Over some decades, the field of nonlinear evolution equations (NEEs) has attracted the attention of many researchers. NEEs are broadly used to describe problems in science, engineering and mathematical physics such as fluid dynamics, plasma physics, hydro magnetic waves, optic fibers, solid state physics and many others. NEEs can also be used to describe the propagation of a nonlinear dispersive waves in inhomogeneous media [1, 2]. It has become an important bottom-line to find the analytical solutions to these types of equations. Several methods for finding the solutions of various NEEs have been proposed and/or improved by many scholars [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71].
The aim of this paper was to apply the sine-Gordon expansion method (SGEM) to find a new solutions to the Lonngren-wave equation .
where α and β are real constants. The equation describes the electric signals in telegraph lines on the basis of the tunnel diode [15, 16]. The Lonngren-wave equation was used as an example by Akcagil and Aydemir  to show the existence of strong connection between the
SGEM is a method for solving different nonlinear partial differential equations that is developed based on wave transformation and te sine-Gordon expansion method . A new hyperbolic function solutions to the Davey-Stewartson equation with power-law nonlinearity was obtained in  by using SGEM. With the aid of symbolic computation, a new transformation was developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation to obtain the solutions of various types of nonlinear differential equations . A considerable investigation has been implemented by Yan  to sine-Gordon-type equations where the equations are systematically solved by using the Jacobi elliptic function expansion method.
The remaining parts of this paper are organized as follows: In Section 2, we discuss the general facts of the SGEM. In Section 3, we apply the SGEM to the Lonngren-wave equation given in Eq. (1). Section 4 is about results, discussion and some remarks. We finally, give the conclusion of this paper in Section 5.
2 General Facts of the SGEM
In this section we discuss the general facts of SGEM.
where u = u(x, t) and m is a real constant.
Applying the wave transformation u = u(x, t) = U(ξ), ξ = μ (x − ct) to Eq. (2), yields the following nonlinear ordinary differential equation (NODE):
where U = U(ξ), ξ is the amplitude of the travelling wave and c is the velocity of the travelling wave. Reconsidering Eq. (3), we can write its full simplification as:
where K is the integration constant.
Substituting K = 0, w(ξ) =
Putting a = 1 in Eq. (5), we have:
Equation (6) is variables separable equation, we obtain the following two significant equations from solving it:
where p is the integral constant.
For the solution of the following nonlinear partial differential equation;
We determine the value n under the terms of NODE by the balance principle. Letting the coefficients of sini(w)cosj(w) to be all zero, yields a system of equations. Solving this system by using Wolfram Mathematica 9 gives the values of Ai, Bi, μ and c. Finally, substituting the values of Ai, Bi, μ and c in Eq. (10), we obtain the new travelling wave solutions to Eq. (9)
Consider the Lonngren-wave equation given in Eq. (1);
Applying the transformation u = u(x, t) = U(ξ), ξ = μ (x − ct) to Eq. (1), we have:
Differentiating Eq. (13) twice, gives:
A0 − c2α A0 + c2β A02 + A1cos(w) − c2α A1cos(w) - 2c2μ2A1sin2(w)cos(w) + 2c2β A0A1cos(w) + c2β A12cos2(w) + A2cos2(w) − c2α A2cos2(w)
−4c2μ2A2cos2(w)sin2(w) + 2c2μ2A2sin4(w) + 2c2β A0A2cos2(w)+
2c2β A1A2cos3(w) + c2β A22cos4(w) + B1sin(w) − c2α B1sin(w) +
c2μ2B1cos2(w)sin(w) − c2μ2B1sin3(w) + 2c2β A0B1sin(w)+
2c2β A1B1cos(w)sin(w)+ 2c2β A2B1cos2(w)sin(w) + c2βB12sin2(w)
+ B2cos(w)sin(w) − c2α B2cos(w)sin(w) + c2μ2B2cos3(w)sin(w)−
5c2μ2B2sin3(w)cos(w) + 2c2β A0B2cos(w)sin(w) + 2c2β A1B2cos2sin(w)
+ 2c2β A2B2cos3sin(w) + 2c2β B1B2sin2(w)cos(w) + c2β B22cos2(w)sin2(w) = 0.
