- Email: [email protected]
Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation
Ocean Engineering 94 (2015) 111–115
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation Abdul-Majid Wazwaz n Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 23 April 2014 Accepted 30 November 2014
We investigate the logarithmic-Boussinesq equation (log-BE) for Gaussian solitary waves. We also derive Gaussian solitary wave solutions for the logarithmic-regularized Boussinesq equation (log-RBE). The logarithmic nonlinearity for both models will be generalized to obtain generalized Gaussian solutions. We show that both logarithmic models are characterized by their Gaussian solitons. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Logarithmic Boussinesq equation Logarithmic regularized Boussinesq equation Gaussian solitons
1. Introduction In science and engineering applications, various shallow water wave models were derived. These shallow water models are widely used in plasma physics, nonlinear fiber optics, fluid mechanics, oceanography, and atmospheric science. Examples of these nonlinear equations are the KdV equation and its modified form, Boussinesq equation, sine-Gordon equation and many others. In reality, the KdV equation, that models shallow water waves, had appeared in a work on water waves published by Boussinesq (1872). Zabusky and Kruskal (1965) realized that the KdV equation arises as the continuum limit of a one dimensional anharmonic lattice used by Fermi–Pasta–Ulam (FPU) to investigate how energy is distributed among the many possible oscillations in the lattice. Zabusky and Kruskal (1965) simulated the collision of solitary waves in a nonlinear crystal lattice and observed that they retain their shapes and speed after collision (Hereman, 2009). Interacting solitary waves merely experience a phase shift, advancing the faster and retarding the slower (Hereman, 2009). The dynamics of shallow water waves, that are seen in various places like sea and ocean beaches, lakes and rivers, are governed by the Boussinesq Equation (BE) (Biswas, 2001). The Boussinesq equations model large scale atmospheric and oceanic flows that are responsible for cold fronts and the jet stream (Adhikari et al., 2010). Boussinesq equations for surface gravity waves have been shown to be effective tools to simulate wave propagation in coastal and ocean regions.
n
Tel.: þ 1 773 298 3397; fax: þ1 773 779 9061. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.oceaneng.2014.11.024 0029-8018/& 2014 Elsevier Ltd. All rights reserved.
In dimensionless form, the good Boussinesq equation reads utt uxx ðu2 Þxx βuxxxx ¼ 0;
ð1Þ
where β ¼ 7 1. The Boussinesq equation (1) appeared not only in the study of the dynamics of thin inviscid layers with free surface but also in the study of the nonlinear string, the shape-memory alloys, the propagation of waves in elastic rods and in the continuum limit of lattice dynamics or coupled electrical circuits (Hereman, 2009; Bona et al., 2002). This equation also arises in other physical applications such as nonlinear lattice waves, iron sound waves in a plasma, and in vibrations in a nonlinear string. Moreover, it was applied to problems in the percolation of water in porous subsurface strata. It is obvious that Eq. (1) includes a second-order time-derivative term and adds a fourth order dispersion term to certain nonlinearity. The Boussinesq equation is completely integrable and gives multiple soliton solutions and has an infinite law of conservation. The improved or regularized Boussinesq equation (RBE) reads utt uxx ðu2 Þxx βuxxtt ¼ 0;
ð2Þ
that contains uxxtt instead of uxxxx. The regularized Boussinesq equation (2) improves the properties of the dispersion relation, and like the good Boussinesq equation (1), it describes unidirectional waves. However, the good Boussinesq equation (1) is completely integrable, whereas the regularized Boussinesq equation (2) is not completely integrable. In coastal engineering, the Boussinesq equation models the simulation of water waves in shallow seas and harbors (Shokri and Dehghan, 2010). Numerical methods for solving partial differential equations are usually used for numerical solutions of the Boussinesq equation such as the Galerkin and the collocation methods. Shokri and Dehghan (2010) presented a numerical simulation for
112
A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
solving the Boussinesq equation using collocation and approximating solution by radial basis functions. Moreover, they used a third-order time discretization and predictor–corrector scheme and the accuracy was improved by using a Not-a-Knot meshless method. However, Dehghan and Shakeri (2008) applied the homotopy perturbation method to study a generalized Boussinesq equation and its application in modelling of flow in porous media from a numerical point of view. It was shown by Dehghan and Shakeri (2008) that the homotopy perturbation method does not require discretization, linearization, small perturbation and reduce the size of calculations compared to other existing techniques. The problem of analyzing the response of a nonlinear lattice to a localized disturbance arises in many applications (James and Pelinovsky, 2014; Cazaneve, 1983; Chatterjee, 1999) such as the study of stress waves in granular media after an impact, or the response of nonlinear transmission lines to a voltage pulse (Chatterjee, 1999). It is well known that nonlinear transmission lines are used for pulse shaping. Several important dynamical phenomena can be captured by the Fermi–Pasta–Ulam (FPU) model (James and Pelinovsky, 2014; Cazaneve, 1983; Chatterjee, 1999) consisting of a chain of particles coupled by a pairwise interaction potential. James and Pelinovsky (2014) presented a useful study that was conducted on a class of fully nonlinear Ferma–Pasta–Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities. It was found that this class exhibits a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of pre-compression. Two generalized KdV equations were derived as formal asymptotic limits of the FPU lattices. The first generalized KdV equation is the so-called logarithmic-KdV (log-KdV) equation defined by vt þ ðvlnjvjÞx þ uxxx ¼ 0;
