Self-Adjointness, Formal Lagrangian and Conservation Laws of Partial Differential Equations

Abstract

The dynamics of many real world important phenomenon in different fields of science and technology are expressed with the aid of nonlinear partial differential equations (NLPDEs). Calculation of exact solutions, Lie point symmetries and conservation laws of such NLPDEs is a useful tool to study the mechanism of the problem deeply. This dissertation is devoted to the determination of new exact solutions, symmetries and conservation laws of some NLPDs used in plasma physics, shallow water waves, fiber optics, and medling fluid. In particular, (1 + n)-dimensional Zakharov-Kuznetsov (ZK) equation and modified Zakharov-Kuznetsov (mZK) equation are studied for obtaining symmetries, exact solutions and conservation laws. New conservation theorem of Ibragimov is applied for the calculation of these conservation laws. Different forms of solutions including bright, dark, singular and solitary wave solutions are extracted with solitary wave anstaz method, 1/G0 -expansion method, G 0 /G -expansion method, modified Kudryshov method, and symmetry reduction technique. As ZK equation contains third order dispersion term Uxxx , so it is suitable for the waves of small amplitude only. In case of increase in the amplitude, the velocity and the soliton width deviate from the prediction of this equation. To solve this problem, higher order dispersion term Uxxxxx is added to ZK equation. Conservation laws for this fifth order ZK equation are obtained with Ibragimov’s method and multiplier method. Moreover, some new exact solutions are also obtained with sine-cosine method and modified Kudryashov methods. These results are very useful to study the waves in plasma Physics. Time regularized long wave (TRLW) equation is applicable to study water waves in the oceans. For this equation, conservation laws are obtained with Ibragimov’s technique and multiplier method. Also exact solutions are achieved by applying modified Kudryashov method and G 0 /G2

-expansion method which are important and useful in oceanography. Higher order nonlinear Schr¨odinger’s equation (NLSE) is important to study the propagation of short light pulses in the monomode optical fibers. A variety of exact analytical solutions to the NLSE are obtained with ansatz method, modified simple equation method and

1/G0 -expansion method. Resonant nonlinear Schr¨odinger’s equation (R-NLSE) is also considered for symmetries, exact solutions and conservation laws. Modified Kudryashov method and exp(φ(ξ))-expansion method are utilized for obtaining these exact solutions and multiplier method is applied for calculating the conservation laws. These results are useful to understand the dynamics of the solitons and Medeling fluids in various nonlinear systems. Finally, conformable time fractional extended ZK equation is studied for exact solutions with modified Kudryashov method and.....-expansion method. Obtained results are used to study the waves in magnetized and dusty plasma.