Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

Abstract

We consider several different bidirectional Whitham equations that have recently appeared in the literature. Each of these models combines the full two‐way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity, providing nonlocal model equations that may be expected to exhibit some of the interesting high‐frequency phenomena present in the Euler equations that standard “long‐wave” theories fail to capture. Of particular interest here is the existence and stability of periodic traveling wave solutions in such models. Using numerical bifurcation techniques, we construct global bifurcation diagrams for each system and compare the global structure of branches, together with the possibility of bifurcation branches terminating in a “highest” singular (peaked/cusped) wave. We also numerically approximate the stability spectrum along these bifurcation branches and compare the stability predictions of these models. Our results confirm a number of analytical results concerning the stability of asymptotically small waves in these models and provide new insights into the existence and stability of large amplitude waves.

A comparative study of bi-directional Whitham systems

“…was put forward by Hur and Pandey in [17], and it was shown to behave somewhat more favorably than (1.3), (1.4) with regard to modulational instability and local well posedness (see also [3]. We will call this system the HP system.…”

“…In particular, it was shown in [9] that it admits periodic traveling-wave solutions and features a highest cusped wave on the bifurcation branch. The modulational stability of its periodic traveling-wave solutions has been investigated numerically in [3], and the system has been studied numerically in the presence of an uneven bottom in [26]. Moreover, it was shown in [14] that the initialvalue problem on the real line is well posed locally-in-time for data that are strictly positive and bounded away from zero.…”

“…However strictly positive solutions, like solitons for example, have always featured prominently in the analysis of such systems. In a recent paper by Claassen and Johnson [3] the well-posedness for more general initial data was questioned. In fact the authors showed numerically that the ASMP system is probably ill-posed in L 2 (T).…”

“…The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.In the current work, we review some of the existing fully dispersive systems, such as found in [1,3,8,17,23,24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems.…”

“…In the current work, we review some of the existing fully dispersive systems, such as found in [1,3,8,17,23,24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems.…”