We collect a set of algebraic equations by equating each summation of the coefficients of the trigonometric terms of the same power to zero in the abovementioned equation. We solve the set of generated to obtained the values of the coefficients. To get the new solitary solutions, u(x, t) to Eq. (1), we substitute in each case the obtained results of the coefficients into Eq. (10) along with n = 2.
- Case-1When we consider following coefficients:these produce new dark solution as:
- Case-2If it is taken asthey produce a new singular solution as:
- Case-3When we takethey give mixed complex singular solution as:
- Case-4give mixed complex rational solution as:
- Case-5which produces the following trigonometric travelling wave solution as:
- Case-6which introduces the following complex mixed solution as:
4 Results and Discussion
The powerful SGEM as one of the prominent methods for obtaining the some new travelling wave solutions to the nonlinear partial differential equations has been used in this paper. This method is based on both important properties of the sine-Gordon equation such as Eqs. (7) and Eq.(8). The SGEM includes trigonometric functions, which will be used later for obtaining novel solutions in Eq.(11). Many new solutions can be obtained by using the properties of these trigonometric functions. This is one of the main properties of SGEM. Therefore, it gives many coefficients to the considered model such as complex, exponential and trigonometric.
In this manuscript, by selecting of some of them, we have obtained the same solution, Eq.(15); moreover, we have found some entirely new complex, exponential, dark and hyperbolic solutions to the model considered when we compared the solutions obtained with the help of exp the (G′/G)-expansion method and the modified extended tanh method used in . These solutions are new physical properties of model equation Eq.(1.1). The effectiveness and the simplicity of the method show that its powerful and reliable mathematical tool that can be applied in solving various NEEs. For computational calculations, we have used the packet programs for drawing graphical surfaces in this paper. To the best of our knowledge, the application of SGEM to the Lonngren-wave equation has not been submitted to the literature beforehand.
P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Physics, 18 (1977) 1212-1217.
A.M. Wazwaz, New solitons and kinks solutions to the Sharmaassolver equation, Appl. Math. Computation 188 (2007) 1205-1213.
C. Cattani, A review on Harmonic Wavelets and their fractional extension, Journal of Advanced Eng. Comp., 2018, 2(4), 224-238.
D. Lu, A.R. Seadawy, M.M. A. Khater, Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods, Results in Physics, 2017, 7, 2028-2035.
A. Ciancio, A. Quartarone, A Hybrid Model For Tumor-Immune Competition, U.P.B. Sci. Bull. Series A, 2013, 75(4), 125-136.
M. Arshad, A.R. Seadawy, D.Lu, Bright-dark solitary wave solutions of generalized higher-order nonlinear Schrödinger equation and its applications in optics, J Electromag Waves and Appl, 2017, 31(16), 1711-1722.
I. Cilingir, H. Demir, New Algorithm for the Lid-driven Cavity Flow Problemwith Boussinesq-Stokes Suspension, Karaelmas J. Science and Eng, 8(2), (2018), 462-472.
T. Caraballo, M. Herrera-Cobos, P. Marín-Rubio, An iterative method for non-autonomous nonlocal reaction-diffusion equations, Appl. Math. Nonlinear Sciences, 2(1) (2017) 73-82.
Baskonus, H.M.; Bulut, H.; Emir, D.G. Regarding New Complex Analytical Solutions for the Nonlinear Partial Vakhnenko-Parkes Differential Equation via Bernoulli Sub-Equation Function Method, Math. Letters, 2015, 1(1), 1-9.
M. Askin, Effect of the Transition Metal Elements on the Relaxation Times in the Agar Solutions, Asian Journal of Chemistry, 19(4), (2007), 3191-3196.
C. Cattani, A. Ciancio, On the fractal distribution of primes and prime-indexed primes by the binary image analysis, Physica A, 2016, 460, 222-229
A. Ciancio, H.M. Baskonus, T.A. Sulaiman, H.Bulut, New structural dynamics of isolated waves via the coupled nonlinear Maccari’s system with complex structure, Indian J of Physics, 92(10), (2018), 1281-1290.
