الفهرس 
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Abstract
The aim of the present thesis is to obtain a variational principle, Lie symmetries, conservation laws, exact solutions and their interpretations for the different types of partial differential equations (PDEs). In additional to using the Bell polynomials to obtain N-soliton solutions, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws for some nonlinear differential equations.
Chapter 1.
We have given a brief overview of the integrable properties of the nonlinear evolution equations (NLEEs), which is a pre-test and the first step of its exact solvability. The Integrability features of soliton equations can be characterized by Painlevé test, Hamiltonian structure, Hirota bilinear form, Lax pair, infinite symmetries, infinite conservation laws, Bäcklund transformation (BT), and so on. We also discussed the importance of finding different solutions, whether exact, analytical or numerical, for each of the nonlinear partial differential equations (NLPDEs) using different methods such as but not limited to the method of extended tanh-function method, the direct algebraic function method, the extended (G′/G)-expansion method, Hirota method and other generalized methods.
Chapter 2.
We consider the extended Zakharov-Kuznetsov (EZK) equation, which describes the nonlinear plasma dust acoustic waves (DAWs) in a magnetized dusty plasma. Dusty plasmas consist of three components: electrons, highly negatively charged dust grains, and two temperature ions (low-temperature ions and high temperature ions). We study the Lie
symmetries, reductions, conservation laws and new exact solutions of EZK equations. Conservation laws for EZK equation is derived by applying the new conservation theorem of Ibragimov. Similarity solution for EZK equation will be obtained using Lie symmetry method. We find the Lie symmetries group of EZK equation, using similarity variables,
get reduction equation, solving the reduction equations and then get the similarity solution. Solitary wave solutions of the EZK equation are derived from the reduction equation. Thus, some new exact explicit solutions of the EZK equation are obtained.
Chapter 3.
We obtain the bilinear form for the Lax-Kadomtsev-Petviashvili (Lax-KP) and the generalized (3 + 1)-dimensional Korteweg-de Vries equations based on the binary Bell polynomials. Accordingly, N-soliton solutions, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws will be constructed to Lax-KP and the generalized (2 + 1)-dimensional Korteweg-de Vries equation ((2 + 1)G - KdV). At the same time, we get another bilinear Bäcklund transformation. Finally, exact solutions were obtained by using the exchange formulas for Hirota’s bilinear operators.
Chapter 4.
We study the coupled Schrödinger-KdV (CS-KdV) equation. The Lie point symmetry generators are derived for the CS-KdV equation. We also show that the CS-KdV equation is nonlinearly self-adjoint and we use this property to construct conservation laws corresponding to the symmetries of the model. Symmetry reduction technique in conjunction with the extended tanh method is used to calculate the solitary wave solution. Also, conservation laws for the CS-KdV equation is derived by applying the new conservation theorem of Ibragimov. The undetermined coefficients and Bernoulli sub-ODE methods are suggested for constructing the topological, non-topological and singular soliton solutions to the model. New breather solitary solutions are constructed for CS-KdV equation by using bilinear form and the extended homoclinic test approach (EHTA).
Chapter 5.
The nonlinear for the small long amplitude waves in two dimensional (2D) shallow water waves propagation with free surface are considered. The shallow water wave problem leads to the nonlinear Hamiltonian model equation. Based on the Bell polynomials approach, the bilinear form, bilinear Bäckluund transformation and multiple wave solutions are obtained. The conservation laws are constructed using two different techniques, namely, the Ibragimov’s theorem and the multiplier method. The Noether’s approach was applied to the nonlinear Hamiltonian model equation to obtain the conservation laws. Conserved quantities of Hamiltonian model equation are illustrated. Finally, with the help of the extended (G′/G)-expansion method, a set of new exact solutions for the nonlinear Hamiltonian model equation are obtained.
Chapter 6.
We investigate the temporal-second-order KdV equation, which describes the propagation of two wave modes with the same dispersion relation but different phase velocities, nonlinearity, and dispersion parameters. The similarity reductions and new exact solutions are obtained via the Kudryashov method and a new version of Kudryashov method. Furthermore, the conservation laws are derived using the new conservation theorem. The bilinear forms and bilinear Bäcklund transformation of the temporal-second-order KdV equation are derived
through the binary Bell polynomial. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota method. The characteristics and interaction of the solitons are discussed graphically. We discuss the effect of α, c_i and a_i(i = 1, 2) on the soliton amplitudes and velocities. Bifurcation method of dynamical systems is employed to
investigate bifurcation of solitary waves in the temporal-second-order KdV equation.
Chapter 7.
Nonlinear propagation of oscillatory and monotonous DIA shocks in dusty plasma with dust charge fluctuations and small isothermal deviation of electrons is considered. The Schamel-Korteweg-de Vries-Burger (S-KdVB) equation can be derived by using the reductive perturbation
(R-P) method. The variational principle and the conservation laws of the S-KdVB equation are constructed by introducing two special functions. A nonlinear self-adjoint classification of the S-KdVB equation is presented. Based on the Ibragimovs theorem, conservation laws for S-KdVB equation are established. We also study the S-KdVB equations by improved tan((ψ(χ))/2)-expansion method. Abundant dark, singular and periodic optical solitons solutions of the model are constructed. Furthermore, additional graphical simulations were performed using Mathematica to see the behavior of these solutions. With the help of bifurcation theory of planar dynamical systems, we study bifurcation and phase portrait analysis of traveling wave solution of the S-KdVB equation.