On local well-posedness for the nonlinear Schrödinger equation with general power nonlinearity
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911002089488384 |
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| author | Arens, Lucia Gritl, Marius |
| author_facet | Arens, Lucia Gritl, Marius |
| contents | The nonlinear Schrödinger equation plays a fundamental role in mathematical physics, particularly in the study of quantum mechanics and Bose-Einstein condensation. This paper explores two distinct approaches to establishing the local well-posedness of solutions: the semigroup theory ansatz and the Strichartz estimates ansatz. Semigroup theory provides a general and elegant framework rooted in functional analysis, allowing for the interpretation of the time evolution of solutions as operator semigroups. Strichartz estimates, developed specifically for dispersive equations, offer an alternative technique based on refined space-time estimates and fixed-point arguments. We systematically analyze and compare both approaches and apply them to nonlinear Schrödinger equations where the nonlinearity is given by $F(u)=λ|u|^p u$ for some $λ\in \mathbb{R}$. So our results extend beyond the physically relevant case $p=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On local well-posedness for the nonlinear Schrödinger equation with general power nonlinearity Arens, Lucia Gritl, Marius Analysis of PDEs The nonlinear Schrödinger equation plays a fundamental role in mathematical physics, particularly in the study of quantum mechanics and Bose-Einstein condensation. This paper explores two distinct approaches to establishing the local well-posedness of solutions: the semigroup theory ansatz and the Strichartz estimates ansatz. Semigroup theory provides a general and elegant framework rooted in functional analysis, allowing for the interpretation of the time evolution of solutions as operator semigroups. Strichartz estimates, developed specifically for dispersive equations, offer an alternative technique based on refined space-time estimates and fixed-point arguments. We systematically analyze and compare both approaches and apply them to nonlinear Schrödinger equations where the nonlinearity is given by $F(u)=λ|u|^p u$ for some $λ\in \mathbb{R}$. So our results extend beyond the physically relevant case $p=2$. |
| title | On local well-posedness for the nonlinear Schrödinger equation with general power nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.10595 |