ð3Þ
that admits Gaussian solitary wave solutions. The logarithmic KdV equation models solitary waves in anharmonic chains with Hertzian interaction forces (James and Pelinovsky, 2014; Chatterjee, 1999). The second generalized KdV equation with Hölder continuous nonlinearity, derived from the fully nonlinear FPU lattice, is the socalled H-KdV equation (James and Pelinovsky, 2014) vt þ
α
ðv vjvj1=α 1 Þx þ vxxx ¼ 0;
α1
ð4Þ
that admits compacton solutions as a result of the Höldercontinuous nonlinearity vjvj1=α 1 . Solitons with Gaussian tails have many attractive features and can occur in a wide range of physical, engineering and optical applications (Christov et al., 1996; Afshari and Hajimiri, 2005; Ahnert and Pikovsky, 2009). The dynamics of solitons propagating in optical fibers has been a major area of research given its potential applicability in all optical applications (Leblond and Mihalache, 2013). Biswas (2001) used the variational principle to obtain the parameter dynamics of Gaussian and super-Gaussian chirped solitons which propagates through birefringent optical fibers that is governed by the dispersion-managed vector nonlinear Schrodinger's equation. Searching for exact solutions of differential equations is significantly important in mathematical physics, scientific applications and engineering because it offers a rich knowledge on the mechanism of the complicated physical phenomena and dynamical processes modelled by these nonlinear evolution equations. A set of systematic methods has been used in the literature to obtain explicit solutions for nonlinear equations. A variety of powerful methods have been used to study the nonlinear evolution equations, such as the Hirota bilinear method (Hirota, 2004; Hereman and Nuseir, 1997; Esfahani and Farah, 2012; Wazwaz, 2006, 2007, 2008, 2009, 2011, 2012, 2013), the Bäcklund transformation method (Christov et al., 1996; Dehghan and Salehi, 2012; Khalique et al., 2008), Darboux transformation (Nesterenko, 2001; Nguyen and Brogliato, 2014), Pfaffian technique, the inverse scattering method, the Painlevé analysis, the generalized symmetry method and other methods. The inverse scattering method of integrable problems is more general than Hirota's bilinear method which yields special solutions. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. Our analysis of the dynamics of Gaussian solitary wave solutions is based on combining the sense of the logarithmic KdV equation on the good and the regularized Boussinesq equations. We firstly introduce the generalized logarithmic Boussinesq equation (log-BE) in the form vtt þ vxx þðvlog jvjn Þxx þ vxxxx ¼ 0;
n Z 1:
ð5Þ
To our understanding, the log-BE equation is introduced for the first time. In a like manner, we use the same sense to establish the generalized logarithmic regularized Boussinesq equation (log-RBE) in
Fig. 1. (a) The Gaussian solitary wave solution vðx; tÞ for k ¼2, 3, 4, c ¼2, 5 r x r 5, (b) One Gaussian soliton solution vðx; tÞ for k ¼ 2; c ¼ 2; 5 r x; t r 5.
A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
equivalent to
Table 1 Results for the parameters αn and βn, where n ¼ 1; 2; 3; …. n 1 2 3 4 ⋮ n
αn
βn c2
1 2 2 k c2 2 2k c2 1 2þ 6 3k c2 1 2þ 4 4k ⋮ c2 n 2 2þ 2n nk
vðx; tÞ ¼ euðx;tÞ ;
Gaussian solitary wave solution
1 2
4k 1
2
2k 3
2
4k 1 2
k ⋮ n
vðx; tÞ ¼ e c
2
2
that carries (7) to
2
vðx; tÞ ¼ e vðx; tÞ ¼ e
c2 =3k þ 1=6 ð3=4k Þðkx ctÞ2
vðx; tÞ ¼ e
2 c2 =4k
utt þuxxxx þ uxx þ u2t þ u2x þ4ux uxxx þ 3u2x uxx þ 3u2xx þ u4x þ 3u2x uxx
2
ð1=2k Þðkx ctÞ
2
þ nuu2x þ nuuxx þ 2nu2x þ nuxx ;
2
2
2
þ 1=4 ð1=k Þðkx ctÞ
⋮ 2
4k
2
ð10Þ
2
=k 1=2 ð1=4k Þðkx ctÞ2
2 c2 =2k
vðx; tÞ ¼ e c
113
2
2
=nk þ ðn 2Þ=2n ðn=4k Þðkx ctÞ2
n Z 1:
ð11Þ
Notice that the first three terms give the first three terms of the Boussinesq equation. To derive the Gaussian solitary wave solution for the generalized log-BE equation, we select to derive a solution for the first few cases for n ¼ 1, 2, 3, and we then set the generalized solution from the pattern that we will derive. Case 1: For n ¼1. Considering the case of a Gaussian pulse, the solution takes the form uðx; tÞ ¼ α1 β 1 ðkx ctÞ2 ;
ð12Þ
where α1 and β1 are constants that will be determined. Substituting (12) into (11), for n ¼1, and solving the resulting equation we find
α1 ¼
c2 k
2
1 ; 2
1
β1 ¼
2
4k
:
ð13Þ
This in turn gives the Gaussian solitary waves solution vðx; tÞ ¼ e c
2
2
2
=k 1=2 ð1=4k Þðkx ctÞ2
;
ð14Þ
that works for every k. Fig. 1 shows the Gaussian solitary wave solutions (14) for k ¼2, 3, 4 and one soliton for k ¼ c ¼ 2. Case 2: For n ¼2. Proceeding as before, we assume that the solution takes the form uðx; tÞ ¼ α2 β 2 ðkx ctÞ2 ;
Fig. 2. The Gaussian solitary wave solutions vðx; tÞ for n¼ 1, 2, 3, k ¼ 2, c ¼2, 5r x r 5.
