EVOLUTION EQUATIONS AND doi:10.3934/eect.2021056 CONTROL THEORY A SPECIAL FORM OF SOLUTION TO HALF-WAVE EQUATIONS Hyungjin Huh * Department of Mathematics Chung-Ang University Seoul 156-756, Republic of Korea (Communicated by Vladimir Georgiev) 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. 1. Introduction. In this paper we are interested in a special form of solutions to the one-dimensional half-wave equations: i∂ t φ + |D|φ = φ 3 , (1) and i∂ t φ + |D|φ = λφ ¯ ψ, i∂ t ψ -|D|ψ = μ ¯ φ 2 , (2) where φ, ψ : R 1+1 → C and ¯ φ is a complex conjugate of φ, λ and μ are complex constants. The operator |D| is defined by F (|D|f )(ξ )= |ξ |F (f )(ξ ), where F is Fourier transform. We propose the one-dimensional half-wave equations (1) and (2) and will show that an special algebraic structure of nonlinearities im- plies that one-dimensional half-wave equations may be reduced to some complex transport equations. Several authors studied the following cubic half-wave equation: i∂ t φ + |D|φ = |φ| 2 φ, φ(x, 0) = φ 0 (x). (3) The associated conserved quantities of the equation (3) are N (φ(t)) = Z R |φ| 2 (x, t)dx = N (φ 0 ), E (φ(t)) = Z R  hJφ,φi- 1 2 |φ| 4  (x, t)dx = E (φ 0 ), 2020 Mathematics Subject Classification. Primary: 35L45; Secondary: 35F25. Key words and phrases. Half-wave equation, Hilbert transform, nonlocal nonlinear transport equation. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (2020R1F1A1A01072197). * Corresponding author: Hyungjin Huh. 1