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The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants. (English) Zbl 1078.35108
Summary: The modified equal width equation \[ u_t+ 3u^2u_x- \alpha u_{xxt}=0 \] and two of its variants are investigated. The strategy here rests mainly on a sine-cosine ansatz and the tanh method. Both schemes work well and reveal exact solutions with distinct physical structures. The obtained solutions include compactons, solitons, solitary patterns, and periodic solutions.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35C05 Solutions to PDEs in closed form
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
Software:
MACSYMA
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References:
[1] Wadati, M., Introduction to solitons, Pramana: J. phys., 57, 5-6, 841-847, (2001)
[2] Wadati, M., The exact solution of the modified kortweg-de Vries equation, J. phys. soc. Japan, 32, 1681-1687, (1972)
[3] Wadati, M., The modified kortweg-de Vries equation, J. phys. soc. Japan, 34, 1289-1296, (1973) · Zbl 1334.35299
[4] Li, B.; Chen, Y.; Zhang, H., Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation, Appl. math. comput., 146, 653-666, (2003) · Zbl 1037.35070
[5] Kivshar, Y.S.; Pelinovsky, D.E., Self-focusing and transverse instabilities of solitary waves, Phys. rep., 331, 117-195, (2000)
[6] Kadomtsev, B.B.; Petviashvili, V.I., Sov. phys. JETP, 39, 285-295, (1974)
[7] Hereman, W.; Takaoka, M., Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA, J. phys. A, 23, 4805-4822, (1990) · Zbl 0719.35085
[8] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelengths, Phys. rev. lett., 70, 5, 564-567, (1993) · Zbl 0952.35502
[9] Dusuel, S.; Michaux, P.; Remoissenet, M., From kinks to compactonlike kinks, Phys. rev. E, 57, 2, 2320-2326, (1998)
[10] Ludu, A.; Draayer, J.P., Patterns on liquid surfaces: cnoidal waves, compactons and scaling, Physica D, 123, 82-91, (1998) · Zbl 0952.76008
[11] Wazwaz, A.M., Partial differential equations: methods and applications, (2002), Balkema Publishers The Netherlands
[12] Wazwaz, A.M., New solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n) equations, Chaos, solitons and fractals, 13, 2, 321-330, (2002) · Zbl 1028.35131
[13] Wazwaz, A.M., Exact specific solutions with solitary patterns for the nonlinear dispersive K(m,n) equations, Chaos, solitons and fractals, 13, 1, 161-170, (2001) · Zbl 1027.35115
[14] Wazwaz, A.M., General compactons solutions for the focusing branch of the nonlinear dispersive K(n,n) equations in higher dimensional spaces, Appl. math. comput., 133, 2-3, 213-227, (2002) · Zbl 1027.35117
[15] Wazwaz, A.M., General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n,n) equations in higher dimensional spaces, Appl. math. comput., 133, 2-3, 229-244, (2002) · Zbl 1027.35118
[16] Wazwaz, A.M., A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. comput. simul., 56, 269-276, (2001) · Zbl 0999.65109
[17] Wazwaz, A.M., Compactons dispersive structures for variants of the K(n,n) and the KP equations, Chaos, solitons and fractals, 13, 5, 1053-1062, (2002) · Zbl 0997.35083
[18] Wazwaz, A.M., Compactons and solitary patterns structures for variants of the KdV and the KP equations, Appl. math. comput., 139, 1, 37-54, (2003) · Zbl 1029.35200
[19] Wazwaz, A.M.; Taha, T., Compact and noncompact structures in a class of nonlinearly dispersive equations, Math. comput. simul., 62, 1-2, 171-189, (2003) · Zbl 1013.35072
[20] Wazwaz, A.M., Existence and construction of compacton solutions, Chaos, solitons and fractals, 19, 3, 463-470, (2004) · Zbl 1068.35124
[21] Wazwaz, A.M., A study on nonlinear dispersive partial differential equations of compact and noncompact solutions, Appl. math. comput., 135, 2-3, 399-409, (2003) · Zbl 1027.35120
[22] Wazwaz, A.M., A construction of compact and noncompact solutions of nonlinear dispersive equations of even order, Appl. math. comput., 135, 2-3, 324-411, (2003) · Zbl 1027.35121
[23] Wazwaz, A.M., Compactons in a class of nonlinear dispersive equations, Math. comput. modell., 37, 3-4, 333-341, (2003) · Zbl 1044.35078
[24] Wazwaz, A.M., Distinct variants of the KdV equation with compact and noncompact structures, Appl. math. comput., 150, 365-377, (2004) · Zbl 1039.35110
[25] Wazwaz, A.M., Variants of the generalized KdV equation with compact and noncompact structures, Comput. math. appl., 47, 583-591, (2004) · Zbl 1062.35120
[26] Wazwaz, A.M., New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos, solitons and fractals, 22, 249-260, (2004) · Zbl 1062.35121
[27] Gardner, L.R.T.; Gardner, G.A.; Ayoub, F.A.; Amein, N.K., Simulations of the EW undular bore, Commun. numer. methods eng., 13, 7, 583-592, (1998) · Zbl 0883.76048
[28] Morrison, P.J.; Meiss, J.D.; Carey, J.R., Scattering of RLW solitary waves, Physica D, 11, 324-336, (1984) · Zbl 0599.76028
[29] Zaki, S.I., Solitary wave interactions for the modified equal width equation, Comput. phys. commun., 126, 219-231, (2000) · Zbl 0951.65098
[30] Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am. J. phys., 60, 7, 650-654, (1992) · Zbl 1219.35246
[31] Malfliet, W.; Hereman, W., The tanh method: II. perturbation technique for conservative systems, Phys. scr., 54, 569-575, (1996) · Zbl 0942.35035
[32] Parkes, E.J.; Duffy, B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. phys. commun., 98, 288-300, (1996) · Zbl 0948.76595
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