where α2 and β2 are constants that will be determined. Substituting (15) into (11), for n¼ 2, and solving the resulting equation we find
α2 ¼
the form vtt þ vxx þ ðvlog jvjn Þxx þ vxxtt ¼ 0;
n Z 1:
ð6Þ
For both regimes, the log-BE equation and the log-RBE equation, an analytical analysis will be presented aiming to determine Gaussian solitary waves for each model. Both equations will be investigated for a variety of values of n, and consequently, generalized solutions will be developed.
ð15Þ
β2 ¼
c2 2
2k 1
;
2
2k
:
ð16Þ
This in turn gives the Gaussian solitary waves solution: vðx; tÞ ¼ e c
2
2
2
=2k 1=2k ðkx ctÞ2
:
ð17Þ
Case 3: For n ¼3. Proceeding as before, we assume that the solution takes the form uðx; tÞ ¼ α3 β 3 ðkx ctÞ2 ;
2. The generalized logarithmic Boussinesq equation In this section we will study the generalized logarithmic Boussinesq equation: vtt þ vxx þ ðvlog jvjn Þxx þ vxxxx ¼ 0;
n Z 1:
ð7Þ
For a Gaussian pulse we choose the form f ðxÞ ¼ e x . To achieve our goal, the Gaussian solitary wave solution for n ¼1 is given by 2
vðx; tÞ ¼ eðα βðkx ctÞ Þ ; 2
ð8Þ
that decays super-exponentially. To proceed in our approach, we first use the transformation uðx; tÞ ¼ lnvðx; tÞ;
ð9Þ
ð18Þ
where α2 and β2 are constants that will be determined. Substituting (18) into (11), for n ¼3, and solving the resulting equation we find
α3 ¼
c2 2
3k
β3 ¼
1 þ ; 6 3 2
4k
:
ð19Þ
This in turn gives the Gaussian solitary waves solution: vðx; tÞ ¼ ec
2
2
2
=3k þ 1=6 ð3=4k Þðkx ctÞ2
:
Table 1 summarizes the results for the parameters where n ¼ 1; 2; 3; ….
ð20Þ
αn and βn,
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A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
Fig. 3. (a) The Gaussian solitary wave solution vðx; tÞ for c¼ 0.5, 0.75, 1, k¼ 1, 5r x r 5. (b) Two single Gaussian solitons vðx; tÞ for k ¼ 1; c ¼ 0:25, and k ¼ 2; c ¼ 1:5, 5r x; t r 5.
Notice that the first three terms give the first three terms of the regularized Boussinesq equation. To derive the Gaussian solitary wave solution for the generalized log-RBE equation, we select to derive a solution for the first few cases for n ¼1, 2, 3, and we then set the generalized solution from the pattern that we will derive. Case 1: For n ¼1. Considering the case of a Gaussian pulse, the solution takes the form
Table 2 Results for the parameters λn and μn, where n ¼ 1; 2; 3; …. n 1 2 3 4 ⋮ n
αn
c2 2
k c2
1 2
2
2k c2
1 6 3k c2 1 2þ 4 4k ⋮ c2 n2 2þ 2n nk
2
þ
βn
Gaussian solitary wave solution
1 4c2 1 2c2 3 4c2 1 c2 ⋮ n 4c2
vðx; tÞ ¼ e c
2
vðx; tÞ ¼ e c
2 =2k2
vðx; tÞ ¼ e c
2
vðx; tÞ ¼ e c
2 =4k2
2
=k 1=2 ð1=4c2 Þðkx ctÞ2
ð1=2c2 Þðkx ctÞ2
2
=3k þ 1=6 ð3=4c2 Þðkx ctÞ2
uðx; tÞ ¼ λ1 μ1 ðkx ctÞ2 ;
þ 1=4 ð1=c2 Þðkx ctÞ2
⋮ vðx; tÞ ¼ e c
2
2
=nk þ ðn 2Þ=2n ðn=4c2 Þðkx ctÞ2
where λ1 and μ1 are constants that will be determined. Substituting (26) into (25), for n ¼1, and solving the resulting equation we find c2
1 ; 2 1 μ1 ¼ 2 : 4c
λ1 ¼ Fig. 2 shows the Gaussian solitary wave solutions for n ¼1, 2, 3.
