Conservation laws and optical solutions of the complex modified Korteweg-de Vries equation

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In this work, conservation laws and optical solutions are obtained to the complex modified Korteweg-de Vries (mKdV) equation. The new conservation theorem is used for obtaining conservation laws. In this regard, the formal Lagrangian and the adjoint equation are given at first. The infinite and finite conservation laws are given for each Lie symmetry of the equation. As a result of the study, we can say that obtained conservation laws are trivial. Then, the unified method is used for solving the given equation. As outcomes, the hyperbolic, trigonometric, and rational function solutions are obtained. The accuracy of all obtained solutions is checked by the symbolic computation software Maple. Finally, the 3D and 2D plots of some obtained solutions are displayed to show the physical appearance of the model. Our obtained results in this work concerning our investigated equation are essential to explain many physical and oceanographic applications involving ocean gravity waves and many other related phenomena. : 02.30.Jr, 11.30.-j, 11.10.Ef