Indian J Phys (July 2014) 88(7):751–755 DOI 10.1007/s12648-014-0466-x
ORIGINAL PAPER
Domain walls to Boussinesq-type equations in (2 + 1)-dimensions H Triki1, A H Kara2* and A Biswas3,4 1
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria 2
School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa
3
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA
4
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia Received: 26 November 2013 / Accepted: 20 February 2014 / Published online: 14 March 2014
Abstract: In this paper, two models with fourth-order dispersion in 2 ? 1 dimensions are investigated. Based on Ansatz method, exact domain wall solutions are derived. Parametric conditions for the existence of the domain wall solutions are given. Lie symmetry analysis also retrieves conserved densities of governing nonlinear evolution equations. Keywords:
Domain walls; Boussinesq type models; Conservation laws
PACS Nos.: 02.30.Jr; 02.30.Ik
1. Introduction Envelope solitons are stable nonlinear wave packets that preserve their shape when propagating in a nonlinear dispersive medium [1] . Their existence is a result of a perfect balance between nonlinearity and dispersion effects under specific conditions. Interest in solutions and integrability of nonlinear partial differential equations (PDEs) has grown steadily in recent years. One of great importance is soliton-type solutions due to their extensive applications in many physics areas such as nonlinear optics [2–5], plasmas [4–8], fluid mechanics [2], condensed matter [3], electro magnetics [3] and many more. Various effective methods have been presented to solve nonlinear problems with constant coefficients and with time-dependent coefficients. Hirota bilinear method [9], Ba¨cklund transformation method [9, 10], subsidiary ordinary differential equation method (sub-ODE) [9], coupled amplitude-phase formulation [9, 11–13], sinecosine method [9] and the solitary wave ansatz method [9] are examples of methods that have been used. These may also be found in the references listed in [14–23]. Note that knowledge of exact solutions of nonlinear PDEs is important from many points of view, e.g., for a better
*Corresponding author, E-mail:
[email protected]
understanding of physical phenomena and calculating certain important physical quantities analytically as well as serving as diagnostics for simulations [18, 19]. In this work we use solitary wave ansatz method for analytic study for two variants of the Boussinesq-type equation with constant coefficients. By virtue of an ansatz, exact domain wall solutions are derived. These domain walls are two-dimensional generalization of a kink wave. In 1-D, kink waves are often informally referred to as shock waves. However in (2 ? 1)-dimensions, an appropriate terminology is domain wall and not kink waves or shock waves [3]. Domain wall solutions of KdV equations are obtained in [24, 25]. The obtained results show that solitary wave ansatz method provides a powerful mathematical tool for solving generalized nonlinear wave equations. Finally, a list of nontrivial conservation laws are presented. We allude the readers to possible alternate approaches to this and other equations in this broad area involving domain walls, viz., the method of paraxial ray approximation [26] and other approaches see [25–32].
2. Theory and method Special attention is paid to domain wall solutions to two variants of Boussinesq-type equation with constant coefficients given by [3] Ó 2014 IACS
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utt k2 ðuxx þ uyy Þ þ aðuxxxx þ uyyyy Þ þ b1 ux utx þ b2 uy uty þ cut ðuxx þ uyy Þ ¼ 0;
H Triki et al.
ð1Þ
where the first term is temporal evolution, k2 is second order dispersion, a is 4th order dispersion and coefficients b1 , b2 and c take account of nonlinear dispersions and utt þ a1 uxxxx þ uyyyy þ a2 uxxyy þ b1 u2n ux x þ b2 u2n uy y ¼ 0 ð2Þ where the first term is temporal evolution, a1 and a2 represent fourth order dispersions and b1 and b2 represent nonlinear dispersions. In general, a; a1 ; a2 ; b1 ; b2 and c are arbitrary nonzero constants and uðx; tÞ is a sufficiently often differentiable function. Eqs. (1) and (2) combine two dispersive terms uxxxx and uyyyy in x-direction and y-direction, respectively. Notably, the domain wall of Eq. (1) is usually supported by a balance between nonlinearity, dispersion and dissipation.
