the Virial Functional

Hamano, Masaru; Ikeda, Masahiro

doi:10.1007/978-3-030-58215-9_12

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…the authors showed in [40,41] the following results (Theorems 4 and 5) Eq. ( 27), where the functional K α,β ω,V is given as…”

Section: (Blow-up or Grow-up)mentioning

confidence: 93%

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Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential

Hamano,

Ikeda

2024

Schrödinger Equation - Fundamentals Aspects and Potential Applications

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This is a sequel to the paper “Characterization of the ground state to the intercritical NLS with a linear potential by the virial functional” by the same authors. We continue to study the Cauchy problem for a nonlinear Schrödinger equation with a potential. In the previous chapter, we investigated some minimization problems and showed global existence of solutions to the equation with initial data, whose action is less than the value of minimization problems and positive virial functional. In particular, we saw that such solutions are bounded. In this chapter, we deal with solutions to the equation with initial data, whose virial functional is negative contrary to the previous paper and show that such solutions are unbounded.

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“…the authors showed in [40,41] the following results (Theorems 4 and 5) Eq. ( 27), where the functional K α,β ω,V is given as…”

Section: (Blow-up or Grow-up)mentioning

confidence: 93%

“…to get a potential V, which deduces scattering and blow-up or grow-up at the same time. It proved in [40] that the condition Eq. ( 16) can be rewritten as the following by using n ω,V .…”

Section: (Blow-up or Grow-up)mentioning

confidence: 99%

Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential

Hamano,

Ikeda

2024

Schrödinger Equation - Fundamentals Aspects and Potential Applications

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“…In this subsection, we prove that the minimization problem r α,β ω,γ (see (1.2) for the definition) has a minimizer. The argument is based on that in [11,20,22,24] (see also [9,10]). First, we show that ( Ḣ1 ;…”

Section: 2mentioning

confidence: 99%

Stability and instability of radial standing waves to NLKG equation with an inverse-square potential

Hamano,

Ikeda

2021

Preprint

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In this paper, we consider radial standing waves to a nonlinear Klein-Gordon equation with a repulsive inverse-square potential. It is known that existence of a "radial" ground state to the stationary problem of the nonlinear Klein-Gordon equation. Here, the "radial" ground state is a solution with the least energy among radial solutions to the stationary problem. We deal with stability and instability of the standing wave for the "radial" ground state.

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“…We recall that the authors [11,12] proved that (NLS γ ) has a radial standing wave u(t, x) = e iωt Q ω,γ (x). In particular, (SP ω,γ ) has a "radial" ground state Q ω,γ .…”

Section: Introductionmentioning

confidence: 99%

“…For γ > 0, the following minimization problem does not have a minimizer and is independent of (α, β) (see [12,Theorem 1.5]): n α,β ω,γ := inf{S ω,γ (f ) : f ∈ H 1 (R d ) \ {0}, K α,β ω,γ (f ) = 0}. So, we do not know that (SP ω,γ ) has the ground state in the usual sense.…”

Section: Introductionmentioning

confidence: 99%

Scattering solutions to nonlinear Schrödinger equation with a long range potential

Hamano¹,

Ikeda²

2021

Preprint

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In this paper, we consider a nonlinear Schrödinger equation with a repulsive inversepower potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy solution among radial solutions to the stationary problem. We prove that if radial initial data below the "radial" ground state has positive virial functional, then the corresponding solution to the nonlinear Schrödinger equation scatters. In particular, we can treat not only short range potentials but also long range potentials. Contents 1. Introduction 1 2. Preliminaries 6 2.1. Notation and definition 6 2.2. Some tools 6 3. Parameter independence of the splitting below the "radial" ground state 8 4. Well-posedness 9 4.1. Local well-posedness 9 4.2. Small data theory 11 4.3. Stability 13 4.4. Final state problem 16 5. Linear profile decomposition 17 6. Scattering 21 Acknowledgements 28 References 29

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Characterization of the Ground State to the Intercritical NLS with a Linear Potential by the Virial Functional

Cited by 6 publications

References 16 publications

Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential

Blow-up Solutions to Nonlinear Schrödinger Equation with a Potential

Stability and instability of radial standing waves to NLKG equation with an inverse-square potential

Scattering solutions to nonlinear Schrödinger equation with a long range potential

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