Collision Effects of Solitary Waves for the Gardner Equation

Abstract

We study the physical and collision properties of the combined KdV–mKdV solitons given by the Gardner equation which possess solitary wave solution characterized by sech function. A collision of the two solitary waves produces 2-soliton solution. We make a physical form of the 2-soliton solution where the fast soliton moves with speed c 1 and the slow soliton moves with speed c 2. In the collision described by the 2-soliton solution, the solitary waves preserve their shapes and speeds, but get a shift in position where the fast soliton overtakes the slow soliton if their speeds have same direction, and two solitons cross head-on if their speeds have opposite direction. For a collision there exist three different types of interactions which depend on the relative ratio $c_1/c_2$ of speeds and the relative orientation of the two solitary waves.