×

Exact special solitary solutions with compact support for the nonlinear dispersive \(K\)(\(m\), \(n\)) equations. (English) Zbl 1088.35547

Summary: The nonlinear dispersive \(K(m,n)\) equations, \[ u_{t}-(u^{m})_{x}-(u^{n})_{xxx} = 0 \] which exhibit compactons: solitons with compact support, are studied. New exact solitary solutions with compact support are found. The two special cases, \(K\)(2, 2) and \(K\)(3, 3), are chosen to illustrate the concrete features of the decomposition method in \(K(m, n)\) equations. General formulas for the solutions of \(K(m, n)\) equations are established.

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
PDF BibTeX XML Cite
Full Text: DOI

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 jpn, 32, 1681, (1972)
[3] Wadati, M., The modified kortweg-de Vries equation, J phys soc jpn, 34, 1289-1296, (1973) · Zbl 1334.35299
[4] Wadati, M.; Toda, M., The exact N-soliton solution of kortweg-de Vries equation, J phys soc jpn, 32, 1403-1411, (1972)
[5] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelengths, Phys rev lett, 70, 5, 564-567, (1993) · Zbl 0952.35502
[6] Wazwaz, A.M., Exact specialsolutions with solitary patterns for the nonlinear dispersive K(m,n) equations, Chaos, solitons & fractals, 13, 161-170, (2002) · Zbl 1027.35115
[7] Wazwaz, A.M., New solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n) equations, Chaos, solitons & fractals, 13, 321-330, (2002) · Zbl 1028.35131
[8] Rosenau, P., Nonlinear dispersion and compact structures, Phys rev lett, 73, 13, 1737-1741, (1994) · Zbl 0953.35501
[9] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys lett A, 230, 5/6, 305-318, (1997) · Zbl 1052.35511
[10] Rosenau, P., On a class of nonlinear dispersive-dissipative interactions, Phys D, 230, 5/6, 535-546, (1998) · Zbl 0938.35172
[11] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122
[12] Adomian, G., A review of the decomposition method in applied mathematics, J math anal appl, 135, 501-544, (1988) · Zbl 0671.34053
[13] Adomian, G., Nonlinear stochastic operator equations, (1986), Academic Press San Diego · Zbl 0614.35013
[14] Wazwaz, A.M., Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, solitons & fractals, 12, 1549-1556, (2001) · Zbl 1022.35051
[15] Yonggui Zhu, Exact special solutions with solitary patterns for Boussinesq-like B(m,n) equations with fully nonlinear dispersion, Chaos, solitons & fractals, 22, 213-220, (2004) · Zbl 1062.35125
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.