Research article Special Issues

Compact structures as true non-linear phenomena

  • Received: 15 December 2018 Accepted: 29 March 2019 Published: 28 May 2019
  • Traveling waves of permanent form with compact support are possible in several nonlinear partial nonlinear differential equations and this, mainly, along two pathways: A pure nonlinearity stronger than quadratic in the higher order gradient terms describing the mathematical model of the phenomena or a special inhomogeneity in quadratic gradient terms of the model. In the present note we perform a rigorous analysis of the mathematical structure of compactification via a generalization of a classical theorem by Weierstrass. Our mathematical analysis allows to explain in a rigorous and complete way the presence of compact structures in nonlinear partial differential equations 1 + 1 dimensions.

    Citation: Emilio N. M. Cirillo, Giuseppe Saccomandi, Giulio Sciarra. Compact structures as true non-linear phenomena[J]. Mathematics in Engineering, 2019, 1(3): 434-446. doi: 10.3934/mine.2019.3.434

    Related Papers:

  • Traveling waves of permanent form with compact support are possible in several nonlinear partial nonlinear differential equations and this, mainly, along two pathways: A pure nonlinearity stronger than quadratic in the higher order gradient terms describing the mathematical model of the phenomena or a special inhomogeneity in quadratic gradient terms of the model. In the present note we perform a rigorous analysis of the mathematical structure of compactification via a generalization of a classical theorem by Weierstrass. Our mathematical analysis allows to explain in a rigorous and complete way the presence of compact structures in nonlinear partial differential equations 1 + 1 dimensions.


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    [1] Ambrose DM, Simpson G, Wright JD, et al. (2012) Ill-posedness of degenerate dispersive equations. Nonlinearity 25: 2655–2680. doi: 10.1088/0951-7715/25/9/2655
    [2] Bray AJ (1994) Theory of phase-ordering kinetics. Adv Phys 43: 357–459. doi: 10.1080/00018739400101505
    [3] Chafee N, Infante EF (1974) A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl Anal 4: 17–37. doi: 10.1080/00036817408839081
    [4] Cirillo ENM, Ianiro N, Sciarra G (2010) Phase coexistence in consolidating porous media. Phys Rev E 81: 061121. doi: 10.1103/PhysRevE.81.061121
    [5] Cirillo ENM, Ianiro N, Sciarra G (2011) Phase transition in saturated porous-media: Pore-fluid segregation in consolidation. Phys D 240: 1345–1351. doi: 10.1016/j.physd.2011.05.015
    [6] Cirillo ENM, Ianiro N, Sciarra G (2012) Kink localization under asymmetric double-well potentials. Phys Rev E 86: 041111. doi: 10.1103/PhysRevE.86.041111
    [7] Cirillo ENM, Ianiro N, Sciarra G (2013) Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media. J Mech Phys Solids 61: 629–651. doi: 10.1016/j.jmps.2012.08.014
    [8] Cirillo ENM, Ianiro N, Sciarra G (2016) Compacton formation under Allen-Cahn dynamics. Proc R Soc A 472: 20150852. doi: 10.1098/rspa.2015.0852
    [9] Cueto-Felgueroso L, Juanes R (2012) Macroscopic phase-field model of partial wetting: Bubbles in a capillary tube. Phys Rev Lett 108: 144502. doi: 10.1103/PhysRevLett.108.144502
    [10] Destrade M, Gaeta G, Saccomandi G (2007) Weierstrass' criterion and compact solitary waves. Phys Rev E 75: 047601. doi: 10.1103/PhysRevE.75.047601
    [11] Destrade M, Jordan PM, Saccomandi G (2009) Compact travelling waves in viscoelastic solids. EPL 87: 48001. doi: 10.1209/0295-5075/87/48001
    [12] Destrade M, G. Saccomandi G (2006) Solitary and compactlike shear waves in the bulk of solids. Phys Rev E 73: 065604. doi: 10.1103/PhysRevE.73.065604
    [13] Destrade M, Saccomandi G (2008) Nonlinear transverse waves in deformed dispersive solids. Wave Motion 45: 325–336. doi: 10.1016/j.wavemoti.2007.07.002
    [14] Durickovic B, Goriely A, Saccomandi G (2009) Compact waves on planar elastic rods. Int J Non-Linear Mech 44: 538–544. doi: 10.1016/j.ijnonlinmec.2008.10.007
    [15] Dusuel S, Michaux P, Remoissenet M (1998) From kinks to compactonlike kinks. Phys Rev E 57: 2320–2326. doi: 10.1103/PhysRevE.57.2320
    [16] Gaeta G, Gramchev T, Walcher S (2007) Compact solitary waves in linearly elastic chains with non-smooth on-site potential. J Phys A 40: 4493–4509. doi: 10.1088/1751-8113/40/17/007
    [17] John F (1974) Formation of singularities in one–dimensional nonlinear wave propagation. Commu Pure Appl Math 27: 377–405. doi: 10.1002/cpa.3160270307
    [18] Pucci E, Saccomandi G (2010) On a special class of nonlinear viscoelastic solids. Math Mech Solids 15: 803–811. doi: 10.1177/1081286509104540
    [19] Renardy M, Rogers RC (2004) An Introduction to Partial Differential Equations, Springer.
    [20] Rogers C, Saccomandi G, Vergori L (2015) Nonlinear elastodynamics of materials with strong ellipticity condition: Carroll-type solutions. Wave Motion 56: 147–164. doi: 10.1016/j.wavemoti.2015.02.009
    [21] Rosenau P (2005) What is... a compacton? Not Am Math Soc 52: 738–739.
    [22] Rosenau P, Hyman JM (1993) Compactons: Solitons with finite wavelength. Phys Rev Lett 70: 564–567. doi: 10.1103/PhysRevLett.70.564
    [23] Rosenau P, Hyman JM, Stanley M (2007) Multidimensional compactons. Phys Rev Lett 98: 024101. doi: 10.1103/PhysRevLett.98.024101
    [24] Rosenau P, Zilburg A (2018) Compactons. J Phys A 51: 343001. doi: 10.1088/1751-8121/aabff5
    [25] Rubin MB, Rosenau P, Gottlieb O (1995) Continuum model of dispersion caused by an inherent material characteristic length. J Appl Phys 77: 4054–4063. doi: 10.1063/1.359488
    [26] Saccomandi G (2004) Elastic rods, Weierstrass' theory and special travelling waves solutions with compact support. Int J Non-Linear Mech 39: 331–339. doi: 10.1016/S0020-7462(02)00192-0
    [27] Saccomandi G, Sgura I (2006) The relevance of nonlinear stacking interactions in simple models of double-stranded DNA. J R Soc Interface 3: 655–667. doi: 10.1098/rsif.2006.0126
    [28] Saccomandi G, Vitolo R (2014) On the mathematical and geometrical structure of the determining equations for shear waves in nonlinear isotropic incompressible elastodynamics. J Math Phys 55: 081502. doi: 10.1063/1.4891602
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