ð26Þ
2
k
ð27Þ
This in turn gives the Gaussian solitary waves solution 3. The generalized logarithmic regularized Boussinesq equation
vðx; tÞ ¼ e c
In this section we will study the generalized logarithmic regularized Boussinesq equation:
that works for every k. Fig. 3 shows the Gaussian solitary wave solutions for a variety of c values. Case 2: For n ¼2. Proceeding as before, we assume that the solution takes the form
vtt þvxx þ ðvlog jvjn Þxx þ vxxtt ¼ 0;
n Z 1:
ð21Þ x2
. Based For a Gaussian pulse we usually select the form f ðxÞ ¼ e on this, the Gaussian solitary wave solution for (21) takes the form vðx; tÞ ¼ eðλ μðkx ctÞ Þ ; 2
ð22Þ
that, like the log-BE equation, decays super-exponentially. To carry out our analysis, we use the transformation uðx; tÞ ¼ lnvðx; tÞ; equivalent to vðx; tÞ ¼ euðx;tÞ ;
ð24Þ
2
=k 1=2 ð1=4c2 Þðkx ctÞ2
;
uðx; tÞ ¼ λ2 μ2 ðkx ctÞ2 ;
ð28Þ
ð29Þ
where λ2 and μ2 are constants that will be determined. Substituting (29) into (25), for n ¼2, and solving the resulting equation we find
λ2 ¼ ð23Þ
2
c2
; 2 2k 1 μ2 ¼ 2 : 2c
ð30Þ
This in turn gives the Gaussian solitary waves solution: 2
=2k ð1=2c2 Þðkx ctÞ2
that carries (21) to
vðx; tÞ ¼ e c
utt þ uxxtt þuxx þ u2t þu2x þ 2ut uxxt þ u2t uxx þ utt uxx
Case 3: For n ¼3. Proceeding as before, we assume that the solution takes the form
þ 4ut ux uxt þ 2ux uxtt þ2u2xt þ u2t u2x þ utt u2x þ nuu2x þ nuuxx þ 2nu2x þ nuxx ;
n Z 1:
ð25Þ
2
uðx; tÞ ¼ λ2 μ2 ðkx ctÞ2 ;
:
ð31Þ
ð32Þ
A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
where λ2 and μ2 are constants that will be determined. Substituting (32) into (25), for n ¼3, and solving the resulting equation we find c2
1 þ ; 6 3 μ3 ¼ 2 : 4c
λ3 ¼
3k
2
ð33Þ
This in turn gives the Gaussian solitary waves solution: vðx; tÞ ¼ ec
2
2
=3k þ 1=6 ð3=4c2 Þðkx ctÞ2
:
Table 2 summarizes the results for the parameters where n ¼ 1; 2; 3; ….
ð34Þ
λn and μn,
4. Discussion By carefully examining the two tables listed earlier, we list the following conclusions:
λn ¼ α n : n βn ¼ 2 ; 4k n μn ¼ 2 : 4c
ð35Þ
Moreover, for αn and λn, the second term is fixed by 12, whereas the first term changes depending on n. In this paper we examined the Gaussian solitons for the generalized logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation. The fundamental dynamics of both equations are characterized by their Gaussian solitons. The relations between the parameters were also derived. The obtained results will be used for further studies on other logarithmic nonlinear equations. References Adhikari, D., Cao, C., Wu, J., 2010. The 2D Boussinesq equations with vertical viscosity and vertical diffusivity. J. Differ. Equ. 249, 1078–1088. Afshari, E., Hajimiri, A., 2005. Nonlinear transmission lines for pulse shaping in silicon. IEEE J. Solid State Circuits 40, 744–752. Ahnert, K., Pikovsky, A., 2009. Compactons and chaos in strongly nonlinear lattices. Phys. Rev. E 79, 026209. Biswas, A., 2001. Dynamics of Gaussian and super-Gaussian solitons in birefringent optical fibers. Prog. Electromagn. Res. 33, 119–139. Bona, J.L., Chen, M., Saut, J.-C., 2002. Boussinesq equations and other systems for small–amplitude long waves in nonlinear dispersive media. I: derivation and linear theory. J. Nonlinear Sci. 12, 283–318.
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Boussinesq, J., 1872. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108. Cazaneve, T., 1983. Stable solutions of the logarithmic Schrodinger equation. Nonlinear Anal.: Theory Methods Appl. 7 (10), 1110–1127. Chatterjee, A., 1999. Asymptotic solution for solitary waves in a chain of elastic spheres. Phys. Rev. E 59, 5912–5919. Christov, C.I., Maugin, G.A., Velarde, M.G., 1996. Well-posed Boussinesq paradigm with purely spatial higher-order derivatives. Phys. Rev. E 54 (4), 3621–3638. Dehghan, M., Salehi, R., 2012. A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation. Appl. Math. Model. 36, 1939–1956. Dehghan, M., Shakeri, F., 2008. Use of He's homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media. J. Porous Media 11 (8), 765–778. Esfahani, A., Farah, L.G., 2012. Local-posedness for the sixth-order Boussinesq equation. J. Math. Anal. Appl. 385, 230–242. Hereman, W., Nuseir, A., 1997. Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simul. 