ðp 1Þtanhp2 s 2p tanhp s pA v2 k2 B21 þ B22 þ ðp þ 1Þtanhpþ2 s þ apA B41 þ B42 ðp 1Þðp 2Þð p 3Þtanhp4 s þ ðp þ 1Þðp þ 2Þð p þ 3Þ tanhpþ4 s n o 2 p2 þ ðp 2Þ2 f p 1gtanhp2 s n o 2 p2 þ ðp þ 2Þ2 f p þ 1g tanhpþ2 s n o o þ 4p3 þ ðp 1Þ2 ðp 2Þ þ ðp þ 1Þ2 ðp þ 2Þ tanhp s þ p2 A2 v b1 B21 þ b2 B22 ð3p 1Þ tanh2p1 s ð3p þ 1Þtanh2pþ1 s þ ðp þ 1Þtanh2pþ3 s ðp 1Þtanh2p3 s þ cp2 A2 v B21 þ B22 ð3p 1Þ tanh2p1 s þ ðp þ 1Þtanh2pþ3 s ðp 1Þtanh2p3 s ð3p þ 1Þtanh2pþ1 s ¼ 0 ð7Þ From Eq. (15), equating exponents p þ 4 and 2p þ 3 gives p þ 4 ¼ 2p þ 3
ð8Þ
so that p¼1
3. Result and discussion 3.1. Variant I We are interested in domain wall solutions for Eq. (1) utt k2 uxx þ uyy þ a uxxxx þ uyyyy þ b1 ux utx þ b2 uy uty þ cut uxx þ uyy ¼ 0 ð3Þ where uðx; y; tÞ represents wave profile, depending on space coordinates x and y, and the time variable t. The subscripts x; y and t denote partial derivatives with respect to these variables, and k; a, b1 , b2 and c are constant coefficients. In order to look for domain wall solution to Eq. (3), the starting assumption is uðx; y; tÞ ¼ A tanhp s
ð4Þ
where s ¼ B1 x þ B2 y vt
ð5Þ
and p[0
ð6Þ
for domain walls to exist. Here A, B1 and B2 are free parameters, while v represents the speed of domain wall. The exponent p is unknown at this point and is determined later. Substituting Eq. (4) into Eq. (3) yields
ð9Þ
It needs to be noted that the same value of p is yielded when the exponents pairs: p þ 2 and 2p þ 1; p and 2p 1; p 2 and 2p 3, respectively, are equated with each other. Thus, the linearly independent functions are tanhpþj s, where j ¼ 0; 2; 4. So, from Eq. (15), each of the coefficients of these linearly independent functions must be zero. Setting their respective coefficients to zero yields the following parametric equations: pA v2 k2 B21 þ B22 ðp 1Þ o n 2apA B41 þ B42 p2 þ ðp 2Þ2 ðp 1Þ p2 A2 v b1 B21 þ b2 B22 ðp 1Þ cp2 A2 v B21 þ B22 ðp 1Þ ¼ 0 ð10Þ 2p2 A v2 k2 B21 þ B22 þ apA B41 þ B42 n o 4p3 þ ðp 1Þ2 ðp 2Þ þ ðp þ 1Þ2 ðp þ 2Þ þ p2 A2 v b1 B21 þ b2 B22 ð3p 1Þ þ cp2 A2 v B21 þ B22 ð3p 1Þ ¼ 0:
ð11Þ
pA v2 k2 B21 þ B22 ðp þ 1Þ 2apA B41 þ B42 n o p2 þ ðp þ 2Þ2 f p þ 1g p2 A2 v b1 B21 þ b2 B22 ð3p þ 1Þ cp2 A2 v B21 þ B22 ð3p þ 1Þ ¼ 0 ð12Þ
Domain walls to Boussinesq-type equations
753
apA B41 þ B42 ðp þ 1Þðp þ 2Þð p þ 3Þ þ p2 A2 v b1 B21 þ b2 B22 ðp þ 1Þ þ cp2 A2 v B21 þ B22 ðp þ 1Þ ¼ 0
ð13Þ
apA B41 þ B42 ðp 1Þðp 2Þð p 3Þ ¼ 0
ð14Þ
To solve, we have consider the case p 1 ¼ 0: This yields p¼1 for which we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ k2 B21 þ B22 4a B41 þ B42
ð15Þ
ð16Þ
ð17Þ which forces the constraint relation ð18Þ
Thus, finally, domain wall solution to wave equation Eq. (3) is given by uðx; y; tÞ ¼ A tanhð B1 x þ B2 y vtÞ
ð19Þ
where velocity and free parameter are given above. It may be noted that this solution exists provided that the constraint relation is satisfied:
