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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2011.08029 |
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| _version_ | 1866913706795859968 |
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| author | Hayashi, Masayuki |
| author_facet | Hayashi, Masayuki |
| contents | We consider the following nonlinear Schrödinger equation of derivative type: \begin{equation} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in \mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_08029 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach Hayashi, Masayuki Analysis of PDEs We consider the following nonlinear Schrödinger equation of derivative type: \begin{equation} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in \mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS. |
| title | Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2011.08029 |