New Lower Bounds on the Radius of Spatial Analyticity for the Third‐Order Nonlinear Schrödinger Equation.
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| Title: | New Lower Bounds on the Radius of Spatial Analyticity for the Third‐Order Nonlinear Schrödinger Equation. |
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| Authors: | Getachew, Tegegne1 (AUTHOR) gcmsc2006@gmail.com, Shi, Xiasheng1 (AUTHOR) shixiasheng@zju.edu.cn |
| Source: | Journal of Applied Mathematics. 7/6/2026, Vol. 2026, p1-9. 9p. |
| Subjects: | Nonlinear Schrödinger equation, Analytic functions, Initial value problems, Mathematical analysis, Gevrey class, Sobolev spaces, Conservation laws (Physics) |
| Abstract: | In this paper, we consider the initial value problem associated with the cubic nonlinear Schrödinger equation with third‐order dispersion ∂tu+iα∂x2u+β∂x3u+iγu2u=0,x∈ℝ,t∈ℝ,ux,0=u0x, where α, β and γ are real constants such that β, γ ≠ 0, u is a complex valued function and the initial data u0 is analytic on ℝ and has a uniform radius of analyticity σ0 in the spatial variable. We show the uniform radius of spatial analyticity σ(T) of solutions at time T cannot decay faster than cT−2 for large T. This improves a recent result by Figueira and Panthee, where they established a decay rate of order cT−(4 + ε) for ε > 0. We used an approximate conservation law in the modified Gevrey spaces, a contraction mapping argument, space‐time estimates and one‐dimensional Sobolev embedding to obtain our improved result. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | In this paper, we consider the initial value problem associated with the cubic nonlinear Schrödinger equation with third‐order dispersion ∂tu+iα∂x2u+β∂x3u+iγu2u=0,x∈ℝ,t∈ℝ,ux,0=u0x, where α, β and γ are real constants such that β, γ ≠ 0, u is a complex valued function and the initial data u0 is analytic on ℝ and has a uniform radius of analyticity σ0 in the spatial variable. We show the uniform radius of spatial analyticity σ(T) of solutions at time T cannot decay faster than cT−2 for large T. This improves a recent result by Figueira and Panthee, where they established a decay rate of order cT−(4 + ε) for ε > 0. We used an approximate conservation law in the modified Gevrey spaces, a contraction mapping argument, space‐time estimates and one‐dimensional Sobolev embedding to obtain our improved result. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 1110757X |
| DOI: | 10.1155/jama/2851779 |