43, 13–27. Hereman, W., 2009. Shallow water waves and solitary waves. In: Meyers, R.A. (Ed.), Encyclopedia of Complexity and Systems Science. Springer Verlag, Heibelberg, Germany. Hirota, R., 2004. The Direct Method in Soliton Theory. Cambridge University Press, Cambridge. James, G., Pelinovsky, D., 2014. Gaussian solitary waves and compactons in Fermi– Pasta–Ulam lattices with Hertzian potentials. Proc. R. Soc. A 470, 20130462. Khalique, C.M., Mahomed, F.M., Muatjetjeja, B., 2008. Lagrangian formulation of a generalized Lane–Emden equation and double reduction. J. Nonlinear Math. Phys. 15, 152–161. Leblond, H., Mihalache, D., 2013. Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 523, 61–126. Nesterenko, V.F., 2001. Dynamics of Heterogeneous Materials. Springer, Berlin, Germany. Nguyen, N.-S., Brogliato, B., 2014. Multiple impacts in dissipative granular chains. In: Lecture Notes in Applied and Computational Mechanics, vol. 72. Springer, Berlin, Germany. Shokri, A., Dehghan, M., 2010. A Not-a-Knot meshless method using radial basis functions and predictor-corrector scheme to the numerical solution of improved Boussinesq equation. Comput. Phys. Commun. 181m, 1990–2000. Wazwaz, A.M., 2006. New hyperbolic schemes for reliable treatment of Boussinesq equation. Phys. Lett. A 358, 409–413. Wazwaz, A.M., 2007. Multiple-soliton solutions for the Boussinesq equation. Appl. Math. Comput. 192, 479–486. Wazwaz, A.M., 2008. New travelling wave solutions to the Boussinesq and the Klein-Gordon equations. Commun. Numer. Sci. Numer. Simul. 13, 889–901. Wazwaz, A.M., 2009. Partial Differential Equations and Solitary Waves Theorem. Springer and HEP, Berlin. Wazwaz, A.M., 2011. Soliton solutions of the dispersive sine-Gordon and the dispersive sinh-Gordon equations with fourth spatial or spatio-temporal derivatives. Phys. Scr. 84, 065007. Wazwaz, A.M., 2012. Solitons and singular solitons for a variety of Boussinesq-like equations. Ocean Eng. 53, 1–5. Wazwaz, A.M., 2013. Multiple soliton solutions for an integrable couplings of Boussinesq equation. Ocean Eng. 73, 38–40. Zabusky, N.J., Kruskal, M.D., 1965. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation Abdul-Majid Wazwaz n Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 23 April 2014 Accepted 30 November 2014
We investigate the logarithmic-Boussinesq equation (log-BE) for Gaussian solitary waves. We also derive Gaussian solitary wave solutions for the logarithmic-regularized Boussinesq equation (log-RBE). The logarithmic nonlinearity for both models will be generalized to obtain generalized Gaussian solutions. We show that both logarithmic models are characterized by their Gaussian solitons. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Logarithmic Boussinesq equation Logarithmic regularized Boussinesq equation Gaussian solitons
1. Introduction In science and engineering applications, various shallow water wave models were derived. These shallow water models are widely used in plasma physics, nonlinear fiber optics, fluid mechanics, oceanography, and atmospheric science. Examples of these nonlinear equations are the KdV equation and its modified form, Boussinesq equation, sine-Gordon equation and many others. In reality, the KdV equation, that models shallow water waves, had appeared in a work on water waves published by Boussinesq (1872). Zabusky and Kruskal (1965) realized that the KdV equation arises as the continuum limit of a one dimensional anharmonic lattice used by Fermi–Pasta–Ulam (FPU) to investigate how energy is distributed among the many possible oscillations in the lattice. Zabusky and Kruskal (1965) simulated the collision of solitary waves in a nonlinear crystal lattice and observed that they retain their shapes and speed after collision (Hereman, 2009). Interacting solitary waves merely experience a phase shift, advancing the faster and retarding the slower (Hereman, 2009). The dynamics of shallow water waves, that are seen in various places like sea and ocean beaches, lakes and rivers, are governed by the Boussinesq Equation (BE) (Biswas, 2001). The Boussinesq equations model large scale atmospheric and oceanic flows that are responsible for cold fronts and the jet stream (Adhikari et al., 2010). Boussinesq equations for surface gravity waves have been shown to be effective tools to simulate wave propagation in coastal and ocean regions.
n
Tel.: þ 1 773 298 3397; fax: þ1 773 779 9061. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.oceaneng.2014.11.024 0029-8018/& 2014 Elsevier Ltd. All rights reserved.
