Explicit Synchronized Solitary Waves for Some Models for the Interaction of Long and Short Waves in Dispersive Media

Abstract A system coupling the cubic, nonlinear Schrodinger equation (NLS) and the Korteweg–de Vries equation (KdV) commonly known as the NLS–KdV system was previously analysed to be inconsistently derived and a few alternative systems were proposed as more suitable models for describing the interaction of long and short waves in dispersive media (Deconinck et al. in J Phys A: Math Theor 49:415501, 2016. https://doi.org/10.1088/1751-8113/49/41/415501 ; Nguyen and Liu in Water Waves 2:327–359, 2020. https://doi.org/10.1007/s42286-020-00038-6 ; Commun Math Sci 21(3):641–669, 2023. https://doi.org/10.4310/CMS.2023.v21.n3.a3 ). In this manuscript, these alternative systems are shown to possess synchronized solitary waves via the method of constrained minimization, as well as by direct computation of the associated systems of ODEs. These synchronized solitary waves have the hyperbolic $$\textrm{sech}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>sech</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -profile typical of dispersive equations.