Compact structures as true non-linear phenomena

Emilio N. M. Cirillo; Giuseppe Saccomandi; Giulio Sciarra; Emilio N. M. Cirillo; Giuseppe Saccomandi; Giulio Sciarra

doi:10.3934/mine.2019.3.434

Mathematics in Engineering

2019, Volume 1, Issue 3: 434-446. doi: 10.3934/mine.2019.3.434

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Compact structures as true non-linear phenomena

1.
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, I–00161, Roma, Italy
2.
Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy
3.
Institut de Recherche en Génie Civil et Mécanique, Ecole Centrale de Nantes, 1, rue de la Noë 44321 Nantes, France

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.
- phase coexistence,
- interface,
- compact waves,
- Weierstrass construction,
- travelling waves
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

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Abstract

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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