Product Representations of Periodic Waves for the Modified Korteweg–de Vries Family of Evolution Equations

抄録

Periodic waves for evolution equations of the modified Korteweg–de Vries (mKdV) family are expressed as products of elliptic functions. By employing the Hirota bilinear formulation, periodic waves of the single component mKdV with positive cubic nonlinearity are obtained as rational functions of Jacobi elliptic functions. Coupled mKdV systems with two and three components are treated. Dark or kink type solitary waves can exist for systems where the equation for each component is in the bright soliton regime. Finally, a coupled system of (2+1) (2 spatial and 1 temporal) dimensional evolution equations is studied. Periodic waves in terms of products of elliptic functions are also derived. The validity of these new solutions is verified independently by a computer algebra software whenever feasible.