3.2. Variant II We investigate a family of a generalized wave equation given by Eq. (2) utt þ a1 uxxxx þ uyyyy þ a2 uxxyy þ b1 u2n ux x þb2 u2n uy y ¼ 0 ð20Þ where a1 ; a2 ; b1 and b2 are non-zero parameters and uðx; y; tÞ is real-valued function of x; y and t. The parameter n is a positive integer. The starting hypothesis for domain wall is same as Eq. (4). We have obtained pv2 A ðp 1Þ tanhp2 s 2p tanhp s þ ðp þ 1Þ tanhpþ2 s 4 þ pA a1 B1 þ B42 þ a2 B21 B22 ðp 1Þðp 2Þð p 3Þ tanhp4 s þ ðp þ 1Þðp þ 2Þð p þ 3Þ tanhpþ4 s n o 2 p2 þ ðp 2Þ2 f p 1g tanhp2 s
2pð2n þ 1Þ tanhpð2nþ1Þ s
o þ½ pð2n þ 1Þ þ 1 tanhpð2nþ1Þþ2 s ¼ 0
ð21Þ
By equating exponents p þ 4 and pð2n þ 1Þ þ 2, we get p þ 4 ¼ pð2n þ 1Þ þ 2
12a B41 þ B42 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ b1 B21 þ b2 B22 þ c B21 þ B22 k2 B21 þ B22 4a B41 þ B42
k2 B21 þ B22 4a B41 þ B42 [ 0
n o 2 p2 þ ðp þ 2Þ2 f p þ 1g tanhpþ2 s n o þ 4p3 þ ðp 1Þ2 ðp 2Þ þ ðp þ 1Þ2 ðp þ 2Þ tanhP sg n þ pA2nþ1 b1 B21 þ b2 B22 ½ pð2n þ 1Þ 1 tanhpð2nþ1Þ2 s
ð22Þ
so that p¼
1 n
ð23Þ
which is also obtained by equating the exponent pairs: p þ 2 and pð2n þ 1Þ; p and pð2n þ 1Þ 2. Thus, linearly independent functions are tanhpþj s where j ¼ 0; 2; 4. So, each of the coefficients of these linearly independent functions must be zero. Now, setting the coefficient of tanhp2 s to zero yields p¼1
ð24Þ
and this value of p is obtained on setting the coefficient of the stand-alone linearly independent function tanhp4 s to zero. Consequently, n¼1
ð25Þ
This shows that for generalized wave equation, domain walls exist only for n ¼ 1. Setting the coefficients of the remaining linearly independent functions to zero gives pv2 Aðp þ 1Þ 2pA a1 B41 þ B42 þ a2 B21 B22 n o p2 þ ðp þ 2Þ2 f p þ 1g 2p2 A2nþ1 b1 B21 þ b2 B22 ð2n þ 1Þ ¼ 0 ð26Þ 4 pA a1 B1 þ B42 þ a2 B21 B22 ðp þ 1Þðp þ 2Þð p þ 3Þ ð27Þ þpA2nþ1 b1 B21 þ b2 B22 ½ pð2n þ 1Þ þ 1 ¼ 0 2p2 v2 A þ pA2nþ1 b1 B21 þ b2 B22 ½ pð2n þ 1Þ 1 þ pA a1 B41 þ B42 þ a2 B21 B22 n o 4p3 þ ðp 1Þ2 ðp 2Þ þ ðp þ 1Þ2 ðp þ 2Þ ¼ 0: ð28Þ Further substitution of values p ¼ 1 and n ¼ 1 into the above system gives 1 4 6 a1 B1 þ B42 þ a2 B21 B22 2 ð29Þ A¼ b1 B21 þ b2 B22
754
H Triki et al.
which shows that domain walls exist for 4 a1 B1 þ B42 þ a2 B21 B22 b1 B21 þ b2 B22 \0
4.1. Variant I ð30Þ
and v¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a1 B41 þ B42 þ a2 B21 B22
so that domain walls exist for a1 B41 þ B42 þ a2 B21 B22 [ 0
ð31Þ
Q¼1 ¼ k2 þ ut ux þ auxxx ; ¼ k2 þ ut uy þ auyyy ; 1 2ut þ u2y þ u2x T1t ¼ 2
(i)
ð32Þ
T1x T1y
ð33Þ
(ii)
Finally, domain wall to considered model is given by uðx; y; tÞ ¼ A tanhð B1 x þ B2 y vtÞ
Other than an obvious case, nontrivial conservation laws are admitted only for the case b1 ¼ b2 ¼ 2, c ¼ 1. These are listed below with the corresponding multipliers.
where free parameter A and speed are given above. This solution exists provided that the parametric conditions are satisfied and when n ¼ 1: These results given by Eqs. (19) and (33) represent respective domain wall solutions to the variants of Boussinesq equations that are studied in this paper. It can be easily observed that from solutions of Eqs. (19) and (33), if B2 ¼ 0, these would collapse to kink solutions that are alternatively refereed to as shock wave solutions or topological solitons. However, in (2?1)-dimensions, domain wall is the most appropriate terminology that describes these solutions [3].
