Solitonic connections in capillarity theory: A review
Abstract view|25|times PDF download|9|times
Abstract
A review is presented here of research to date on the application of model parameter-dependent constitutive laws for which capillarity systems admit underlying solitonic structure with their characteristic key properties such as invariance under Bäcklund transformations and admittance of Painlevé reduction. The classical Korteweg capillarity system and its extensions are considered. Reductions to the canonical solitonic nonlinear Schrodinger and its resonant nonlinear Schrödinger equation extension containing a de Broglie-Bohm potential are exhibited in turn for certain model constitutive relations. A capillarity analogue of the classical Kármán-Tsien model law of gasdynamics is shown to have a key role in such canonical reductions. A novel geometric link between a Korteweg capillarity system and the classical Da Rios system of hydrodynamics is recorded. Invariance of capillarity systems under multi-parameter Bäcklund transformations is detailed and applied. Gausson and q-gaussion phenomena in certain capillarity systems is described with concomitant classes of exact solutions. A Lagrangian encapsulation of a Korteweg capillarity system is presented whereby reduction is made to the canonical Boussinesq equation.
Document Type: Invited review
Cited as: Rogers, C. Solitonic connections in capillarity theory: A review. Capillarity, 2024, 12(3): 80-88. https://doi.org/10.46690/capi.2024.09.03
Keywords
Full Text:
PDFReferences
Ablowitz, M. J., Clarkson, P. A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge, UK, Cambridge University Press, 1991. Antanovskii, L. K. Microscale theory of surface tension. Physical Review E, 1996, 54: 6285-6290.
Antanovskii, L. K., Rogers, C., Schief, W. K. A note on a capillarity model and the nonlinear Schrodinger equation. ¨ Journal of Physics A: Mathematical and General, 1997, 30: L555-L557.
Bass, L., Nimmo, J. J. C., Rogers, C., et al. Electrical structures of interfaces: A Painlevé II model. Proceedings of the Royal Society A, 2010, 466: 2117-2130.
Bateman, H. The lift and drag functions for an elastic fluid in two-dimensional irrotational flow. Proceedings of the National Academy of Sciences USA, 1938, 24: 246-251.
Bateman, H. The transformation of partial differential equations. Quarterly of Applied Mathematics, 1943, 1: 281-295.
Bialynicki-Birula, I., Mycielski, J. Gaussons: Solutions of the logarithmic Schrödinger equation. Physica Scripta, 1978, 20: 530-544.
Boussinesq, J. Théorie des ondos et des remous qui propagent ´ lelong d’un canal rectangulaire horizontal en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de Mathématiques Pures et Appliquées, 1872, 17: 55-108.
Castell, S. P., Rogers, C. Application of invariant transformations in one-dimensional non-steady gasdynamics. Quarterly of Applied Mathematics, 1974, 32: 241-251.
Cekirge, H. M., Varley, E. Large amplitude waves in bounded media I: Reflexion and transmission of large amplitisds shockless pulses at an interface. Proceedings of the Royal Society A, 1973, 273: 261-313.
Chaplygin, S. A. On gas jets. Scientific Memoirs, Moscow University Mathematical Physics, 1904, 21: 1-121.
Clarkson, P. A., Mansfield, E. L., Milne, A. E. Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system. Philosophical Transactions of the Royal Society A, 1996, 354: 1807-1835.
Clarkson, P. A., Winternitz, P. Symmetric reduction and exact solutions of nonlinear partial differential equations, in The Painlevé Property, edited by R. Conte, Springer New York, New York, pp. 591-660, 1999.
Clements, D. L., Rogers, C. On the theory of stress concentration for shear-strained prismatical bodies with a nonlinear stress-strain law. Mathematika, 1975, 22: 34-42.
Coburn, N. The Kármán-Tsien pressure-volume relation in two-dimensional supersonic flow of compressible fluids. Quarterly of Applied Mathematics, 1945, 3: 106-116.
Cornolti, F., Luccesi, M., Zambon, B. Elliptic Gaussian beam self-focussing in nonlinear media. Optics Communications, 1990, 75: 129-135.
Da Martino, S., Falanga, M., Lauro, G., et al. Kinetic derivation of the hydrodynamic equations for capillary fluids. Physical Review E, 2004, 70: 067301.
Gell-Mann, M., Tsallis, C. Non-Extensive Entropy: Interdisciplinary Applications. Oxford, UK, Oxford University Press, 2004.
Goncharenko, A. M., Logvin, Yu. A., Samson, A. M., et al. Ermakov Hamiltonian systems in nonlinear optics of elliptic Gaussian beams. Physics Letters A, 1991, 160(2): 138-142.
