Multiple Soliton Solutions for a Variety of Coupled Modified Korteweg--de Vries Equations

Recently, many nonlinear coupled evolution equations, such as the coupled Korteweg–de Vries (KdV) equation, the coupled Boussinesq equation, and the coupled mKdV equation, appear in scientific applications [1 – 13]. The coupled evolution equations attracted a considerable research work in the literature. The aims of these works have been the determination of soliton solutions and the proof of complete integrability of these coupled equations [14 – 24]. Various methods have been used to investigate the nonlinear evolution and the coupled nonlinear evolution equations. Examples of the methods that have been used are the Hirota bilinear method, the Hietarinta approach, the Bäcklund transformation method, the Darboux transformation, the Pfaffian technique, the inverse scattering method, the Painlevé analysis, the generalized symmetry method, and other methods. The Hirota bilinear method [1 – 7], the Hietarinta approach [8, 9], and the Hereman simplified form [10 – 12] are rather heuristic and significant. These approaches possess powerful features that make them practical for the determination of multiple soliton solutions [13 – 24] for a wide class of nonlinear evolution equations. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. It is interesting to point out that the soliton solution should demonstrate a wave of permanent form. The soliton solution is localized, which means that the solution either decays exponentially to zero such as the KdV solitons, or converges to a constant at infinity such as the sine-Gordon equation. Since we will talk about the multiple soliton solutions, we point out that the soliton interacts with other solitons preserving its character. We also add that the soliton solution u(x, t), along with its derivatives, tends to zero as | x |→ ∞. This clearly shows that the soliton reside in Hilbert space, and it results from initial-boundary value problems. Concerning the modified KdV equation, it describes nonlinear wave propagation in systems with polarity symmetry. The mKdV equation is used in electrodynamics, wave propagation in size quantized films, and in elastic media. It is used to describe acoustic waves in anharmonic lattices and Alfvén waves in collisionless plasma. The mKdV equation differs from the KdV equation only because of its cubic nonlinearity. The mKdV equation is completely integrable and can be solved by the inverse scattering method. In this work, a variety of coupled mKdV equations will be investigated for complete integrability and for the determination of multiple soliton solutions. The coupled mKdV equations that we selected are