Analytical study on the balancing principle for the nonlinear Klein–Gordon equation with a fractional power potential

1 Introduction

Nonlinear models play key role in various filed of physics such as field theory, condensed matter physics, and hydrodynamics. (1 + 1)- and (2 + 1)-dimensional models can be seen more than (3 + 1)-dimensional models. Especially, the behavior of solutions of the sine-Gordon (Alfimov et al., 2000) and nonlinear Schrödinger (Richard et al., 2007) equations for spatially two and three-dimensional cases was numerically and analytically studied in detail by Ekomasov and Salimov (2014; 2015).

For the analytical solution of nonlinear models is tried to obtain via one of the computational methods such as the tanh method (Zhou et al., 2003; Ozis and Koroglu, 2008), sine–cosine method (Ma et al., 2010), $\raisebox{1ex}{$G^{\prime }$}\!\left/ \!\raisebox{-1ex}{$G$}\right.$-expansion method (Mirzazadeh et al., 2014; Ozis and Aslan, 2010), the Jacobi elliptic function expansion method (Yong et al., 2009), auxiliary equation method (Sirendaoreji, 2007; Lv et al., 2009; Yomba, 2008; Abdou, 2008; Lim et al., 2001; Pınar and Öziş, 2013a; 2013b), sub-equation method (Zhang, 2009; Lia et al., 2008; Yomba, 2006; 2005) etc.. The mentioned methods depending on the ansatz are chosen and the main problem is how many terms of the finite series i.e. ansatz is considered. Now, the general methodology for the mentioned methods is given as following:

For general, (1 + 1)-dimensional nonlinear model is considered

The analytical solutions of the Eq.(2) is considered as a finite series which is known ansatz

As it is seen that the main problem of these type methodologies is determining $N$ and generally $N$ is determined by a “balancing principle” and there is a classical determination but it cannot be applied all types of nonlinear models. Pınar and Öziş (2015a) proposed different determination for $N$ and it is more general than exist one.

The value of $N$ is always considered as a positive integer and to satisfy this condition, the variable transformations are done (Pınar and Öziş, 2015a). The review of all balancing principles is given by Pınar and Öziş (2015a) and they pointed that there is no unique balancing principle for every type nonlinear models.

Now the novel and generalized balancing principle is introduced in this work. It is suitable for all type of nonlinearities, results as positive integer and also, the power is, in most cases, the least number that meets the final expansion (Eq. (3)). To obtain integer degree ansatz, the novel balancing principle (Eq. (4)) given by modular balance formula

In the following section we exemplify some situations that do not work with the existing balancing principles either in the literature or in their work, Eq. (4) is lower than the anchor in which the power determined by the balancing principle proposed is. Therefore, the computation cost is also reduced via optimum degree of the ansatz.

In this work Lorentz-invariant model is solved via Bernoulli approximation method which has the given procedure and generalized BBM equation is solved via the extended auxiliary equation method.

2 The solution of Lorentz-invariant model

Nonlinear wave equations, especially most known are nonlinear Klein–Gordon equations, are used in several areas of physics and engineering such as hydrodynamics, condensed state physics, and field theory (Scott,2004; Braun et al., 2004; Knight et al., 2013; Dauxois and Peyrard, 2010; Saadatmand and Kurosh,2013; Sirendaoreji, 2007) and a quantified version of the relative energy–momentum relationship. Although (1 + 1) and (2 + 1) dimensional models (Knight et al., 2013; Saadatmand and Kurosh, 2013; Gonzalez et al., 2007; Efremidis et al., 2007; Fokas, 2006; Biswas et al., 2012) are the most studied, these equations can be easily generalized to higher dimensional space, e.g., for spherical symmetry.

Now one of the most important models of physics is a Lorentz-invariant model that has solutions in the form of plane waves and is a modification of the previously considered equations (Ekomasov and Salimov, 2014):

When $m=3,k=2,n=s=13$, Eq. (5) is rewritten

For Eq. (7), reduced form of Eq. (6), with the novel proposed balancing principle (i.e. Eq.(4)) $N\equiv 0\equiv 2(mod2)$ is obtained.

Hence, the solution of the Eq. (7) can be given by$$u\left(\zeta \right)=\sum _{i=0}^{N=2}{a}_i{z}^i\left(\zeta \right)=a_0+a_{1}z\left(\zeta \right)+a_2{z}^2\left(\zeta \right),\zeta =\mu r+ct$$where $z\left(\zeta \right)$is considered as a solution of the variable coefficient Bernoulli differential equation

For $n=2$, as a result of nonlinear algebraic system, the parameters are obtained$$P\left(\zeta \right)=\frac{2\mu }{r\left(c^2-{\mu }^2\right)},Q\left(\zeta \right)=-\frac{exp\left(-\frac{4\zeta \mu }{r\left(c^2-{\mu }^2\right)}\right)\left(2a_2\mu exp\left(\frac{4\zeta \mu }{r\left(c^2-{\mu }^2\right)}\right)+a_{1}r{C}_1\left({\mu }^2-c^2\right)\right)}{r\left(c^2-{\mu }^2\right)a_1},a_2=\frac{a_{1}r{C}_1\left(c^2-{\mu }^2\right)\sqrt{3}}{6exp\left(\frac{4\zeta }{r\left(c^2-{\mu }^2\right)}\right)},$$

