Travelling Solitary Wave Solutions to Higher Order Korteweg-de Vries Equation

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1. Introduction

Nonlinear wave phenomena appear in various scientific and engineering fields [1] [2] [3] [4] , such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. The higher order Korteweg-de Vries equation is one of important equations for description localized structures in the modern physics, such as two-layer fluid [5] , steady-state solitary waves in a fluid [6] , a three-layer fluid with a constant buoyancy frequency in an each layer [7] [8] , which means the investigation of the travelling wave solutions for nonlinear partial differential equations plays an important role. The Korteweg-de Vries (KdV) equation [9] [10] is first used to describe the nonlinear long internal waves in a fluid stratified by both density and current. The higher-order Korteweg-de Vries equation is investigated with different methods [11] [12] [13] , which is written as follows.

${u}_{t}+{u}_{xxxxx}+\gamma u{u}_{xxx}+\beta {u}_{x}{u}_{xx}+\alpha {u}^{2}{u}_{x}=0$ (1)

where the subscripts denote the partial derivatives of x and t. The γ, β and α are constant parameters; u represents a real scalar function u(x, t).

2. The Tanh-Function Method

Consider a given evolution equation with independent variables x and t in the form are given as

$P\left(u,{u}_{x},{u}_{t},{u}_{xt},{u}_{xx},{u}_{tt},\cdots \right)=0$. (2)

By using the wave variable $\xi =x+\kappa t$ and substituting into Equation (2), we obtain the following ordinary differential equation

$Q\left(u,{u}_{\xi},{u}_{\xi \xi},{u}_{\xi \xi \xi},\cdots \right)=0$. (3)

where ${u}_{\xi},{u}_{\xi \xi},\cdots $ denotes the derivative with respect to the same sole variable $\xi $.

The fact that the solutions of many nonlinear equations can be expressed as a finite series of tanh-function motivates us to seek for the solutions

$u\left(x,t\right)=U\left(\xi \right)={\displaystyle \underset{i=0}{\overset{m}{\sum}}{a}_{i}\mathrm{tanh}{\left(\xi \right)}^{i}}$ (4)

where $\xi $ is a function about x and t, ${a}_{0},{a}_{1},{a}_{2},\cdots $ are constant parameters. m can be obtained by balancing the derivative term of the highest order with the nonlinear term as the follow

$\text{O}\left(\frac{{\text{d}}^{p}u}{\text{d}{\xi}^{p}}\right)=m+p,\text{\hspace{0.17em}}\text{\hspace{0.17em}}p=1,2,3,\cdots $

and

$\text{O}\left({u}^{q}\frac{{\text{d}}^{p}u}{\text{d}{\xi}^{p}}\right)=\left(q+1\right)m+p,\text{\hspace{0.17em}}\text{\hspace{0.17em}}q=0,1,2,\cdots ;p=1,2,3,\cdots $

Usually m is a positive integer, however, once in a while, the value of m is a negative or a fraction, the other kinds of expression will introduced. In the following, we illustrate the method by using it to solve the higher-order Korteweg-de Vries equations.

3. The Tanh-Function Method Solutions for Higher-Order Korteweg-de Vries Equation

Putting the variable $\xi =x+\kappa t$ into Equation (1) and we find

$k{u}_{\xi}+{u}_{\xi \xi \xi \xi \xi}+\gamma u{u}_{\xi \xi \xi}+\beta {u}_{\xi}{u}_{\xi \xi}+\alpha {u}^{2}{u}_{\xi}=0$; (5)

which is an ordinary differential equation. By using the above method, m = 2 is obtained by balancing the derivative term of the highest order with the nonlinear term.

