A special form of solution to half-wave equations

Hyungjin Huh

doi:10.3934/eect.2021056

Article Contents

2022, Volume 11, Issue 5: 1605-1612. Doi: 10.3934/eect.2021056

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A special form of solution to half-wave equations

Hyungjin Huh^,

Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea

^* Corresponding author: Hyungjin Huh
^* Corresponding author: Hyungjin Huh

Received: March 31, 2021

Revised: July 31, 2021

Early access: October 2021

Published: October 2022

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (2020R1F1A1A01072197)

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Abstract

We investigate a special form of solution to the one-dimensional half-wave equations with particular forms of nonlinearities. Using the special form of solution involving Hilbert transform, the half-wave equations reduce to nonlocal nonlinear transport equation which can be solved explicitly.

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Mathematics Subject Classification: Primary: 35L45; Secondary: 35F25.

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References

[1]	J. Bellazzini, V. Georgiev, E. Lenzmann and N. Visciglia, On traveling solitary waves and absence of small data scattering for nonlinear half-wave equations, Comm. Math. Phys., 372 (2019), 713–732. doi: 10.1007/s00220-019-03374-y.
[2]	J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19 pp. doi: 10.1063/1.4726198.
[3]	D. Chae, A. Córdoba, D. Córdoba and M. A. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194 (2005), 203–223. doi: 10.1016/j.aim.2004.06.004.
[4]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267–1282. doi: 10.3934/cpaa.2014.13.1267.
[5]	K. Fujiwara, A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity, Math. Methods Appl. Sci., 41 (2018), 4955–4966. doi: 10.1002/mma.4944.
[6]	K. Fujiwara, S. Machihara and T. Ozawa, Well-posedness for the Cauchy problem for a system of semirelativistic equations, Comm. Math. Phys., 338 (2015), 367–391. doi: 10.1007/s00220-015-2347-3.
[7]	K. Fujiwara, S. Machihara and T. Ozawa, On a system of semirelativistic equations in the energy space, Commun. Pure Appl. Anal., 14 (2015), 1343–1355. doi: 10.3934/cpaa.2015.14.1343.
[8]	P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, 5 (2012), 1139–1155. doi: 10.2140/apde.2012.5.1139.
[9]	P. Gérard, E. Lenzmann, O. Pocovnicu and P. Raphaël, A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, 4 (2018), paper no. 7,166 pp. doi: 10.1007/s40818-017-0043-7.
[10]	K. Hidano and C. Wang, Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation, Selecta Math. (N.S.), 25 (2019), paper no. 2, 28 pp. doi: 10.1007/s00029-019-0460-4.
[11]	T. Inui, Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity, Proc. Amer. Math. Soc., 144 (2016), 2901–2909. doi: 10.1090/proc/12938.
[12]	J. N. Pandey, The Hilbert Transform of Schwartz Distributions and Applications, Wiley, New York, 1996.
[13]	Q. Shi and C. Peng, Wellposedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonlinear Anal., 178 (2019), 133–144. doi: 10.1016/j.na.2018.07.012.