S.M. El-Shaboury, M.K. Ammar, W.M. Yousef, Analytical solutions of the relative orbital motion in unperturbed and in J- perturbed elliptic orbits, Appl. Math. Nonlinear Sciences, 2(2), (2017), 403-414
S. Akcagil, T.T. Aydemir, Comparison Between the (G′/G)-Expansion Method and the Modified Extended tanh Method, Open Physics 14, 88 (2016)
T.A. Sulaiman, A.I. Aliyu, A. Yusuf, A Solution of Telegraph Equation by Natural Decomposition Method International Conference on Mathematics and Mathematics Education (ICMME-2016), Elazig/Turkey, 12-14 May 2016.
K.E. Lonngren, H.C.S. Hsuan, W.F. Ames, On the Soliton, Invariant and Shock Solutions of a Fourth-Order Nonlinear Equation, Journal of Mathematical Analysis and Applications, 52, 538 (1875).
M I Rabinovich, D I Trubetskov, Introduction in Theory of Waves, Nauka, Moscow, (1984).
C. Yan, A Simple Transformation for Nonlinear Waves, Physics Letters A 22(4), 77 (1996)
H.M. Baskonus, New Acoustic Wave Behaviors to the Davey-Stewartson Equation with Power Nonlinearity Arising in Fluid Dynamics, Nonlinear Dynamics 86(1), 177 (2016)
Z. Yan, A New Sine-Gordon Equation Expansion Algorithm to Investi- gate Some Special Nonlinear Differential Equations, Academia Sinica 23, 300 (2004)
S. Liu, Z. Fu and S. Liu, Exact Solutions to Sine-Gordon-Type Equations, Physics Letters A, 351 (2006), 59-63.
Z. Yan and H. Zhang, New Explicit and exact Travelling Wave Solutionsfor a System of Variant Boussinesq equations in Mathematical Physics, Physics Letters A, 252 (1999), 291-296.
Y. Zhen-Ya, Z. Hong-oing and F. En-Gui, New Explicit and Travelling Wave Solutions for a Class of Nonlinear Evolution Equations, Acta Physica Sinica, 48(1) (1999), 1-5.
Araci, S.; Ozer, O. Extended q-Dedekind-type Daehee- Changhee sums associated with extended q-Euler polynomials, Adv. Differen. Eq, 2015, 2015(1), 272-276.
H.M. Baskonus and H. Bulut, Analytical Studies on the (1 + 1)-dimensional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation Defined by Seismic Sea Waves, Waves in Random and Complex Media, 25(4), 1-13, 2015.
Aslan, I. Exact and explicit solutions to nonlinear evolution equations using the division theorem, Appl. Math. and Comp. 2011, 217, 8134-8139.
Sulaiman, T.A.; Bulut, H.; Yokus, A.; Baskonus, H.M. On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering, Indian J of Physics 2018, 1-10.
V.B. Awati, M. Jyoti, Homotopy analysis method for the solution of lubrication of a long porous slider, Appl. Math. Nonlinear Sciences, 1(2) (2016) 507-516
Cattani. C.; Sulaiman, T.A.; Baskonus, H.M.; Bulut, H. On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfeľd-Sokolov systems, Opt. and Quan. Elect. 2018, 50(138), 1-11.
O. Ozer; Pekin A, An Algorithm For Explicit Form of Fundamental Units of Certain Real Quadratic Fields and Period Eight, European J. Pure and Applied Mathematics, 2015, 8(3), 343-356.
Cattani. C.; Sulaiman, T.A.; Baskonus, H.M.; Bulut, H. Solitons in an inhomogeneous Murnaghan’s rod, Europ Phys J Plus, 2018, 133(228), 1-11.
Baskonus, H.M. New complex and hyperbolic function solutions to the generalized double combined Sinh-Cosh-Gordon equation, AIP Conf. Proc. 2017, 1798(020018), 1-10.
Seadawy, A. Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma, Comput. Math. Appl, 2016, 71, 201-206.
