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 , the Hirota’s bilinear method , Bäcklund transformation , the Darboux transformation , the simplest equation method [5, 6], the (G′/G)-expansion method [7, 8], the reduction mKdV equation method , the F-expansion method , the sine-cosine method , the extended tanh method , 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 
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 . In particularly, it describes the propagation of ion acoustic waves of small amplitude without Landau damping in plasma physics . 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].
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
the system (2) transforms to
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
This equation can be solved in the following manner. Integrating (6) with respect to z yields
where c2 is an arbitrary constant of integration. We multiply (7) by the integrating factor U′ and obtain
Now integration of (8) with respect to z gives us
where c3 is a constant. The solution of (9) can be presented in form of the Jacobi elliptic function  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
where 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
3 Conservation laws
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 … respectively. The differential operator Di denotes the total derivative operator with respect to xi defined by
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
for all solutions of (11).
The functions Qα(x, u, u(1), …) are multipliers yielding a conservation law of the Euler-Lagrange equation if
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
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
where f1, f2, depend of t and f3 is a function of y − t. We can now write down the associated nonlocal conserved vector of (1) as:
Since the functions f1, f2 and f3 appearing in the multipliers are arbitrary, as a result (1) has infinitely many nonlocal conservation laws.
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.
TM thanks the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-Mass) and North-West University, Mafikeng Campus for financial support.
Ablowitz M.J., Clarkson P.A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991 Google Scholar
Gu C.H., Soliton Theory and Its Application, Zhejiang Science and Technology Press, Zhejiang, 1990 Google Scholar
Matveev V.B., Salle M.A., Darboux Transformation and Soliton, Springer, Berlin, 1991 Google Scholar
Bruzón M., Recio, E., Garrido, T.M., Marquez, A.P., Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation, Open Phys., 2017, 15, 433-439. CrossrefWeb of ScienceGoogle Scholar
Wang M., Li, X., Zhang J., The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 2008, 372, 417-423. CrossrefWeb of ScienceGoogle Scholar
Wazwaz A.M., The tanh and sine-cosine method for compact and noncompact solutions of nonlinear Klein Gordon equation, Appl. Math. Comput., 2005, 167, 1179-1195. Google Scholar
Wazwaz A.M., New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations, Appl.Math. Comput., 2006, 182, 1642-1650. Google Scholar
Yasar E., San S., Ozkan, Y.S., Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq 3equation, Open Phys., 2016, 14, 37-43. Google Scholar
Olver P.J., Applications of Lie Groups to Differential Equations (2nd ed.), Springer-Verlag, Berlin, 1993 Google Scholar
Ibragimov N.H., CRC Handbook of Lie Group Analysis of Differential Equations, Vols 1-3, CRC Press, Boca Raton, Florida, 1994-1996 Google Scholar
Bluman G.W., Anco S.C., Symmetry and Integration Methods for Differential Methods, Springer-Verlag, New York, 2002 Google Scholar
Knops R.J., Stuart C.A., Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Ration. Mech. Anal., 1984, 86, 234-249. Google Scholar
Kudryashov N.A., Analytical Theory of Nonlinear Differential Equations, IKI, Moscow-Igevsk, 2004 Google Scholar
Anco S.C., Bluman G.W., Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, European J. Appl. Math., 2002, 13, 545-566. Google Scholar
Bruzón M., Garrido T., De la Rosa, R., Conservation laws and exact solutions of a generalized Benjamin-Bona-Mahony-Burgers equation, Chaos Solitons Fractals, 2016, 89, 578-583. CrossrefWeb of ScienceGoogle Scholar
About the article
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.
© 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