SOLITARY AND E-SHOCK WAVES IN A RESONANT SYSTEM BETWEEN LONG AND SHORT WAVES

In order to examine solitary and E-shock waves in a resonant system between long and short waves, a reduced set of ordinary differential equations (ODEs) are considered by a simple traveling-wave transformation. It is then shown that analytical solutions can be obtained systematically by means of the modified Hirota's method. A variety of numerical solutions including oscillatory solitary and E-shock waves are also found for various values of the parameters. In particular, characteristic properties of the solutions are examined for the case when the ODEs hold the Painleve property. In this connection, integrability of the original partial differential equations (PDEs) in the nearly integrable region is discussed together with the Lyapunov exponent for the soliton-like solutions.