4. Theory and methods: conservation laws
Q ¼ ux 1 3k2 þ 4ut u2x þ u 3utt þ 4uy uyt 3k2 uyy T3x ¼ 6 þ 2ut uyy þ 3auyyyy þ 2ux uxt Þ 3au2xx þ 6aux uxxx Þ; 1 T3y ¼ ð3auyyy ux þ 3k2 uuxy 2uut uxy 3auyy uxy 6 þ uy 3k2 þ 4ut ux 2uuxt þ 3auxyy 3auuxyyy Þ; 1 T3t ¼ 3ut ux þ 2u2y ux þ 2u3x 3uuxt 2uuy uxy 2uux uxx 6
(iii)
In order to determine conserved densities and fluxes, we resort to invariance and multiplier approach based on well known result that Euler–Lagrange operator annihilates a total divergence [33]. Firstly, if ðT t ; T x ; T y Þ is a conserved vector corresponding to a conservation law, then
(iv)
Dt T t þ Dx T x þ Dy T y ¼ 0
T4x ¼
along the solutions of differential equation, Gðt; x; y; u; ut ; ux ; . . .Þ ¼ 0. Moreover, if there exists a nontrivial differential function Q, called a ‘multiplier’, such that Eu ½QG ¼ 0;
Q ¼ ut
1 T2x ¼ 4u2t ux 3auxt uxx þ 3aux uxxt þ ut 3k2 ux 2uuxt þ 3auxxx 6 þu 2utt ux þ 3k2 uxt 3auxxxt ; 1 T2y ¼ 4u2t uy 3auyt uyy þ 3auy uyyt þ ut 3k2 uy 2uuyt þ 3auyyy 6 þu 2utt uy þ 3k2 uyt 3auyyyt ; 1 T2t ¼ 3u2t þ 2ut u2y þ u2x þ u uyy þ uxx 6 þu 2uy uyt 3k2 uyy þ 3auyyyy þ 2ux uxt 3k2 uxx þ 3auxxxx
Q ¼ uy
1 3auxy uxx þ 3aux uxxy þ uy 3k2 þ 4ut ux þ 3auxxx 6 u 2uyt ux þ 3k2 þ 2ut uxy þ 3auxxxy ; 1 3k2 þ 4ut u2y 3au2yy þ 6auy uyyy T4y ¼ 6 þu 3utt þ 2uy uyt þ 4ux uxt 3k2 uxx þ 2ut uxx þ 3auxxxx ; 1 T4t ¼ 3ut uy þ 2u3y þ uy 2uuyy þ 2u2x u 3uyt þ 2ux uxy 6
then QG is a total divergence, i.e., QG ¼ Dt T t þ Dx T x þ Dy T y ; for some (conserved) vector ðT t ; T x ; T y Þ and Eu is the respective Euler–Lagrange operator. Thus, a knowledge of each multiplier Q leads to a conserved vector determined by, inter alia, a Homotopy operator [33, 34]. If u and its derivatives tend to zero as x and y goes to R 1 infinity, R 1 t the conserved quantities are obtained by 1 1 T dxdy.
4.2. Variant II For this case, we obtain a number of nontrivial conserved vectors; below, we list only the conserved densities T t . Derivative dependent multipliers are only obtainable for for n ¼ 0. (i) Q ¼ 1; T1t ¼ ut (ii) Q ¼ t; T2t ¼ u þ tut (iii) Q ¼ x; T3t ¼ xut
Domain walls to Boussinesq-type equations
(iv) Q ¼ xt; T4t ¼ xu þ txut (v) Q ¼ y; T5t ¼ yut (iv) Q ¼ yt; T6t ¼ yu þ tyut
5. Conclusions In this work have utilised ansatz method for analytic study of two variants of Boussinesq-type equation with constant coefficients. By virtue of an intelligent guess, exact domain wall solutions have been derived. The obtained results show that solitary wave ansatz method provides a useful tool for analysing generalized nonlinear wave equations. Finally, a list of nontrivial conservation laws are presented through which one can compute conserved quantities and can secure an in-depth concept of these models.
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