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Stability of Algebraic Solitons for Nonlinear Schrödinger Equations of Derivative Type: Variational Approach

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Abstract

We consider the following nonlinear Schrödinger equation of derivative type:

$$\begin{aligned} i \partial _t u + \partial _x^2 u +i |u|^{2} \partial _x u +b|u|^4u=0 , \quad (t,x) \in {{\mathbb {R}}}\times {{\mathbb {R}}}, \ b \in {{\mathbb {R}}}. \end{aligned}$$
(1)

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 (1) 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 \(b>-\frac{3}{16}\), Eq. (1) has algebraically decaying solitons, which we call algebraic solitons, as well as exponentially decaying solitons. In this paper, we study stability properties of solitons for (1) 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 in (1) shows an interesting mathematical example because stable algebraic solitons are not known in the context of double power NLS equations.

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Notes

  1. Equation (1.1) for \(b=0\) is equivalent to (DNLS) under the transformation

    $$\begin{aligned} \psi (t,x) = u (t,x)\exp \left( -\frac{i}{2} \int _{-\infty }^{x} |u (t,y)|^2 \mathrm{d}y\right) . \end{aligned}$$

  2. See (1.17) for more details.

  3. The rotations and space translations appearing in (1.12) come from the invariant of Eq. (1.1).

  4. The essential spectrum of \(S_{\omega ,c}''(\phi _{\omega ,c})\) is given by \(\sigma _\mathrm{ess}\left( S_{\omega ,c}''(\phi _{\omega ,c})\right) =\left[ \omega -c^2/4,\infty \right) \), which gives the lack of coercivity property for the case \(c=2\sqrt{\omega }\) (see [10] for more details).

  5. This can be regarded as a certain extension of the argument in [38] to a two-parameter family of solitons.

  6. Equation (1.1’) is also used in previous works [17, 39].

  7. Although there is a link of soliton profiles in between (1.1) and (1.17), stability properties may change (see [7, Remark 1]).

  8. If \(b\le -\frac{3}{16}\), for any initial data \(u_0\in H^1({{\mathbb {R}}})\), the corresponding \(H^1({{\mathbb {R}}})\)-solution is global and bounded.

  9. After this work was completed, it was proved in [1] that (DNLS) is globally well-posed in Sobolev spaces without the smallness assumption of the mass.

  10. As can be seen in the proof, \(\delta \) depends on \(\varepsilon \) and \((\omega ,c)\).

  11. One can also show this formula without using the Taylor expansion since the function g is the quadratic function.

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Acknowledgements

The results of this paper were mostly obtained when the author was a PhD student at Waseda University. The author would like to thank his thesis adviser Tohru Ozawa for constant encouragements. The author is also grateful to Masahito Ohta for fruitful discussions, and to Noriyoshi Fukaya for helpful comments on the first manuscript. This work was supported by JSPS KAKENHI Grant Numbers JP17J05828, JP19J01504, and Top Global University Project, Waseda University.

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  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Masayuki Hayashi

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Hayashi, M. Stability of Algebraic Solitons for Nonlinear Schrödinger Equations of Derivative Type: Variational Approach. Ann. Henri Poincaré 23, 4249–4277 (2022). https://doi.org/10.1007/s00023-022-01195-9

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