Remarks on solitary waves in equations with nonlocal cubic terms

2024/06/21 by Johanna Ulvedal Marstrander, Marstrander, Johanna Ulvedal
Engineering · Mathematics · #35A15 #76B15 #76B25 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.15148

openalex publication_date 2024/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form ∂_t u + ∂_x(Λ^s u + uΛ^r u²) = 0, where Λ^s, Λ^r are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, s>0, while the operator on the nonlinear part is assumed to act slightly smoother, r

Related

• Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities
• Solitary waves for dispersive equations with Coifman-Meyer nonlinearities
• Existence of solitary-wave solutions to nonlocal equations
• Decay of solitary waves
• Instability of nonlinear dispersive solitary waves