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December  2020, 14(6): 1057-1105. doi: 10.3934/ipi.2020055

Interactions of semilinear progressing waves in two or more space dimensions

Department of Mathematics, Purdue University, 150 North University Street, West Lafayette Indiana, 47907, USA

Received  January 2020 Revised  July 2020 Published  August 2020

Fund Project: The author is supported by the Simons Foundation grant #349507, Antônio Sá Barreto

We analyze the behavior of the singularities of solutions of semilinear wave equations after the interaction of three transversal conormal waves. Our results hold for space dimensions two and higher, and for arbitrary $ {{C}^{\infty }} $ nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang. We also indicate possible applications to inverse problems.

Citation: Antônio Sá Barreto. Interactions of semilinear progressing waves in two or more space dimensions. Inverse Problems & Imaging, 2020, 14 (6) : 1057-1105. doi: 10.3934/ipi.2020055
References:
[1]

M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math., 118 (1983), 187-214.  doi: 10.2307/2006959.  Google Scholar

[2]

M. Beals, Vector fields associated with the nonlinear interaction of progressing waves, Indiana Univ. Math. J., 37 (1988), 637-666.  doi: 10.1512/iumj.1988.37.37031.  Google Scholar

[3]

M. Beals, Singularities of conormal radially smooth solutions to nonlinear wave equations, Comm. in P. D. E., 13 (1988), 1355-1382.  doi: 10.1080/03605308808820579.  Google Scholar

[4]

M. Beals, Propagation and interaction of singularities in nonlinear hyperbolic problems, Progress in Nonlinear Differential Equations and their Applications, 3, Birkhäuser Boston, Inc., Boston, MA, 1989. doi: 10.1007/978-1-4612-4554-4.  Google Scholar

[5]

M. Beals, Regularity of nonlinear waves associated with a cusp, Microlocal Analysis and Nonlinear Waves (Minneapolis, MN, 1988–1989), IMA Vol. Math. Appl., 30, Springer, New York, 1991, 9–27. doi: 10.1007/978-1-4613-9136-4_2.  Google Scholar

[6]

J.-M. Bony, Localization et propagation des singularités pour les équations nonlinéaires,, Journées des E.D.P., St. Jean-de-Monts, (1978).  Google Scholar

[7]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux derivées partielles nonlinéaires, Ann. Sci. Ec. Norm. Sup., 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[8]

J.-M. Bony, Interaction des singularités pour les équations aux dérivées partielles nonlinéaires, Sem. Goulaouic-Meyer-Schwartz Exp. 2, (1981/1982).  Google Scholar

[9]

J.-M. Bony, Propagation et interaction des singularités pour les solutions des équations aux dérivées partielles non-linéaires, Proceedings of the International Congress of Mathematicians, 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1133–1147.  Google Scholar

[10]

J-M. Bony, Interaction des singularités pour les équations de Klein-Gordon non linéaires, Goulaouic-Meyer-Schwartz Seminar, Exp. No. 10, École Polytech., Palaiseau, 1984, 28 pp.  Google Scholar

[11]

J-M. Bony, Second microlocalization and propagation of singularities for semilinear hyperbolic equations, Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), Academic Press, Boston, MA, 1986, 11–49.  Google Scholar

[12]

J. Y. Chemin, Interaction de trois ondes dans les équations semi-linéaires strictement hyperboliques d'ordre 2, Communications in Partial Differential Equations, 12:11 (1987), 1203-1225.  doi: 10.1080/03605308708820525.  Google Scholar

[13]

X. Chen, M. Lassas, L. Oksanen and G. Paternain, Detection of Hermitian connections in wave equations with cubic nonlinearity, preprint, arXiv: 1902.05711. Google Scholar

[14]

J-M. Delort, Conormalité des ondes semi-linéaires le long des caustiques, Amer. J. Math., 113 (1991), 593-651.  doi: 10.2307/2374842.  Google Scholar

[15]

J. J. Duistermaat, Fourier integral operators, Series Progress in Mathematics, 130  Google Scholar

[16]

A. Fiorenza, M. R. Formica, T. Roskovec and F. Soudský, Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks, preprint, arXiv: 1812.04281. Google Scholar

[17]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232.  doi: 10.1016/0022-1236(90)90011-9.  Google Scholar

[18] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511721441.  Google Scholar
[19]

L. Holt, Singularities produced in conormal wave interactions, Trans. Amer. Math. Soc., 347 (1995), 289-315.  doi: 10.1090/S0002-9947-1995-1264146-3.  Google Scholar

[20]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume III, Springer Verlag, 1994.  Google Scholar

[21]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume IV, Springer Verlag, 1994.  Google Scholar

[22]

M. Joshi and A. Sá Barreto, The generation of semilinear singularities by a swallowtail caustic, Amer. J. Math., 120 (1998), 529-550.  doi: 10.1353/ajm.1998.0023.  Google Scholar

[23]

Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Inventiones Mathematicae, 212.3 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.  Google Scholar

[24]

G. Lebeau, Équations des ondes semi-linéaires. Ⅱ. Contrôle des singularités et caustiques nonlinéaires, Invent. Math., 95 (1989), 277-323.  doi: 10.1007/BF01393899.  Google Scholar

[25]

M. LassasG. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Comm. Math. Phys., 360 (2018), 555-609.  doi: 10.1007/s00220-018-3135-7.  Google Scholar

[26]

R. B. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.  Google Scholar

[27]

