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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

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Volume 16, Issue 1

Issues

Volume 13 (2015)

On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics

Tanki Motsepa
  • International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Chaudry Masood Khalique
  • Corresponding author
  • International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa
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  • Other articles by this author:
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Published Online: 2018-05-04 | DOI: https://doi.org/10.1515/phys-2018-0030

Abstract

In this paper, we study a (2+1) dimensional KdV-mKdV equation, which models many physical phenomena of mathematical physics. This equation has two integral terms in it. By an appropriate substitution, we convert this equation into two partial differential equations, which do not have integral terms and are equivalent to the original equation. We then work with the system of two equations and obtain its exact travelling wave solutions in form of Jacobi elliptic functions. Furthermore, we employ the multiplier method to construct conservation laws for the system. Finally, we revert the results obtained into the original variables of the (2+1) dimensional KdV-mKdV equation.

Keywords: (2+1) dimensional KdV-mKdV equation; travelling wave solutions; Jacobi elliptic functions; conservation laws; multiplier method

PACS: 02.30.Jr; 02.30.Hq; 04.20.Jb

1 Introduction

In the last few decades nonlinear partial differential equations (NPDEs) have been vastly studied by researchers as these equations model many physical phenomena of the real world. This is apparent from many research papers which have appeared in the literature. One of the main interest of the researchers is to look for closed form solutions of NPDEs. Nonetheless, this is not a simple task. Despite of this fact, many researchers have developed novel approaches to solve NPDEs. These approaches include the inverse scattering transform method [1], the Hirota’s bilinear method [2], Bäcklund transformation [3], the Darboux transformation [4], the simplest equation method [5, 6], the (G′/G)-expansion method [7, 8], the reduction mKdV equation method [9], the F-expansion method [10], the sine-cosine method [11], the extended tanh method [12], the exp-function expansion method [13, 14, 15], the multiple exp-function method [16, 17] the Lie symmetry method [18, 19, 20, 21], etc.

In this paper we study a NLPDE known as (2+1) dimensional KdV-mKdV equation [22]

ut+uxxy+4uuy4u2uy+2uxx1uy2uxx1(2uuy)=0,(1)

which models many physical phenomena of mathematical physics. The (1+1) dimensional KdV-mKdV equation describes the wave propagation of bounded particle with a harmonic force [23]. In particularly, it describes the propagation of ion acoustic waves of small amplitude without Landau damping in plasma physics [24]. The propagation of thermal pulse through single crystal of sodium fluoride in solid physics can also be explained by this equation [25, 26].

In the study of differential equations (DEs) conservation laws play a very important role. Also, in exploring the existence, uniqueness and stability of solutions to DEs, researchers have employed conservation laws [27, 28, 29]. Conservation laws can be engaged in determining exact solutions of NLPDEs [30, 31].

To study the (2+1) dimensional KdV-mKdV (1), we start by letting v=x1(uy2uuy)=(uy2uuy)dx. This transforms (1) to a system of two partial differential equations (PDEs)

ut+uxxy+4uuy4u2uy+2uxv=0,vxuy+2uuy=0.(2)

The outline of the paper is as follows. In Section 2, firstly, we use the travelling wave variable to reduce (2) to two ordinary differential equations (ODEs) and then obtain its exact explicit solutions. We compute conservation laws of (1) with the help of multiplier method in Section 3. Finally, in Section 4 we give concluding remarks.

2 Exact solutions of (2)

We first use the travelling wave variable z = x + by + ct to reduce the PDEs (2) to two ordinary differential equations (ODEs). Thus by letting

u(x,y,t)=U(z),v(x,y,t)=V(z),wherez=x+by+ct,(3)

the system (2) transforms to

bU(z)+4b(1U(z))U(z)U(z)+2V(z)U(z)+cU(z)=0,(4a)

2bU(z)U(z)bU(z)+V(z)=0,(4b)

which is a system of nonlinear ODEs. We now decouple the above system by solving the second equation for V. Integrating Eq. (4b) with respect to z, we obtain

V(z)=bU(z)2+bU(z)+c1,(5)

where c1 is an arbitrary constant of integration. The substitution of the value of V from (5) into Eq. (4a) gives the third-order nonlinear ODE

bU(z)6bU(z)2U(z)+6bU(z)U(z)+cU(z)+2c1U(z)=0.(6)

This equation can be solved in the following manner. Integrating (6) with respect to z yields

bU2bU3+3bU2+(c+2c1)U+c2=0,(7)

where c2 is an arbitrary constant of integration. We multiply (7) by the integrating factor U′ and obtain

bUU2bU3U+3bU2U+(c+2c1)UU+c2U=0.(8)