$u\left(\xi \right)={a}_{0}+{a}_{1}\mathrm{tanh}\left(\xi \right)+{a}_{2}\mathrm{tanh}{\left(\xi \right)}^{2}$ (6)

substituting (6) into (5) yields a set of algebraic equations for a_{0}, a_{1}, a_{2}, γ, β and α. Collecting all terms with the same power of
$\mathrm{tanh}\left(\xi \right)$ together, equating each coefficient to zero, we obtain a set of simultaneous algebraic equations as follows:

$16{a}_{1}+\alpha {a}_{0}^{2}{a}_{1}+2{a}_{1}{a}_{2}\beta -2{a}_{1}{a}_{0}\beta +{a}_{1}\kappa =0$

$2\alpha {a}_{1}^{2}{a}_{0}+272{a}_{2}+2\alpha {a}_{0}^{2}{a}_{2}-2{a}_{1}^{2}\beta +4{a}_{2}^{2}\beta -2{a}_{1}^{2}\gamma -16{a}_{0}{a}_{2}\gamma +2{a}_{2}\kappa =0$

$-136{a}_{1}-\alpha {a}_{0}^{2}{a}_{1}+\alpha {a}_{1}^{3}+6\alpha {a}_{0}{a}_{1}{a}_{2}-14{a}_{1}{a}_{2}\beta +8{a}_{0}{a}_{1}\gamma -18{a}_{1}{a}_{2}\gamma -{a}_{1}\kappa =0$

$\begin{array}{l}-\text{2}\alpha {a}_{1}^{2}{a}_{0}-1232{a}_{2}-\text{2}\alpha {a}_{0}^{2}{a}_{2}+\text{4}\alpha {a}_{1}^{2}{a}_{2}+\text{4}\alpha {a}_{2}^{2}{a}_{0}+4{a}_{1}^{2}\beta \\ -20{a}_{2}^{2}\beta +8{a}_{1}^{2}\gamma +40{a}_{0}{a}_{2}\gamma -16{a}_{0}{a}_{2}\gamma -2{a}_{2}\kappa =0\end{array}$

$240{a}_{1}-\alpha {a}_{1}^{3}-6\alpha {a}_{0}{a}_{1}{a}_{2}+5\alpha {a}_{2}^{2}{a}_{1}+22{a}_{1}{a}_{2}\beta -6{a}_{0}{a}_{1}\gamma +48{a}_{1}{a}_{2}\gamma =0$

$\begin{array}{l}1680{a}_{2}-\text{4}\alpha {a}_{1}^{2}{a}_{2}-\text{4}\alpha {a}_{2}^{2}{a}_{0}+\text{2}\alpha {a}_{2}^{3}-2{a}_{1}^{2}\beta +28{a}_{2}^{2}\beta \\ \text{\hspace{0.05em}}-6{a}_{1}^{2}\gamma -24{a}_{0}{a}_{2}\gamma +40{a}_{2}^{2}\gamma =0\end{array}$

$-120{a}_{1}-5\alpha {a}_{2}^{2}{a}_{1}-10{a}_{1}{a}_{2}\beta -30{a}_{1}{a}_{2}\gamma =0$

$-720{a}_{2}-2\alpha {a}_{2}^{3}-12{a}_{2}^{2}\beta -24{a}_{2}^{2}\gamma =0$

Solving the algebraic equations, we get the results:

1) The first case ${a}_{\text{1}}=0$

${a}_{0}=\frac{4\gamma -\sqrt{-136\alpha -2\alpha {a}_{2}\beta +16{\gamma}^{2}-\alpha \kappa}}{\alpha}$ , ${a}_{2}=\frac{-6\beta -12\gamma -\sqrt{{\left(6\beta +12\gamma \right)}^{2}-1440\alpha}}{2\alpha}$ ,

${u}_{1}\left(x,t\right)={a}_{0}+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (7)

${a}_{0}=\frac{4\gamma +\sqrt{-136\alpha -2\alpha {a}_{2}\beta +16{\gamma}^{2}-\alpha \kappa}}{\alpha}$ , ${a}_{2}=\frac{-6\beta -12\gamma +\sqrt{{\left(6\beta +12\gamma \right)}^{2}-1440\alpha}}{2\alpha}$