In dimensionless form, the good Boussinesq equation reads utt uxx ðu2 Þxx βuxxxx ¼ 0;
ð1Þ
where β ¼ 7 1. The Boussinesq equation (1) appeared not only in the study of the dynamics of thin inviscid layers with free surface but also in the study of the nonlinear string, the shape-memory alloys, the propagation of waves in elastic rods and in the continuum limit of lattice dynamics or coupled electrical circuits (Hereman, 2009; Bona et al., 2002). This equation also arises in other physical applications such as nonlinear lattice waves, iron sound waves in a plasma, and in vibrations in a nonlinear string. Moreover, it was applied to problems in the percolation of water in porous subsurface strata. It is obvious that Eq. (1) includes a second-order time-derivative term and adds a fourth order dispersion term to certain nonlinearity. The Boussinesq equation is completely integrable and gives multiple soliton solutions and has an infinite law of conservation. The improved or regularized Boussinesq equation (RBE) reads utt uxx ðu2 Þxx βuxxtt ¼ 0;
ð2Þ
that contains uxxtt instead of uxxxx. The regularized Boussinesq equation (2) improves the properties of the dispersion relation, and like the good Boussinesq equation (1), it describes unidirectional waves. However, the good Boussinesq equation (1) is completely integrable, whereas the regularized Boussinesq equation (2) is not completely integrable. In coastal engineering, the Boussinesq equation models the simulation of water waves in shallow seas and harbors (Shokri and Dehghan, 2010). Numerical methods for solving partial differential equations are usually used for numerical solutions of the Boussinesq equation such as the Galerkin and the collocation methods. Shokri and Dehghan (2010) presented a numerical simulation for
112
A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
solving the Boussinesq equation using collocation and approximating solution by radial basis functions. Moreover, they used a third-order time discretization and predictor–corrector scheme and the accuracy was improved by using a Not-a-Knot meshless method. However, Dehghan and Shakeri (2008) applied the homotopy perturbation method to study a generalized Boussinesq equation and its application in modelling of flow in porous media from a numerical point of view. It was shown by Dehghan and Shakeri (2008) that the homotopy perturbation method does not require discretization, linearization, small perturbation and reduce the size of calculations compared to other existing techniques. The problem of analyzing the response of a nonlinear lattice to a localized disturbance arises in many applications (James and Pelinovsky, 2014; Cazaneve, 1983; Chatterjee, 1999) such as the study of stress waves in granular media after an impact, or the response of nonlinear transmission lines to a voltage pulse (Chatterjee, 1999). It is well known that nonlinear transmission lines are used for pulse shaping. Several important dynamical phenomena can be captured by the Fermi–Pasta–Ulam (FPU) model (James and Pelinovsky, 2014; Cazaneve, 1983; Chatterjee, 1999) consisting of a chain of particles coupled by a pairwise interaction potential. James and Pelinovsky (2014) presented a useful study that was conducted on a class of fully nonlinear Ferma–Pasta–Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities. It was found that this class exhibits a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of pre-compression. Two generalized KdV equations were derived as formal asymptotic limits of the FPU lattices. The first generalized KdV equation is the so-called logarithmic-KdV (log-KdV) equation defined by vt þ ðvlnjvjÞx þ uxxx ¼ 0;
ð3Þ
that admits Gaussian solitary wave solutions. The logarithmic KdV equation models solitary waves in anharmonic chains with Hertzian interaction forces (James and Pelinovsky, 2014; Chatterjee, 1999). The second generalized KdV equation with Hölder continuous nonlinearity, derived from the fully nonlinear FPU lattice, is the socalled H-KdV equation (James and Pelinovsky, 2014) vt þ
α
ðv vjvj1=α 1 Þx þ vxxx ¼ 0;
α1
ð4Þ
that admits compacton solutions as a result of the Höldercontinuous nonlinearity vjvj1=α 1 . Solitons with Gaussian tails have many attractive features and can occur in a wide range of physical, engineering and optical applications (Christov et al., 1996; Afshari and Hajimiri, 2005; Ahnert and Pikovsky, 2009). The dynamics of solitons propagating in optical fibers has been a major area of research given its potential applicability in all optical applications (Leblond and Mihalache, 2013). Biswas (2001) used the variational principle to obtain the parameter dynamics of Gaussian and super-Gaussian chirped solitons which propagates through birefringent optical fibers that is governed by the dispersion-managed vector nonlinear Schrodinger's equation. Searching for exact solutions of differential equations is significantly important in mathematical physics, scientific applications and engineering because it offers a rich knowledge on the mechanism of the complicated physical phenomena and dynamical processes modelled by these nonlinear evolution equations. A set of systematic methods has been used in the literature to obtain explicit solutions for nonlinear equations. A variety of powerful methods have been used to study the nonlinear evolution equations, such as the Hirota bilinear method (Hirota, 2004; Hereman and Nuseir, 1997; Esfahani and Farah, 2012; Wazwaz, 2006, 2007, 2008, 2009, 2011, 2012, 2013), the Bäcklund transformation method (Christov et al., 1996; Dehghan and Salehi, 2012; Khalique et al., 2008), Darboux transformation (Nesterenko, 2001; Nguyen and Brogliato, 2014), Pfaffian technique, the inverse scattering method, the Painlevé analysis, the generalized symmetry method and other methods. The inverse scattering method of integrable problems is more general than Hirota's bilinear method which yields special solutions. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. Our analysis of the dynamics of Gaussian solitary wave solutions is based on combining the sense of the logarithmic KdV equation on the good and the regularized Boussinesq equations. We firstly introduce the generalized logarithmic Boussinesq equation (log-BE) in the form vtt þ vxx þðvlog jvjn Þxx þ vxxxx ¼ 0;
n Z 1:
ð5Þ
To our understanding, the log-BE equation is introduced for the first time. In a like manner, we use the same sense to establish the generalized logarithmic regularized Boussinesq equation (log-RBE) in
Fig. 1. (a) The Gaussian solitary wave solution vðx; tÞ for k ¼2, 3, 4, c ¼2, 5 r x r 5, (b) One Gaussian soliton solution vðx; tÞ for k ¼ 2; c ¼ 2; 5 r x; t r 5.
