Long-wave Short-wave Interaction Equations Research Articles

Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime

Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime

Transverse linear instability of solitary waves for coupled long-wave–short-wave interaction equations

In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.

Non-existence and existence of localized solitary waves for the two-dimensional long-wave–short-wave interaction equations

In this study, we establish the non-existence and existence results for the localized solitary waves of the two-dimensional long-wave–short-wave interaction equations. Both the non-existence and existence results are based on Pohozaev-type identities. We prove the existence of solitary waves by showing that the solitary waves are the minimizers of an associated variational problem.

Numerical computations for long-wave short-wave interaction equations in semi-classical limit

This paper presents and compares various numerical techniques for the long-wave short-wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time-splitting approach combined with the Runge–Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi-classical limit.

Long-wave short-wave resonance case for a generalized Davey–Stewartson system

It is observed that the generalized Davey–Stewartson equations are not valid for a long-wave short-wave resonance case. In the case where the phase velocity of the long longitudinal wave is equal to the group velocity of the short transverse wave, new ( 2 + 1 ) dimensional evolution equations, called the long-wave short-wave interaction equations, are derived to describe the resonance case. The special solutions of the long-wave short-wave interaction equations are also obtained in terms of Jacobian elliptic functions.

To the complete integrability of long-wave–short-wave interaction equations

It is shown that the nonlinear partial differential equations governing the interaction of long and short waves are completely integrable. The methodology used is that of Ablowitz et al. [M. J. Ablowitz, A. Ramani, and A. Segur, Lett. Nuovo Cimento 23, 333 (1980); M. J. Ablowitz, A. Ramani, and H. Segur, J. Math. Phys. 21, 715, 1006 (1980)], though in the last section of our paper the problem also has been discussed in the light of the procedure due to Weiss et al. [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983)] and a Backlünd transformation has been obtained.