Haar, A. Über adjungierte Variationsprobleme und adjungierte Extremalflachen. Mathematische Annalen, 1928, 100: 481-502.
Hasimoto, H. A soliton on a vortex filament. Journal of Fluid Mechanics, 1972, 51: 477-485.
Hirota, R. Exact solution of the Korteweg-de Vries equation for multiple collision of solitons. Physical Review Letters, 1971, 27: 1192-1194.
Hirota, R., Satsuma, J. Nonlinear evolution equations generated from the Bäcklund transformation for the Boussinesq equation. Progress in Theoretical Physics, 1977, 57: 797-807.
Konopelchenko, B. G. Introduction to Multi dimensional Integrable Equations: The Inverse Spectral Transform in 2+1-Dimensions. New York, USA, Springer New York, 1992.
Konopelchenko, B. G., Rogers, C. On (2+1) dimensional nonlinear systems of Loewner-type. Physics Letters A, 1991, 158: 391-397.
Konopelchenko, B. G., Rogers, C. On generalised Loewner systems: Novel integrable equations in 2+1-dimensions. Journal of Mathematical Physics, 1993, 22: 34-42.
Konopelchenko, B. G., Schief, W. K., Rogers, C. The 2+1-dimensional sine Gordon system: Its auto-Backlund ¨ transformation. Physics Letters A, 1992, 72: 39-48.
Lee, J. H., Pashaev, O. K., Rogers, C., et al. The resonant nonlinear Schrödinger equation in cold plasma physics. Application of Bäcklund-Darboux transformations and superposition principles. Journal of Plasma Physics, 2007, 73(2): 257-272.
Loewner, C. A transformation theory of partial differential equations of gasdynamics. National Advisory Committee on Aeronautics Technical Note, 1950, 2065: 1-56.
Loewner, C. Generation of systems of partial differential equations by composition of infinitesimal Bäcklund transformations. Journal d’Analyse Mathematique, 1952, 2: 219-242.
Movsesian, L. A. On an invariant transformation of equations of one-dimensional unsteady motion of an ideal comprossible fluid. Prikladnaya Matematika: Mekhanika, 1967, 37: 137-141.
Nabelek, P. V., Zakharov, V. E. Solutions to the Kaup-Broer system and its (2+1)-dimensional integrable generalization via the dressing method. Physica D: Nonlinear Phenomena, 2020, 409: 132478.
Neuber, H. Kerbspannungslehre: Theorie der Spannungskonzentration Genaue Berechnung der Festigkeit. Berlin, Germany, Springer-Verlag, 1958. Nimmo, J. J. C. A class of solutions of the Konopelchenko-Rogers equations. Physics Letters A, 1992, 168: 113-119.
Pashaev, O. K., Lee, J. H., Rogers, C. Soliton resonances in a generalized nonlinear Schrödinger equation. Journal of Physics A: Mathematical and Theoretical, 2008, 41: 452001.
Rasin, A. G., Schiff, J. Bäcklund transformations for the Boussinesq equation and merging solitons. Journal of Physics A: Mathematical and Theoretical, 2017, 50: 325202.
Ricca, R. L. Rediscovery of Da Rios equations. Nature, 1991, 352: 561-562.
Rogers, C. Reciprocal relations in non-steady one-dimensional gasdynamics. Zeitschrift für angewandte Mathematik und Physik, 1968, 19: 58-63.
Rogers, C. Invariant transformations in non-steady gasdynamics and magneto-gasdynamics. Zeitschrift für angewandte Mathematik und Physik, 1969, 20: 370-382.
Rogers, C. The construction of invariant transformations in plane rotational gasdynamics. Archives for Rational Mechanics and Analysis, 1972, 47: 36-46.
Rogers, C. A novel Ermakov-Painlevé II system. N+1-dimensional coupled NLS and elastodynamic reductions. Studies in Applied Mathematics, 2014a, 133: 214-231.
Rogers, C. Integrable substructure in a Korteweg capillarity model. A Kármán-Tsien type constitutive relation. Journal of Nonlinear Mathematical Physics, 2014b, 21(1): 74-88.
Rogers, C. Gausson-type representations in nonlinear physics: Ermakov modulation. Physica Scripta, 2014c, 80: 105208.
Rogers, C. The Korteweg capillarity system. Integrable reduction via gauge and reciprocal links. Zeitschrift für Angewandte Mathematik und Mechanik, 2016, 96: 813-823.
Rogers, C. Reciprocal gausson phenomena in a Korteweg capillarity system. Meccanica, 2019, 54: 1515-1523.
