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doi: 10.3934/eect.2021056
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## A special form of solution to half-wave equations

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

* Corresponding author: Hyungjin Huh

Received  April 2021 Revised  August 2021 Early access October 2021

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

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.

Citation: Hyungjin Huh. A special form of solution to half-wave equations. Evolution Equations and Control Theory, doi: 10.3934/eect.2021056
##### 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.

show all references

##### 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.
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