${u}_{2}\left(x,t\right)={a}_{0}+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (8)

${a}_{0}=\frac{4\gamma -\sqrt{-136\alpha -2\alpha {a}_{2}\beta +16{\gamma}^{2}-\alpha \kappa}}{\alpha}$ , ${a}_{2}=\frac{-6\beta -12\gamma +\sqrt{{\left(6\beta +12\gamma \right)}^{2}-1440\alpha}}{2\alpha}$

${u}_{3}\left(x,t\right)={a}_{0}+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (9)

${a}_{0}=\frac{4\gamma +\sqrt{-136\alpha -2\alpha {a}_{2}\beta +16{\gamma}^{2}-\alpha \kappa}}{\alpha}$ , ${a}_{2}=\frac{-6\beta -12\gamma -\sqrt{{\left(6\beta +12\gamma \right)}^{2}-1440\alpha}}{2\alpha}$ ,

${u}_{4}\left(x,t\right)={a}_{0}+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (10)

2) The second case

${a}_{\text{0}}=\frac{\text{840}+\text{10}{a}_{2}\beta +15{a}_{2}\gamma}{\text{16}\gamma}$ , ${a}_{\text{1}}=\sqrt{\frac{\text{10}{a}_{0}\gamma +6{a}_{2}\left(2\beta +3\gamma -\alpha {a}_{0}\right)-1320}{\alpha}}$ , ${a}_{2}=\frac{-3\gamma +3\sqrt{{\gamma}^{2}-64\alpha}}{2\alpha}$

${u}_{5}\left(x,t\right)={a}_{0}+{a}_{1}\mathrm{tanh}\left(x+\kappa t\right)+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (11)

${a}_{\text{0}}=\frac{\text{840}+\text{10}{a}_{2}\beta +15{a}_{2}\gamma}{\text{16}\gamma}$ , ${a}_{\text{1}}=-\sqrt{\frac{\text{10}{a}_{0}\gamma +6{a}_{2}\left(2\beta +3\gamma -\alpha {a}_{0}\right)-1320}{\alpha}}$ , ${a}_{2}=\frac{-3\gamma -3\sqrt{{\gamma}^{2}-64\alpha}}{2\alpha}$

${u}_{6}\left(x,t\right)={a}_{0}+{a}_{1}\mathrm{tanh}\left(x+\kappa t\right)+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (12)

${a}_{\text{0}}=\frac{\text{840}+\text{10}{a}_{2}\beta +15{a}_{2}\gamma}{\text{16}\gamma}$ , ${a}_{\text{1}}=\sqrt{\frac{\text{10}{a}_{0}\gamma +6{a}_{2}\left(2\beta +3\gamma -\alpha {a}_{0}\right)-1320}{\alpha}}$ , ${a}_{2}=\frac{-3\gamma -3\sqrt{{\gamma}^{2}-64\alpha}}{2\alpha}$

${u}_{7}\left(x,t\right)={a}_{0}+{a}_{1}\mathrm{tanh}\left(x+\kappa t\right)+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (13)

${a}_{\text{0}}=\frac{\text{840}+\text{10}{a}_{2}\beta +15{a}_{2}\gamma}{\text{16}\gamma}$ , ${a}_{\text{1}}=-\sqrt{\frac{\text{10}{a}_{0}\gamma +6{a}_{2}\left(2\beta +3\gamma -\alpha {a}_{0}\right)-1320}{\alpha}}$ , ${a}_{2}=\frac{-3\gamma +3\sqrt{{\gamma}^{2}-64\alpha}}{2\alpha}$

${u}_{8}\left(x,t\right)={a}_{0}+{a}_{1}\mathrm{tanh}\left(x+\kappa t\right)+{a}_{2}\mathrm{tanh}{\left(x+\kappa t\right)}^{2}$ (14)