Now integration of (8) with respect to z gives us

U2U4+2U3+U2b(c+2c1)+2c2Ub+2c3b=0,(9)

where c3 is a constant. The solution of (9) can be presented in form of the Jacobi elliptic function [32] and is a bit cumbersome to write here. However, by imposing the asymptotic boundary conditions U, U′, U″ → 0 for ∣ z ∣→ ∞, we obtain c2 = 0 and c3 = 0 and one can write a special solution of (9) given by

U(z)=4(ck)2eθ4b2kc+4b(ck)(kc+eθ)+(kc)3/2e2θ,

where k=2c1,θ=(kc)/b(z+l) and l is a constant. Thus, the exact solutions of (1) can be presented in the form of the Jacobi elliptic functions and its special solution is

u(t,x,y)=4(ck)2eθ4b2kc+4b(ck)(kc+eθ)+(kc)3/2e2θ,(10)

where

θ=kcb(x+by+ct+l).

Profile of the solution (10)
Figure 1

Profile of the solution (10)

3 Conservation laws

We now derive the conservation laws for (1). For this purpose the multiplier method will be employed [33]. But first we recall some basic definitions and results that we use later in the section.

Let

Fα(x,u,u(1),,u(k))=0,α=1,,m,(11)

be a kth-order system of PDEs with x = (x1,x2, …, xn) and u = (u1, u2, …, um) the n independent and m dependent variables, respectively. The subscripts in u(1), u(2), …, u(k) denote all first, second, …, kth-order partial derivatives and uiα=Di(uα),uijα=DjDi(uα), … respectively. The differential operator Di denotes the total derivative operator with respect to xi defined by

Di=xi+uiαuα+uijαujα+,i=1,,n.(12)

The operator

Euδδuα=uα+s1(1)sDi1Disui1i2isα,α=1,,m,(13)

is called the Euler-Lagrange operator.

The nmtuple T =(T1, T2, …, Tn), with Tk ∈ 𝓐, k = 1, …, n, where 𝓐 is the space of differential functions, is a conserved vector of (11) if Tk satisfies

DkTk=0(14)

for all solutions of (11).

The functions Qα(x, u, u(1), …) are multipliers yielding a conservation law of the Euler-Lagrange equation if

QαFα=DiTi(15)

holds identically. For our equation we assume multipliers of the zeroth-order,

that is, Qα = Qα (t, x, y, u, v). The multiplier Qα is a solution of the determining equation

Eu(QαFα)=0.(16)

The expansion of the determining equation provides us with an over-determining system of PDEs, whose solution gives us the multipliers. The conservation laws are then obtained using the homotopy formula [33, 34].

3.1 Conservation laws of (1)

In this subsection we derive the conservation laws for (1) using the multiplier approach. For the coupled system (2), we look for the zeroth-order multipliers of the form, Q1 = Q1 (t, x, y, u, v) and Q2 = Q2 (t, x, y, u, v) that are given by

Q1=u2f1(t)uf1(t)12yf1(t)+f2(t)+2uf3(yt)f3(yt),Q2=uf1(t)12f1(t)+f3(yt),

where f1, f2, depend of t and f3 is a function of yt. We can now write down the associated nonlocal conserved vector of (1) as:

Tt=12u2f1(t)12uf1(t)+uf3(yt),Tx={(u2u)(uy2uuy)dx12uxuy+12uuxy12uxy}f1(t)+(f2(t)12yf1(t))(uy2uuy)dx+uxyf3(yt)+(2u1)f3(yt)(uy2uuy)dx,Ty=12u(u3+uuxx2u2)f1(t)12u(uyy)f1(t)+u(u1)f2(t)+uf3(yt).

Remark 2

Since the functions f1, f2 and f3 appearing in the multipliers are arbitrary, as a result (1) has infinitely many nonlocal conservation laws.

4 Conclusions

The (2+1) dimensional KdV-mKdV (1) was investigated from the point of view of solutions and conservation laws. We introduced a new variable v and wrote the (1) as a system of two PDEs, which did not have integral terms. The travelling wave variable was then utilized to reduce the system of PDEs to two nonlinear ODEs. The resulting system of ODEs was decoupled and solved directly. As a result we obtained travelling wave solutions of (1) in the form of Jacobi elliptic functions. These newly obtained solutions are very important in explaining physical situations of some real world problems that are related to the equation. Furthermore, conservation laws of (1) were also computed using the multiplier method. This resulted in infinitely many nonlocal conserved vectors. The significance of conservation laws was mentioned in Section 1 of the paper.

Acknowledgement

TM thanks the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-Mass) and North-West University, Mafikeng Campus for financial support.

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About the article

Tel.: +27 18 389 2009


Received: 2017-08-28

Accepted: 2017-10-30

Published Online: 2018-05-04


Citation Information: Open Physics, Volume 16, Issue 1, Pages 211–214, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0030.

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© 2018 T. Motsepa and C. M. Khalique. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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