A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
equivalent to
Table 1 Results for the parameters αn and βn, where n ¼ 1; 2; 3; …. n 1 2 3 4 ⋮ n
αn
βn c2
1 2 2 k c2 2 2k c2 1 2þ 6 3k c2 1 2þ 4 4k ⋮ c2 n 2 2þ 2n nk
vðx; tÞ ¼ euðx;tÞ ;
Gaussian solitary wave solution
1 2
4k 1
2
2k 3
2
4k 1 2
k ⋮ n
vðx; tÞ ¼ e c
2
2
that carries (7) to
2
vðx; tÞ ¼ e vðx; tÞ ¼ e
c2 =3k þ 1=6 ð3=4k Þðkx ctÞ2
vðx; tÞ ¼ e
2 c2 =4k
utt þuxxxx þ uxx þ u2t þ u2x þ4ux uxxx þ 3u2x uxx þ 3u2xx þ u4x þ 3u2x uxx
2
ð1=2k Þðkx ctÞ
2
þ nuu2x þ nuuxx þ 2nu2x þ nuxx ;
2
2
2
þ 1=4 ð1=k Þðkx ctÞ
⋮ 2
4k
2
ð10Þ
2
=k 1=2 ð1=4k Þðkx ctÞ2
2 c2 =2k
vðx; tÞ ¼ e c
113
2
2
=nk þ ðn 2Þ=2n ðn=4k Þðkx ctÞ2
n Z 1:
ð11Þ
Notice that the first three terms give the first three terms of the Boussinesq equation. To derive the Gaussian solitary wave solution for the generalized log-BE equation, we select to derive a solution for the first few cases for n ¼ 1, 2, 3, and we then set the generalized solution from the pattern that we will derive. Case 1: For n ¼1. Considering the case of a Gaussian pulse, the solution takes the form uðx; tÞ ¼ α1 β 1 ðkx ctÞ2 ;
ð12Þ
where α1 and β1 are constants that will be determined. Substituting (12) into (11), for n ¼1, and solving the resulting equation we find
α1 ¼
c2 k
2
1 ; 2
1
β1 ¼
2
4k
:
ð13Þ
This in turn gives the Gaussian solitary waves solution vðx; tÞ ¼ e c
2
2
2
=k 1=2 ð1=4k Þðkx ctÞ2
;
ð14Þ
that works for every k. Fig. 1 shows the Gaussian solitary wave solutions (14) for k ¼2, 3, 4 and one soliton for k ¼ c ¼ 2. Case 2: For n ¼2. Proceeding as before, we assume that the solution takes the form uðx; tÞ ¼ α2 β 2 ðkx ctÞ2 ;
Fig. 2. The Gaussian solitary wave solutions vðx; tÞ for n¼ 1, 2, 3, k ¼ 2, c ¼2, 5r x r 5.
where α2 and β2 are constants that will be determined. Substituting (15) into (11), for n¼ 2, and solving the resulting equation we find
α2 ¼
the form vtt þ vxx þ ðvlog jvjn Þxx þ vxxtt ¼ 0;
n Z 1:
ð6Þ
For both regimes, the log-BE equation and the log-RBE equation, an analytical analysis will be presented aiming to determine Gaussian solitary waves for each model. Both equations will be investigated for a variety of values of n, and consequently, generalized solutions will be developed.
ð15Þ
β2 ¼
c2 2
2k 1
;
2
2k
:
ð16Þ
This in turn gives the Gaussian solitary waves solution: vðx; tÞ ¼ e c
2
2
2
=2k 1=2k ðkx ctÞ2
:
ð17Þ
Case 3: For n ¼3. Proceeding as before, we assume that the solution takes the form uðx; tÞ ¼ α3 β 3 ðkx ctÞ2 ;
2. The generalized logarithmic Boussinesq equation In this section we will study the generalized logarithmic Boussinesq equation: vtt þ vxx þ ðvlog jvjn Þxx þ vxxxx ¼ 0;
n Z 1:
ð7Þ
For a Gaussian pulse we choose the form f ðxÞ ¼ e x . To achieve our goal, the Gaussian solitary wave solution for n ¼1 is given by 2
vðx; tÞ ¼ eðα βðkx ctÞ Þ ; 2
ð8Þ
that decays super-exponentially. To proceed in our approach, we first use the transformation uðx; tÞ ¼ lnvðx; tÞ;
ð9Þ
ð18Þ
where α2 and β2 are constants that will be determined. Substituting (18) into (11), for n ¼3, and solving the resulting equation we find
α3 ¼
c2 2
3k
β3 ¼
1 þ ; 6 3 2
4k
:
ð19Þ
This in turn gives the Gaussian solitary waves solution: vðx; tÞ ¼ ec
2
2
2
=3k þ 1=6 ð3=4k Þðkx ctÞ2
:
Table 1 summarizes the results for the parameters where n ¼ 1; 2; 3; ….
ð20Þ
αn and βn,
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A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
Fig. 3. (a) The Gaussian solitary wave solution vðx; tÞ for c¼ 0.5, 0.75, 1, k¼ 1, 5r x r 5. (b) Two single Gaussian solitons vðx; tÞ for k ¼ 1; c ¼ 0:25, and k ¼ 2; c ¼ 1:5, 5r x; t r 5.