Equations (7)-(14) are the solutions in the $\mathrm{tanh}\left(x+\kappa t\right)$ form for higher order Korteweg-de Vries Equation (1). The numerical simulations ${u}_{1}\left(x,t\right)$ and ${u}_{2}\left(x,t\right)$ are shown with $\alpha =-1$ , $\beta =\text{2}$ , $\gamma =0.4$ , $\kappa =1$ , the range of x and t are $x\in \left[-1,1\right]$ and $t\in \left[-1,1\right]$ in Figure 1 and Figure 2, respectively. Simulation ${u}_{5}\left(x,t\right)$ of Equation (11) with $\alpha =0.01$ , $\beta =-8$ , $\gamma =3$ , $\kappa =1$ , the range of x and t are $x\in \left[-10,10\right]$ and $t\in \left[-10,10\right]$ , which is shown in Figure 3. Simulation ${u}_{6}\left(x,t\right)$ for Equation (12), with $\alpha =0.01$ , $\beta =-5$ , $\gamma =3$ , $\kappa =1$. The range of x and t are $x\in \left[-10,10\right]$ and $t\in \left[-10,10\right]$ , which is shown in Figure 4. From the figures, we know that the amplitude of the traveling wave solutions changes with time. All figures are smooth and no singularity in the given time and given field. These solutions help us find the peak and the deep location in physics system.

Figure 1. Simulation ${u}_{1}\left(x,t\right)$ of Equation (7) with $\alpha =-1$ , $\beta =\text{2}$ , $\gamma =0.\text{4}$ , $\kappa =\text{1}$. The range of x and t are $x\in \left[-1,1\right]$ and $t\in \left[-1,1\right]$. The valley appears in the middle.

Figure 2. Simulation ${u}_{2}\left(x,t\right)$ for Equation (8), with $\alpha =-1$ , $\beta =\text{2}$ , $\gamma =0.\text{4}$ , $\kappa =\text{1}$. The range of x and t are $x\in \left[-1,1\right]$ and $t\in \left[-1,1\right]$. The peak appears in the middle.

Figure 3. Simulation ${u}_{5}\left(x,t\right)$ of Equation (11) with $\alpha =0.01$ , $\beta =-8$ , $\gamma =3$ , $\kappa =\text{1}$. The range of x and t are $x\in \left[-10,10\right]$ and $t\in \left[-10,10\right]$. The jump appears in the middle.

Figure 4. Simulation ${u}_{6}\left(x,t\right)$ for Equation (12), with $\alpha =0.01$ , $\beta =-5$ , $\gamma =3$ , $\kappa =\text{1}$. The range of x and t are $x\in \left[-10,10\right]$ and $t\in \left[-10,10\right]$. The jump appears in the middle.

4. Summary

Some new analytical solutions of the higher-order Korteweg-de Vries Equation (1) are obtained by successfully employing tanh-function method in this paper, which can be employed to discuss some interest physical phenomena, such as two-layer fluid, steady-state solitary waves in a fluid, three-layer fluid with a constant buoyancy frequency in an each layer. This tanh-function method is based on a previous work [14] [15] [16] [17] [18] . It is very interest to find these novel analytical evolution solutions, which is useful in establishing the connection between the longstanding rheological mechanics energy conservation problem and two-layer fluid, steady-state solitary waves in a fluid.

References

[1] Bhrawy, A., Biswas, A., Javidi, M., Ma, W., Pinar, Z. and Yildirim, A. (2013) New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup-Kuperschmidt Equations. Results in Mathematics, 63, 675-686.

https://doi.org/10.1007/s00025-011-0225-7

[2] Trogdon, T., Olver, S. and Deconinck, B. (2012) Numerical Inverse Scattering for the Korteweg-de Vries and Modified Korteweg-de Vries Equations. Physica D: Nonlinear Phenomena, 241, 1003-1025.

https://doi.org/10.1016/j.physd.2012.02.016

[3] Bozhkov, Y., Dimas, S. and Ibragimov, N. (2013) Conservation Laws for a Coupled Variable-Coefficient Modified Korteweg-de Vries System in a Two-Layer Fluid Model. Communications in Nonlinear Science and Numerical Simulation, 18, 1127-1135.

