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In this manuscript, consideration is given to the existence of periodic traveling-wave solutions to the $abcd$-system. This system was derived by Bona, Saut, and Chen to describe small amplitude, long wavelength gravity waves on the surface of water. These exact solutions are formulated in terms of the Jacobi elliptic function cnoidal. The existence of explicit traveling-wave solutions is very useful in theoretical investigations such as stability of solutions, as well as other numerical analysis of the system., Comment: 19 pages, 6 figures, submitted to Water Waves

Some systems were recently put forth by Nguyen et. al. as models for studying the interaction of long and short waves in dispersive media. These systems were shown to possess synchronized Jacobi elliptic solutions as well as synchronized solitary wave solutions under certain constraints, i.e., vector solutions where the two components are proportional to one another. In this paper, the exact periodic traveling wave solutions to these systems in general are found to be given by Jacobi elliptic functions. Moreover, these cnoidal wave solutions are unique. Thus, the explicit synchronized solutions under some conditions obtained by Nguyen et. al. are also indeed unique., Comment: 21 pages, 4 figures, to appear in AIMS Mathematics. This is the final version, incorporating the referee comments

We use profile decomposition to characterize 2-soliton solutions of the KdV equation as global minimizers to a constrained variational problem involving three of the polynomial conservation laws for the KdV equation., Comment: References added, exposition improved, revised argument in section 5, results unchanged

Multisoliton solutions of the KdV equation satisfy nonlinear ordinary differential equations which are known as stationary equations for the KdV hierarchy, or sometimes as Lax-Novikov equations. An interesting feature of these equations, known since the 1970's, is that they can be explicitly integrated, by virtue of being finite-dimensional completely integrable Hamiltonian systems. Here we use the integration theory to investigate the question of whether the multisoliton solutions are the only nonsingular solutions of these ordinary differential equations which vanish at infinity. In particular we prove that this is indeed the case for $2$-soliton solutions of the fourth-order stationary equation.

It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Kortewegde Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves and such a system has been studied over the last few decades. However, questions about the validity of the system in the study of water waves were raised in our previous work where we presented our analysis using the fifth-order KdV as the starting point. In this paper, these questions are settled unequivocally as we show that the NLS-KdV system or even the linear Schrodinger-KdV system cannot be resulted from the full Euler equations formulated in the study of water waves., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1604.04317

The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves seems attractive and such a system has been studied over the last decades. We evaluate the validity of this system, discussing two main problems. First, only the system coupling the linear Schr\"odinger equation with KdV has been derived in the literature. Second, the time variables appearing in the equations are of a different order. It appears that in the manuscripts that study the coupled NLS-KdV system, an assumption has been made that the coupled system can be derived, justifying its mathematical study. In fact, this is true even for the papers where the asymptotic derivation with the problems described above is presented. In addition to discussing these inconsistencies, we present some alternative systems describing the interaction of long and short waves.