A nonlinear model describing a short wavelong wave interaction in a viscoelastic medium
Authors:
Paulo Amorim and João Paulo Dias
Journal:
Quart. Appl. Math. 71 (2013), 417432
MSC (2010):
Primary 35M31
Published electronically:
October 1, 2012
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Abstract: In this paper we introduce a system coupling a nonlinear Schrödinger equation with a system of viscoelasticity, modeling the interaction between short and long waves, acting for instance on media such as plasmas or polymers. We prove the existence and uniqueness of local (in time) strong solutions and the existence of global weak solutions for the corresponding Cauchy problem. In particular we extend previous results in [Nohel et. al., Commun. Part. Diff. Eq., 13 (1988)] for the quasilinear system of viscoelasticity. We finish with some numerical computations to illustrate our results.
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 1.
 P. Amorim and M. Figueira, Convergence of semidiscrete approximations of Benney equations, C. R. Acad. Sci. Paris, Ser. I. 347 (2009) 11351140. MR 2566991 (2010i:65139)
 2.
 P. Amorim and M. Figueira, Convergence of numerical schemes for short wave long wave interaction equations, J. Hyperbolic Differ. Equ. 8 (2011), no. 4, 777811. MR 2864548
 3.
 D. J. Benney, A general theory for interactions between short and long waves, Stud. Appl. Math. 56 (1977) 8194. MR 0463715 (57:3657)
 4.
 F. Caetano, On the existence of weak solutions to the Cauchy problem for a class of quasilinear hyperbolic equations with a source term. Rev. Mat. Complut. 17 (2004), no. 1, 147167. MR 2063946 (2005c:35198)
 5.
 G.Q. Chen, and C.M. Dafermos, Global solutions in for a system of conservation laws of viscoelastic materials with memory. J. Partial Differential Equations 10 (1997), no. 4, 369383. MR 1486717 (99b:35131)
 6.
 J. ChristensenDalsgaard, Helioseismology, Rev. Mod. Phys. 74 (2002), 10731129.
 7.
 C. M. Dafermos, Solutions in for a conservation law with memory. Analyse mathématique et applications, 117128, GauthierVillars, Montrouge, 1988. MR 956955 (89j:35083)
 8.
 C. M. Dafermos, Hyperbolic conservation laws with memory. Differential equations (Xanthi, 1987), 157166, Lecture Notes in Pure and Appl. Math., 118, Dekker, New York, 1989. MR 1021711 (90k:35163)
 9.
 C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping. Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids. Z. Angew. Math. Phys. 46 (1995), Special Issue, S294S307. MR 1359325 (96i:35080)
 10.
 J.P. Dias and M. Figueira, A remark on the existence of global BV solutions for a nonlinear hyperbolic wave equation. Quart. Appl. Math. 60 (2002), no. 2, 245250. MR 1900492 (2003c:35118)
 11.
 J.P. Dias and M. Figueira, Existence of weak solutions for a quasilinear version of Benney equations. J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 555563 MR 2339808 (2008m:35331)
 12.
 J.P. Dias, M. Figueira, and H. Frid, Vanishing viscosity with short wavelong wave interactions for systems of conservation laws, Arch. Rational Mech. Anal., 196 (2010) 9811010. MR 2644446 (2011e:35215)
 13.
 J.P. Dias and H. Frid, Short wavelong wave interactions for compressible NavierStokes equations. SIAM J. Math. Anal., 43 (2010) 764787. MR 2784875
 14.
 J.P. Dias, M. Figueira, and F. Oliveira, Existence of local strong solutions for a quasilinear Benney system. C. R. Math. Acad. Sci. Paris 344 (2007), no. 8, 493496. MR 2324484 (2008c:35302)
 15.
 J.P. Dias, M. Figueira, and F. Oliveira, On the Cauchy problem describing an electronphonon interaction. Chin. Ann. Math. Ser. B. 32 (2011), no. 4, 483496. MR 2820202
 16.
 R. J. DiPerna, Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82 (1983), no. 1, 2770. MR 684413 (84k:35091)
 17.
 T. Kato, Quasilinear equations of evolution, with applications to partial differential equations. pp. 2570. Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975. MR 0407477 (53:11252)
 18.
 A.I. Leonov and A.N. Prokunin, Nonlinear phenomena in flows of viscoelastic polymer fluids, Chapman & Hall, London, 1994.
 19.
 R.C. MacCamy, A model for onedimensional nonlinear viscoelasticity, Quart. Appl. Math., 35 (1977), 2133. MR 0478939 (57:18395)
 20.
 J.A. Nohel, R.C. Rogers and A.E. Tzavaras, Weak solutions for a nonlinear system in viscoelasticity. Comm. Partial Differential Equations 13 (1988), no. 1, 97127. MR 914816 (89h:35063)
 21.
 F. Oliveira, Stability of the solitons for the onedimensional ZakharovRubenchik equation. Phys. D 175 (2003), no. 34, 220240. MR 1963861 (2004c:35390)
 22.
 D. Serre and J. Shearer, Convergence with physical viscosity for nonlinear elasticity, unpublished preprint, 1993.
 23.
 L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: HeriotWatt Symposium, vol. IV, pp. 136212, Pitman, Boston (1979). MR 584398 (81m:35014)
 24.
 M. Tsutsumi and S. Hatano, Wellposedness of the Cauchy problem for the long waveshort wave resonance equations, Nonlinear Anal. 22 (1994), no. 2, 155171. MR 1258954 (95b:35204)
 25.
 M. Tsutsumi and S. Hatano, Wellposedness of the Cauchy problem for Benney's first equations of long wave short wave interactions, Funkcial. Ekvac. 37 (1994), no. 2, 289316 MR 1299867 (95k:35196)
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Additional Information
Paulo Amorim
Affiliation:
Centro de Matemática e Aplicações Fundamentais, FCUL, Av. Prof. Gama Pinto 2, 1649003 Lisboa, Portugal
Email:
pamorim@ptmat.fc.ul.pt
João Paulo Dias
Affiliation:
Centro de Matemática e Aplicações Fundamentais, FCUL, Av. Prof. Gama Pinto 2, 1649003 Lisboa, Portugal
Email:
dias@ptmat.fc.ul.pt
DOI:
http://dx.doi.org/10.1090/S0033569X2012012984
PII:
S 0033569X(2012)012984
Received by editor(s):
May 30, 2011
Published electronically:
October 1, 2012
Additional Notes:
The authors were supported by FCT, through Financiamento Base 2008ISFL1209 and the FCT grant PTDC/MAT/110613/2009. The first author was also supported by FCT through a Ciência 2008 fellowship
Article copyright:
© Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.