Rogers, C. On the Lagrangian version of the Korteweg capillarity system: integrability aspects. Ricerche di Matematica, 2022a, 71: 29-39.
Rogers, C. A Nonlinear Progress to Modern Soliton Theory. Cambridge, UK, Cambridge Scholars Publishing, 2022b.
Rogers, C., An, H. Ermakov-Ray-Reid systems in 2+1- dimensional rotating shallow water theory. Studies in Applied Mathematics, 2010, 125: 275-299.
Rogers, C., An, H. A non-isothermal spinning magneto-gasdynamic cloud system A Hamiltonian Ermakov integrable reduction. Noti di Matematika, 2012, 32(1): 175-191.
Rogers, C., Chow, K. W., Conte, R. On a capillarity model and the Davey-Stewartson I system: Quasi-doubly periodic wave patterns. Il Nuovo Cimento B, 2007a, 122: 105-112.
Rogers, C., Clarkson, P. A. Ermakov-Painlevé II symmetry reduction of a Korteweg capillarity system. Symmetry, Integrability and Geometry: Methods and Applications, 2017, 13: 018.
Rogers, C., Clarkson, P. A. Ermakov-Painlevé II reduction in cold plasma physics. Application of a Bäcklund transformation. Journal of Nonlinear Mathematical Physics, 2018, 25: 247-261.
Rogers, C., Malomed, B. On Madelung systems in nonlinear optics: A reciprocal invariance. Journal of Mathematical Physics, 2018, 59: 051506.
Rogers, C., Malomed, B., An, H. Ermakov-Ray-Reid reductions of variational approximations in nonlinear optics. Studies in Applied Mathematics, 2012, 129: 389-413.
Rogers, C., Malomed, B., Chow, W., et al. Ermakov-Ray-Reid systems in nonlinear optics. Journal of Physics A: Mathematical and Theoretical, 2010, 43: 455214.
Rogers, C., Pashaev, O. On a 2+1-dimensional Whitham-Broer-Kaup system: A resonant NLS connection. Studies in Applied Mathematics, 2011, 127: 141-152.
Rogers, C., Ruggeri, T. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete & Continuous Dynamical Systems, 2014, 19: 2297-2312.
Rogers, C., Schief, W. K. Intrinsic geometry of the NLS equation and its auto-Bäcklund Transformation. Studies in Applied Mathematics, 1998, 101: 267-287.
Rogers, C., Schief, W. K. The resonant nonlinear Schrödinger equation via an integrable capillarity model. Il Nuovo Cimento B, 1999, 114: 1409-1412.
Rogers, C., Schief, W. K. Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.
Rogers, C., Schief, W. K. Integrable structure in the theory of stress concentration in shear-strained nonlinear elastic materials. Studies in Applied Mathematics, 2010a, 25: 39-53.
Rogers, C., Schief, W. K. Bäcklund transformations and superposition principles in nonlinear elastodynamics. Studies in Applied Mathematics, 2010b, 124: 137-149.
Rogers, C., Schief, W. K. On q-Gaussian integrable Hamiltonican reductions in anisentropic magnetogasdynamics. Acta Applicandae Mathematicae, 2014a, 132: 515-525.
Rogers, C. Schief, W. K. The classical Korteweg capillarity system: geometry and invariant transformations. Journal of Physics A: Mathematical and Theoretical, 2014b, 47: 345201.
Rogers, C., Schief, W. K., Chow, K. W. On a novel class of model constitutive laws in nonlinear elasticity. Construction via Loewner theory. Theoretical and Mathematical Physics, 2007b, 152: 1030-1042.
Rogers, C., Shadwick, W. F. Bäcklund Transformations and Their Applications. New York, USA, Academic Press, 1982.
Rogers, C., Yip, L. P., Chow, K. W. A resonant Davey-Stewartson capillarity model system: Solitonic generation. International Journal of Nonlinear Science and Numerical Simulation, 2009, 10(3): 397-405.
Scott, A. C. The application of Bäcklund transformations to physical problems, in Bäcklund Transformations, edited by R. M. Miura, Springer-Verlag, Berlin, pp. 80-105, 1975.
Toda, M. Studies of a nonlinear lattice. Physics Reports, 1975, 8: 1-125.
Tsien, H. S. Two-dimensional subsonic flow of compressible fluids. Journal of the Aeronautical Sciences, 1939, 6: 399- 407.
Von Kármán, T. Compressibility effects in aerodynamics. Journal of the Aeronautical Sciences, 1941, 8: 337-356.
Refbacks
- There are currently no refbacks.
Copyright (c) 2024 The Author(s)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.