Hence the solution of the Bernoulli differential equation is$$z\left(\zeta \right)=2\left({\mu }^2-5\mu +6\right)a_{1}exp\left(\frac{2\zeta }{r\left(c^2-{\mu }^2\right)}\right)/\left(\begin{array}{c}{c}^2{a}_{1}r{C}_{1}exp\left(\frac{2\left(\mu -2\right)\zeta }{r\left(c^2-{\mu }^2\right)}\right)\left(\mu -3+{\mu }^3-3{\mu }^2\right)\\ +2a_1{C}_2\left(6-5\mu +{\mu }^2\right)+4a_{2}exp{\left(\frac{\zeta }{r\left(c^2-{\mu }^2\right)}\right)}^2\left({\mu }^2-4\mu +4\right)\end{array}\right)$$

The solution is given for large times by Fig. 1 and behavior of the amplitude in the center of the pulson for Eq. (6) at small times and large times are seen in Fig. 2.

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Fig. 1

The solution of the Eq.(6) for $a_0=2e^{-r}cos\left(\frac{t\pi }{2}\right),a_1=0.0001t,C_1=2,C_2=1,\mu =1,c=-2$

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Fig. 2

Behavior of the amplitude in the center of the pulson for Eq. (6) at small times and large times, respectively.

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To be sure, also we need to check that the stabilization of the oscillation by Eq.(6)

The plot of the time evolution of the quantity $c\left(t\right)$for Eq. (6) is given by Fig. 3.

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Fig. 3

Time evolution of the quantity $c\left(t\right)$for Eq. (6).

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For the following example, in (Pınar and Ozis, 2013b), ansatz is taken first degree, in (Layeni and Akinola, 2010) ansatz is $2/n$-th degree and in (Wazwaz and Helal, 2005) the degree of ansatz is $\frac{2}{n-1}$-th for the general Benjamin–Bona–Mahony (BBM) equation. In (Wazwaz and Helal, 2005), it is said that degree of ansatz should be integer. This means that when $n=3$, the result is $N=1$ and in the similar manner if $n=2$, $N=2$is obtained. For general equations, appropriate $n$ is chosen so that $N$ is an integer. So, we do not have the general equation solution. By Pınar and Öziş (2015a), ansatz with fractional degree is used as a result new solutions for the general equation is obtained. Hence, to obtain ansatz with integer degree, the novel balancing principle (Eq. (4)) given by modular balance formula is considered.

3 The generalized Benjamin–Bona–Mahony (BBM) equation

The generalized BBM equation (Nickel, 2007; Wang et al., 2014), which is an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude, is considered;

For Eq. (11), we obtain zero-degree ansatz by which the trivial solution of Eq.(11) is hold. In addition, Eq. (11) is an integrable equation. If Eq. (11) is integrated respect to $\zeta$, then we obtain

Using the novel balancing principle (Eq.(4)), $2/n$- degree ansatz for Eq. (12) is obtained. For $2/n$- degree ansatz, we have several cases as following.

We get solution of Eq. (13) using the second degree ansatz and the extended auxiliary equation ${\left(z^{\prime }\left(\zeta \right)\right)}^2=a_2{z}^2\left(\zeta \right)+a_6{z}^6\left(\zeta \right)$, the solution is plotted in Fig. 4.

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Fig. 4

This solution of Case 1 is obtained using second degree ansatz.

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We get solution of Eq. (14) using the first degree ansatz and the extended auxiliary equation ${\left(z^{\prime }\left(\zeta \right)\right)}^2=a_2{z}^2\left(\zeta \right)+a_6{z}^6\left(\zeta \right)$, the solution is plotted in Fig. 5.

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Fig. 5

This solution of Case 2 is obtained using first degree ansatz.

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Now, we compare same $n$ value with different ansatzs. If we take for $n=1$, first and second degree ansatzes with the extended auxiliary equation ${\left(z^{\prime }\left(\zeta \right)\right)}^2=a_2{z}^2\left(\zeta \right)+a_4{z}^4\left(\zeta \right)+a_6{z}^6\left(\zeta \right)$

Here, we have important question, if we take non-integer degree ansatz, what kind of solutions are obtained. For this example, if we take $n=3$, ansatz is $2/3$-degree but it is equivalence to 2 with respect to $\left(mod2\right)$. Using the extended auxiliary equation ${\left(z^{\prime }\left(\zeta \right)\right)}^2=a_6{z}^6\left(\zeta \right)$ and $u\left(\zeta \right)=a_0+a_1{z}^{2/3}\left(\zeta \right)$ ansatz, we obtain

The solutions which are obtained using second degree ansatz and $2/3$-degree ansatz have same behaviors.

As mentioned above, the value of $N$ is always considered as a positive integer and to satisfy this condition, the variable transformations or if the equation is integrable, the equation is integrated considering the integration constant is zero, etc.. Instead of all these reductions, with the novel principle the problem is solved.

4 Conclusion

The common balancing principles (i.e. determines the power $N$ of ansatz (cf. Eq.(4)) usually by balancing the highest order linear term in the equation with the highest order nonlinear term) work only for positive integer values. In this study, a prosperous balancing principle has been proposed that works for the positive integer one and with least power of the ansatz. To verify our objective well-known problems in the literature is given which works parallel to our goal. In addition, the proposed balancing principle make possible for finding new travelling wave solutions or analytical solutions of the nonlinear problems due to introducing a novel ansatz(s) as in the given examples. The future work will focus on symmetries of the Lorentz-invariant model.