Emrullah Yaşar

New travelling wave solutions to the Ostrovsky equation

In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods.

Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute 𝜆-symmetries and the associated reduction algorithm. The knowledge of a 𝜆−symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.

Invariant solutions and conservation laws to nonconservative FP equation

We generate conservation laws for the one dimensional nonconservative Fokker-Planck (FP) equation, also known as the Kolmogorov forward equation, which describes the time evolution of the probability density function of position and velocity of a particle, and associate these, where possible, with Lie symmetry group generators. We determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied. As the Fokker-Planck equation is evolution type, no recourse to a Lagrangian formulation is made. Moreover, we obtain invariant solutions for the FP equation via potential symmetries.

Group properties and conservation laws for nonlocal shallow water wave equation

Abstract Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.

Conservation Laws and Soliton Solutions of the (1+1)-Dimensional Modified Improved Boussinesq Equation

Abstract In this work, we consider the (1+1)-dimensional modified improved Boussinesq (IMBq) equation. As the considered equation is of evolution type, no recourse to a Lagrangian formulation is made. However, we showed that by utilising the partial Lagrangian method and multiplier method, one can construct a number of local and nonlocal conservation laws for the IMBq equation. In addition, by using a solitary wave ansatz method, we obtained exact bright soliton solutions for this equation. The parameters of the soliton envelope (amplitude, widths, velocity) were obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the (2+1)-dimensional Sawada–Kotera (SK) equation and (3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirotas perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations

Abstract In this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov’s new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer–Chree (PC) equation and the Kaup–Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

Abstract In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

On the Conservation Laws and Exact Solutions of a Modified Hunter-Saxton Equation

We study the modified Hunter-Saxton equation which arises in modelling of nematic liquid crystals. We obtain local conservation laws using the nonlocal conservation method and multiplier approach. In addition, using the relationship between conservation laws and Lie-point symmetries, some reductions and exact solutions are obtained.

Application of the Jacobi method and integrating factors to a class of Painlevé–Gambier equations

In this work, we consider the motion of chain ball drawing with constant force in the frictionless surface which is a class of the Painleve–Gambier equations. We apply Jacobis method which enables us to obtain Lagrangians of any second-order differential equation. It is comprised that the Lagrangian obtained by Musielaks method is the particular case of the many Lagrangians that can be obtained by Jacobis method. In addition, we obtain integrating factors and first integrals for the equation in question by Ibragimovs variational derivative approach.