Solitons and other solutions to Wu – Zhang system

Mohammad Mirzazadeh, Mehmet Ekici, Mostafa Eslami, Edamana Vasudevan Krishnan, Sachin Kumar, Anjan Biswas Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran mirzazadehs2@gmail.com Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey ekici-m@hotmail.com Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran meslami.edu@gmail.com Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Muscat, Oman krish@squ.edu.om Centre for Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda 151001, Punjab, India sachin1jan@yahoo.com Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA biswas.anjan@gmail.com Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia


Introduction
The study of nonlinear evolution equations (NLEEs) forms the basic fabric for various areas of mathematical physics and engineering.There are various forms of NLEEs that are studied for this purpose .The nonlinear Schrödinger's equation, for example, is studied in nonlinear optics.In the context of plasma physics, Zakharov-Kuznetsov where P is a polynomial.The essence of the extended trial equation method can be presented in the following steps: Step 1.To find the traveling wave solutions of Eq. ( 1), we introduce the new wave variable u(x, t) = U (ξ), ξ = kx − vt, where k and v are constants to be determined later.Substituting Eq. ( 2) into Eq.( 1), we obtain the following ordinary differential equations (ODEs): Q(U, U , U , . . . ) = 0.
Step 4. Reduce Eq. ( 5) to the elementary integral form Using a complete discrimination system for polynomial to classify the roots of Φ(Ψ ), we solve the infinite integral (8) and obtain the exact solutions to Eq. ( 3).Furthermore, we can write the exact traveling wave solutions to Eq. (1), respectively.

Application to the Wu-Zhang system
In this section, we apply the extended trial equation method for finding exact solutions of the Wu-Zhang system [29,31] in the form For our purpose, we introduce the following transformations: where l is a constant to be determined later.Substituting Eq. ( 10) into Eq.( 9), integrating once the resulting equation with respect to ξ, and choosing constant of integration to zero, we obtain Inserting Eq. ( 11) into Eq.( 12), we get ODEs as follows: Substituting Eqs. ( 4) and ( 6) into Eq.( 13) and using the balance principle, we find If we take σ = 4, ρ = 0, and ς = 1 in Eq. ( 14), then where µ 4 = 0, ζ 0 = 0. Substituting Eqs. ( 15) and ( 16) into Eq.( 13), collecting the coefficients of Ψ , and solving the resulting algebraic equations system, we obtain Substituting these results into Eqs.( 5) and ( 8), we get where Integrating Eq. ( 17), we obtain the solutions to Eq. ( 13) as follows.

Lie symmetry analysis
In this section, we will perform Lie classical method [5,13,24] on system of Eqs.(9).
Let us consider one parameter Lie group of transformation with small parameter 1.The associated vector field can be written as Now, applying the second prolongation pr 2 V of V to system of Eq. ( 9), we find that the coefficient functions ξ, τ , η 1 , and η 2 must satisfy the invariance condition into Eqs.(37), then using the system of equations ( 9), and equating the coefficients of the various derivative terms, we obtain a system of NLEEs.Solving this system, we obtain following form of infinitesimals: where c 1 , c 2 , c 3 , and c 4 are arbitrary real constants.
Corresponding vector fields are Nonlinear Anal.Model.Control, 22(4):441-458 Commutator table for vector fields (38) is as follows: For the reduction of system of equations ( 9), let us consider following vector fields: where µ and λ are arbitrary constants.
For each case, one can get the similarity variables using characteristic equations: Vector field V 1 Using (39), we obtain similarity variables are as follows: where σ is new independent variable, and F , G are new dependent variable.Substituting the similarity variables (40) into system of equations ( 9), we obtain following system of ODEs where denotes derivatives with respect to σ.
Integrating the first equation of (41), we obtain where C 1 is arbitrary constant.Using (42) into second equation of (41), we have https://www.mii.vu.lt/NA We obtain following solutions of Eq. ( 43): Using ( 44) and ( 42) in (40), we obtain following solutions of main system of equations ( 9): Vector field Corresponding similarity variable are where ζ is new independent variable, and H, J are new dependent variables.Using (45) in ( 9), we obtain the following system of ODEs: where denotes derivatives with respect to ζ.
Integrating first equation of system (46), we have where C 1 is constant of integration.
Integrating second equation of system (46) and using (47) in that, we have where C 2 is constant of integration.Solving Eq. ( 48), we obtain following solutions: with Corresponding solutions of main system of equations ( 9) are as follows: Vector field V 2 + λV 4 Corresponding similarity variables are where ρ and P , Q are new independent and dependent variables, respectively.Substituting (49) in ( 9), we have where denotes derivatives with respect to ρ.
https://www.mii.vu.lt/NAIntegrating first equation of (50), we have where C 1 is constant of integration.Now integrating the second of system (50) and using (51), we have where C 2 is constant of integration.Solving the ODE (52), we obtain where a 1 , a 2 , and a 3 are arbitrary constants.Now using (53) and (51), we have following solutions of system ( 9): where a 1 , a 2 , and a 3 are arbitrary constants.
Vector field V 3 Corresponding similarity variables are where θ is new independent variable, and R, S are new dependent variables.Using (54) in ( 9), we obtain where denotes derivative with respect to θ.
where a 1 , a 2 are arbitrary constants.Corresponding solution of system of equations ( 9) is

Mapping method
In this section, we give an analysis of the mapping method, which will be employed in this paper.Consider a nonlinear coupled PDE with two dependent variables u and v and two independent variables x and t given by where subscripts denote partial derivatives with respect to the corresponding independent variables, and F is a polynomial function of the indicated variables.
Step 1. Assume that Eq. ( 56) has a travelling wave solution (TWS) in the form where ξ = x − ηt, A i , B i , and η are arbitrary constants, l 1 and l 2 are integers, and f i represents integer powers of f .The first derivative of f with respect to ξ denoted by f can be expressed in powers of f in the form where p, q, and r are arbitrary constants.
The motivation for Eq. ( 58) was that the squares of the first derivatives of JEFs can be expressed in even powers of themselves.
Step 2. Substituting Eq. (57) into Eq.( 56), the PDE reduces to an ODE.Balancing the highest order derivative term and the highest order nonlinear term of the ODE, the values of l 1 and l 2 can be found.
Step 3. Substituting for u and v and using Eq.(58), the ODE gives rise to a set of algebraic equations by setting the coefficients of various powers of f to zero.
Step 4. From the values of the parameters A i , B i , p, q, and r, the solution of Eq. (56) can be derived.
Thus, a mapping relation is established through Eq. (57) between the solution to Eq. (58) and that of Eq. (56).
It is to be noted that if the values of l 1 and l 2 are integers, we can use the method directly to get a variety of solutions in terms of hyperbolic functions or JEFs.If they are non integers, the equation may still have solutions as rational expressions involving hyperbolic functions or JEFs.
Applying this method to Eq. ( 13), we can assume the solution in the form Substituting Eq. (59) into Eq.( 13) and using Eq.(58), we arrive at a set of algebraic equations by equating various powers of f to zero.From these equations, we obtain algorithms recover are periodic-singular solutions, cnoidal waves which, as a special case, leads to solitary waves and shock wave solutions.The spectrum of solutions that are reported in this paper will be of immense value in the context of dispersive long waves.In future, there are additional avenues that will be explored.This model will be studied with fractional temporal evolution, time-dependent coefficients as well as stochastic coefficients.These modifications will lead to a closer to reality situations.The results of those research will be disseminated elsewhere.