Unlukal, C.; Penel, M.; Penel, B. Risk Assessment with Failure Mode and Effect Analysis and Gray Relational Analysis Method in Plastic Enjection Prosess, ITM Web of Conf., 2018, 22(01023), 1-10
Sulaiman, T.A.; Yokus, A.; Gulluoglu, N.; Baskonus, H.M.; Bulut, H. Regarding the Numerical and Stability Analysis of the Sharma-Tosso-Olver Equation, ITM Web of Conf., 2018, 22(01036), 1-9.
M.Dewasurendra, K.Vajravelu, On the Method of Inverse Mapping for Solutions of Coupled Systems of Nonlinear Differential Equations Arising in Nanofluid Flow, Heat and Mass Transfer, Appl. Math. Nonlinear Sciences 3(1) (2018) 1-14
H. Bulut, Classification of exact solutions for generalized form of K(m, n) equation, Abstract and Applied Analysis, 2013 (2013), 1-11.
H. M. Baskonus, H.Bulut, F. B. M Belgacem, Analytical Solutions for Nonlinear Long-Short Wave Interaction Systems with Highly Complex Structure, J. Comput. Appl. Math., 312, 257 (2017)
Ilhan, O.A.; Sulaiman, T.A.; Bulut, H.; Baskonus, H.M. On the new wave Solutions to a Nonlinear Model Arising in Plasma Physics, Europ Phys J Plus, 2018, 133(27), 1-6.
Ozer, O. A Note On Structure of Certain Real Quadratic Number Fields, Iranian Journal of Science and Technology, 2017, 41(3), 759-769.
Baskonus, H.M.; Sulaiman, T.A.; Bulut, H.; Akturk, T. Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schrödinger equation with -potential, Superlattice and Microstructures, 2018, 115, 19-29.
S. Duran; M. Askin, T.A. Sulaiman, New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science, An Int.J. Optimization and Control: Theories and Applications, 2017, 7(3), 240-247.
Baskonus, H.M.; Askin, M., Travelling Wave Simulations to the Modified Zakharov-Kuzentsov Model Arising In Plasma Physics, 6th International Youth Science Forum “LITTERIS ET ARTIBUS”, Computer Science and Engineering, Lviv, Ukraine, 24-26 November 2016.
C.M.Khalique, I.E.Mhlanga, Travelling waves and conservation laws of a (2 + 1)-dimensional coupling system with Korteweg-de Vries equation, Appl. Math. Nonlinear Sciences 3(1) (2018) 241-254
Baskonus, H.M.; Koc, D.A.; Bulut, H. New travelling wave prototypes to the nonlinear Zakharov-Kuznetsov equation with power law nonlinearity, Nonlin. Sci. Letters A: Math., Phys. and Mech. 2016, 7(2), 67-76.
W.X. Ma; J.Li; C.M. Khalique, A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2 + 1)-Dimensions, Complexity, 2018, 2018(9059858), 1-7.
Bulut, H.; Atas, S.S.; Baskonus, H.M. Some novel exponential function structures to the Cahn-Allen equation, Cogent Physics, 2016, 3(1240886), 1-8.
Yokus, A.; Sulaiman, T.A.; Gulluoglu, M.T.; Bulut, H. Stability Analysis, Numerical and Exact Solutions of the (1 + 1)-Dimensional NDMBBM Equation, ITM Web of Conf., 2018, 22(01064), 1-10.
Yokus, A.; Sulaiman, T. A.; Baskonus, H. M.; Atmaca, S. P. On the exact and numerical solutions to a nonlinear model arising in mathematical biology, ITM Web of Conf., 2018, 22(01061), 1-10.
F.T. Akyildiz, K. Vajravelu, Galerkin-Chebyshev Pseudo Spectral Method and a Split Step New Approach for a Class of Two dimensional Semi-linear Parabolic Equations of Second Order, Appl. Math. Nonlinear Sciences 3(1) (2018) 255-264
H.M. Baskonus, H. Bulut, On the Complex Structures of Kundu-Eckhaus Equation via Improved Bernoulli Sub-Equation Function Method, Waves in Random and Complex Media, 25(4), 720 (2015).
