The initial-value problem for a Gardner-type equation

Abstract

Discussed here is a regularized version(0.1)ut+ux+uux+Au2ux−uxxt=0, $${u}_{t}+{u}_{x}+u{u}_{x}+A{u}^{2}{u}_{x}-{u}_{\mathit{xxt}}=0,$$ of the classical Gardner equationut+ux+uux+Au2ux+uxxx=0, $${u}_{t}+{u}_{x}+u{u}_{x}+A{u}^{2}{u}_{x}+{u}_{\mathit{xxx}}=0,$$ that arises in hydrodynamics and plasma physics. Considered here are well-posedness issues for the initial-value problem for the regularized equation posed on all of R $\mathbb{R}$ .

Keywords

• gardner equation
• bore propagation
• long-wave models
• unidirectional water-wave models
• plasma physics models
• 30d10
• 30d15
• 30h99
• 35b44
• 35b65
• 35e15
• 35q51
• 35q53