https://doi.org/10.1016/j.cnsns.2012.09.015

[4] Salkuyeh, D. and Bastani, M. (2013) Solution of the Complex Modified Korteweg-de Vries Equation by the Projected Differential Transform Method. Applied Mathematics and Computation, 219, 5105-5112.

https://doi.org/10.1016/j.amc.2012.11.062

[5] Kakutani, T. and Yamasaki, N. (1978) Solitary Waves in a Two-Layer Fluid. Journal of the Physical Society of Japan, 45, 674-679.

https://doi.org/10.1143/JPSJ.45.674

[6] Gear, A. and Grimshaw, R. (1983) A Second-Order Theory for Solitary Waves in Shallow Fluids. Physics of Fluids, 26, 14-29.

https://doi.org/10.1063/1.863994

[7] Grimshaw, R. (1997) Internal Solitary Waves. In: Advances in Coastal and Ocean Engineering, World Scientific Publishing Company, Singapore, Vol. 3, 1-30.

https://doi.org/10.1142/9789812797568_0001

[8] Talipova, T., Pelinovsky, E., Lamb, K., et al. (1999) Cubic Effects at the Intense Internal Wave Propagation. Doklady Earth Sciences, 365, 241-244.

[9] Lax, P.D. (1976) Almost Periodic Solutions of the KdV Equation. SIAM (Society for Industrial and Applied Mathematics) Review, 18, 351-375.

https://doi.org/10.1137/1018074

[10] Miura, R.M. (1976) The Korteweg-de Vries Equation: A Survey of Results. SIAM (Society for Industrial and Applied Mathematics) Review, 18, 412-479.

https://doi.org/10.1137/1018076

[11] Lee, C. and Beardsley, R. (1974) The Generation of Long Nonlinear Internal Waves in a Weakly Stratified Shear Flow. Journal of Geophysical Research, 79, 453-462.

https://doi.org/10.1029/JC079i003p00453

[12] Morris, H.C. (1977) Soliton Solutions and the Higher Order Korteweg-de Vries Equations. Journal of Mathematical Physics, 18, 530-532.

https://doi.org/10.1063/1.523297

[13] Yoshimasa, M. (1980) Bilinearization of Nonlinear Evolution Equations. II. Higher-Order Modified Korteweg-de Vries Equations. Journal of the Physical Society of Japan, 49, 787.

https://doi.org/10.1143/JPSJ.49.787

[14] Kudryashov, N.A. (2005) Simplest Equation Method to Look for Exact Solutions of Nonlinear Differential Equations. Chaos, Solitons and Fractals, 24, 1217-1231.

https://doi.org/10.1016/j.chaos.2004.09.109

[15] Kudryashov, N.A. and Loguinova, N.B. (2008) Extended Simplest Equation Method for Nonlinear Differential Equations. Applied Mathematics and Computation, 205, 396-402.

https://doi.org/10.1016/j.amc.2008.08.019

[16] Wang, T., Ren, Y. and Zhao, Y. (2006) Exact Solutions of (3+1)-Dimensional Stochastic Burgers Equation. Chaos, Solitons and Fractals, 29, 920-927.

https://doi.org/10.1016/j.chaos.2005.08.056

[17] Ahmet, B. and Adem, C. (2011) The Tanh-Coth Method Combined with the Riccati Equation for Solving Nonlinear Coupled Equation in Mathematical Physics. Journal of King Saud University—Science, 23, 127-132.

https://doi.org/10.1016/j.jksus.2010.06.020

[18] Li, Z. and Liu, Y. (2002) RATH: A Maple Package for Finding Travelling Solitary Wave Solutions to Nonlinear Evolution Equations. Computer Physics Communications, 148, 256-266.

https://doi.org/10.1016/S0010-4655(02)00559-3