I. Cilingir; H. Demir, Application of the Hybrid Differential Transform Method to the nonlinear equations, Applied Mathematics, 2012, 3(3), 1-10.
A.R. Seadawy, Ionic acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equations in quantum plasma, Mathematical Methods and Applied Sciences, 2017, 40, 1598-1607.
H.M. Baskonus, H. Bulut, New Complex Hyperbolic Function Solutions for the (2 + 1)-Dimensional Dispersive Long Water-Wave System, Mathematical and Computational Applications, 21(2), 1 (2016),
O. Ozer, Determination of Fundamental Units of Real Quadratic Number Fields Related with Specific Continued Fraction Expansions. Egyptian Computer Science Journal, 2018, 42(2), 1-12.
Bulut, H.; Sulaiman, T.A., Baskonus, H.M., Yazgan, T., Novel Hyperbolic Behaviors to Some Important Models Arising in Quantum Science, Opt. and Quan. Elect., 2017, 49(349), 1-16.
M. Rosa, M.L. Gandarias, Multiplier method and exact solutions for a density dependent reaction-diffusion equation, Appl. Mathematics and Nonlinear Sciences 1(2), 2016 311-320.
H. M. Baskonus, H. Bulut, Analytical Studies on the (1 + 1)-dimensional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation Defined by Seismic Sea Waves, Waves in Random and Complex Media, 25(4), 576-586, 2015.
H. M. Baskonus, Complex Soliton Solutions to the Gilson-Pickering Model, Axioms, 8(1), 18, 2019.
H. Demir, Temporal differential transform and spatial finite difference methods for unsteady heat conduction equations with anisotropic diffusivity, Gazi University Journal of Science, 2014, 27(4), 1063-1076.
H. Bulut, Comparison between The Alternating Group Explicit Method and Adomian Decomposition Method for Solution of Coupled Viscous Burgers Equation, Nonlinear Science Letters A, 1(2), 161-172, (2010).
V.B. Awati, M. Jyoti, Homotopy analysis method for the solution of lubrication of a long porous slider, Appl. Math. Nonlinear Sciences, 1(2) (2016) 507-516
H.M. Baskonus, New complex and hyperbolic function solutions to the generalized double combined Sinh-Cosh-Gordon equation, AIP Conf. Proc. 1798(020018) (2017) 1-10.
A. Biswas, M.O. Al-Amr, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, M. Belic, Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution, Optik 165 (2018) 233-239.
A.F. Qasim, M.O. Al-Amr, Approximate solution of the Kersten-Krasil’shchik coupled Kdv-MKdV system via reduced differential transform method, Eurasian Journal of Science and Engineering 4 (2) (2018) 1-9.
M.O. Al-Amr, Exact solutions of the generalized (2 + 1)-dimensional nonlinear evolution equations via the modified simple equation method, Comput. Math. Appl. 69 (5) (2015) 390-397.
A.J. Al-Sawoor, M.O. Al-Amr, A new modification of variational iteration method for solving reaction-diffusion system with fast reversible reaction, J. Egyptian Math. Soc. (2014), Vol. 22, No. 3, pp. 396-401.
A.J. Al-Sawoor, M.O. Al-Amr, Numerical solution of a reaction-diffusion system with fast reversible reaction by using Adomian’s decomposition method and He’s variational iteration method, Al-Rafidain J. Comput. Sci. Math. 9 (2) (2012) 243-257.
M.O. Al-Amr, S. El-Ganaini, New exact traveling wave solutions of the (4 + 1)-dimensional Fokas equation, Comput. Math. Appl. 74 (2017) 1274-1287.
M.O. Al-Amr, Exact solutions of a family of higher-dimensional space-time fractional KdV type equations, Computer Science and Information Technology 8 (6) (2018) 131-141.
A.J. Al-Sawoor, M.O. Al-Amr, Reduced differential transform method for the generalized Ito system, Int. J. Enhanc. Res. Sci. Tech. Eng. 2 (11) (2013) 135-145.