The Linearized Classical Boussinesq System on the Half-Line

2020/09/20 by Craig Johnston, Johnston, C. M., Clarence T. Gartman +3
Mathematics · Physics and Astronomy · #35G16. Secondary: 35G61 #35G31 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Primary: 35G46

paper · pdf · doi:10.48550/arxiv.2009.09532

openalex publication_date 2020/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.

Citations

• Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

Related

• The Boussinesq equation on the half-line
• Fokas's Uniform Transform Method for linear systems
• Existence of Classic Solution of the Boussinesq Equation
• The Numerical Unified Transform Method for Initial-boundary Value Problems on the Half-line
• Well-posedness and nonlinear smoothing for the "good" Boussinesq\n equation on the half-line