Notice that the first three terms give the first three terms of the regularized Boussinesq equation. To derive the Gaussian solitary wave solution for the generalized log-RBE equation, we select to derive a solution for the first few cases for n ¼1, 2, 3, and we then set the generalized solution from the pattern that we will derive. Case 1: For n ¼1. Considering the case of a Gaussian pulse, the solution takes the form
Table 2 Results for the parameters λn and μn, where n ¼ 1; 2; 3; …. n 1 2 3 4 ⋮ n
αn
c2 2
k c2
1 2
2
2k c2
1 6 3k c2 1 2þ 4 4k ⋮ c2 n2 2þ 2n nk
2
þ
βn
Gaussian solitary wave solution
1 4c2 1 2c2 3 4c2 1 c2 ⋮ n 4c2
vðx; tÞ ¼ e c
2
vðx; tÞ ¼ e c
2 =2k2
vðx; tÞ ¼ e c
2
vðx; tÞ ¼ e c
2 =4k2
2
=k 1=2 ð1=4c2 Þðkx ctÞ2
ð1=2c2 Þðkx ctÞ2
2
=3k þ 1=6 ð3=4c2 Þðkx ctÞ2
uðx; tÞ ¼ λ1 μ1 ðkx ctÞ2 ;
þ 1=4 ð1=c2 Þðkx ctÞ2
⋮ vðx; tÞ ¼ e c
2
2
=nk þ ðn 2Þ=2n ðn=4c2 Þðkx ctÞ2
where λ1 and μ1 are constants that will be determined. Substituting (26) into (25), for n ¼1, and solving the resulting equation we find c2
1 ; 2 1 μ1 ¼ 2 : 4c
λ1 ¼ Fig. 2 shows the Gaussian solitary wave solutions for n ¼1, 2, 3.
ð26Þ
2
k
ð27Þ
This in turn gives the Gaussian solitary waves solution 3. The generalized logarithmic regularized Boussinesq equation
vðx; tÞ ¼ e c
In this section we will study the generalized logarithmic regularized Boussinesq equation:
that works for every k. Fig. 3 shows the Gaussian solitary wave solutions for a variety of c values. Case 2: For n ¼2. Proceeding as before, we assume that the solution takes the form
vtt þvxx þ ðvlog jvjn Þxx þ vxxtt ¼ 0;
n Z 1:
ð21Þ x2
. Based For a Gaussian pulse we usually select the form f ðxÞ ¼ e on this, the Gaussian solitary wave solution for (21) takes the form vðx; tÞ ¼ eðλ μðkx ctÞ Þ ; 2
ð22Þ
that, like the log-BE equation, decays super-exponentially. To carry out our analysis, we use the transformation uðx; tÞ ¼ lnvðx; tÞ; equivalent to vðx; tÞ ¼ euðx;tÞ ;
ð24Þ
2
=k 1=2 ð1=4c2 Þðkx ctÞ2
;
uðx; tÞ ¼ λ2 μ2 ðkx ctÞ2 ;
ð28Þ
ð29Þ
where λ2 and μ2 are constants that will be determined. Substituting (29) into (25), for n ¼2, and solving the resulting equation we find
λ2 ¼ ð23Þ
2
c2
; 2 2k 1 μ2 ¼ 2 : 2c
ð30Þ
This in turn gives the Gaussian solitary waves solution: 2
=2k ð1=2c2 Þðkx ctÞ2
that carries (21) to
vðx; tÞ ¼ e c
utt þ uxxtt þuxx þ u2t þu2x þ 2ut uxxt þ u2t uxx þ utt uxx
Case 3: For n ¼3. Proceeding as before, we assume that the solution takes the form
þ 4ut ux uxt þ 2ux uxtt þ2u2xt þ u2t u2x þ utt u2x þ nuu2x þ nuuxx þ 2nu2x þ nuxx ;
n Z 1:
ð25Þ
2
uðx; tÞ ¼ λ2 μ2 ðkx ctÞ2 ;
:
ð31Þ
ð32Þ
A.-M. Wazwaz / Ocean Engineering 94 (2015) 111–115
where λ2 and μ2 are constants that will be determined. Substituting (32) into (25), for n ¼3, and solving the resulting equation we find c2
1 þ ; 6 3 μ3 ¼ 2 : 4c
λ3 ¼
3k
2
ð33Þ
This in turn gives the Gaussian solitary waves solution: vðx; tÞ ¼ ec
2
2
=3k þ 1=6 ð3=4c2 Þðkx ctÞ2
:
Table 2 summarizes the results for the parameters where n ¼ 1; 2; 3; ….
ð34Þ
λn and μn,
4. Discussion By carefully examining the two tables listed earlier, we list the following conclusions:
λn ¼ α n : n βn ¼ 2 ; 4k n μn ¼ 2 : 4c
ð35Þ
Moreover, for αn and λn, the second term is fixed by 12, whereas the first term changes depending on n. In this paper we examined the Gaussian solitons for the generalized logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation. The fundamental dynamics of both equations are characterized by their Gaussian solitons. The relations between the parameters were also derived. The obtained results will be used for further studies on other logarithmic nonlinear equations. References Adhikari, D., Cao, C., Wu, J., 2010. The 2D Boussinesq equations with vertical viscosity and vertical diffusivity. J. Differ. Equ. 249, 1078–1088. Afshari, E., Hajimiri, A., 2005. Nonlinear transmission lines for pulse shaping in silicon. IEEE J. Solid State Circuits 40, 744–752. Ahnert, K., Pikovsky, A., 2009. Compactons and chaos in strongly nonlinear lattices. Phys. Rev. E 79, 026209. Biswas, A., 2001. Dynamics of Gaussian and super-Gaussian solitons in birefringent optical fibers. Prog. Electromagn. Res. 33, 119–139. Bona, J.L., Chen, M., Saut, J.-C., 2002. Boussinesq equations and other systems for small–amplitude long waves in nonlinear dispersive media. I: derivation and linear theory. J. Nonlinear Sci. 12, 283–318.
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