R. B. Melrose, Interaction of progressing waves through a nonlinear potential, Séminaire Équations aux Dérivées Partielles (Polytechnique), (1983-1984), Exp. No. 12, 1–13.  Google Scholar

[28]

R. B. Melrose, Semilinear waves with cusp singularities, Journées "Équations aux derivées partielles" (Saint Jean de Monts, 1987), Exp. No. X, École Polytech., Palaiseau, 1987, 10 pp.  Google Scholar

[29]

R. B. Melrose, Conormality, cusps and nonlinear interaction, Microlocal Analysis and Nonlinear Waves (Minneapolis, MN, 1988–1989), IMA Vol. Math. Appl., 30, Springer, New York, 1991,155–166. doi: 10.1007/978-1-4613-9136-4_11.  Google Scholar

[30]

R. B. Melrose and N. Ritter, Interaction of nonlinear progressing waves for semilinear wave equations, Ann. Math., 121 (1985), 187-213.  doi: 10.2307/1971196.  Google Scholar

[31]

R. B. Melrose and N. Ritter, Interaction of progressing waves for semilinear wave equations.Ⅱ, Ark. Mat., 25 (1987), 91-114.  doi: 10.1007/BF02384437.  Google Scholar

[32]

R. B. Melrose and A. Sá Barreto, Non-linear interaction of a cusp and a plane, Comm. Partial Differential Equations, 20 (1995), 961-1032.  doi: 10.1080/03605309508821121.  Google Scholar

[33]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.  doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[34]

A. Piriou, Calcul symbolique non linéaire pour une onde conormale simple, Ann. Inst. Fourier (Grenoble), 38 (1988), 173-187.  doi: 10.5802/aif.1153.  Google Scholar

[35]

J. Rauch and M. Reed, Singularities produced by the nonlinear interaction of three progressing waves; Examples, Comm. Partial Differential Equations, 7 (1982), 1117-1133.  doi: 10.1080/03605308208820246.  Google Scholar

[36]

J. Rauch and M. Reed, Non-linear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J., 49 (1982), 379-475.   Google Scholar

[37]

J. Rauch and M. Reed, Classical conormal solutions of semilinear systems, Comm. Partial Differential Equations, 13 (1988), 1297-1335.  doi: 10.1080/03605308808820577.  Google Scholar

[38]

A. Sá Barreto, Interactions of conormal waves for fully semilinear wave equations, J. Funct. Anal., 89 (1990), 233-273.  doi: 10.1016/0022-1236(90)90094-2.  Google Scholar

[39]

A. Sá Barreto, Second microlocal ellipticity and propagation of conormality for semilinear wave equations, J. Funct. Anal., 102 (1991), 47-71.  doi: 10.1016/0022-1236(91)90135-R.  Google Scholar

[40]

A. Sá Barreto, Evolution of semilinear waves with swallowtail singularities, Duke Math. J., 75 (1994), 645-710.  doi: 10.1215/S0012-7094-94-07520-0.  Google Scholar

[41]

A. Sá Barreto, G. Uhlmann and Y. Wang, Inverse scattering for critical semilinear wave equations, preprint, arXiv: 2003.03822. Google Scholar

[42]

A. Sá Barreto and Y. Wang, Singularities generated by the triple interaction of semilinear conormal waves, preprint, arXiv: 1809.09253. Google Scholar

[43]

E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc., 87 (1958), 159-172.  doi: 10.1090/S0002-9947-1958-0092943-6.  Google Scholar

[44]

G. Uhlmann and Y. Wang, Determination of space-time structures from gravitational perturbations, Comm. Pure Appl. Math., 73 (2020), 1315-1367.  doi: 10.1002/cpa.21882.  Google Scholar

show all references

References:
[1]

M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math., 118 (1983), 187-214.  doi: 10.2307/2006959.  Google Scholar

[2]

M. Beals, Vector fields associated with the nonlinear interaction of progressing waves, Indiana Univ. Math. J., 37 (1988), 637-666.  doi: 10.1512/iumj.1988.37.37031.  Google Scholar

[3]

M. Beals, Singularities of conormal radially smooth solutions to nonlinear wave equations, Comm. in P. D. E., 13 (1988), 1355-1382.  doi: 10.1080/03605308808820579.  Google Scholar

[4]

M. Beals, Propagation and interaction of singularities in nonlinear hyperbolic problems, Progress in Nonlinear Differential Equations and their Applications, 3, Birkhäuser Boston, Inc., Boston, MA, 1989. doi: 10.1007/978-1-4612-4554-4.  Google Scholar

[5]

M. Beals, Regularity of nonlinear waves associated with a cusp, Microlocal Analysis and Nonlinear Waves (Minneapolis, MN, 1988–1989), IMA Vol. Math. Appl., 30, Springer, New York, 1991, 9–27. doi: 10.1007/978-1-4613-9136-4_2.  Google Scholar

[6]

J.-M. Bony, Localization et propagation des singularités pour les équations nonlinéaires,, Journées des E.D.P., St. Jean-de-Monts, (1978).  Google Scholar

[7]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux derivées partielles nonlinéaires, Ann. Sci. Ec. Norm. Sup., 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[8]

J.-M. Bony, Interaction des singularités pour les équations aux dérivées partielles nonlinéaires, Sem. Goulaouic-Meyer-Schwartz Exp. 2, (1981/1982).  Google Scholar

[9]

J.-M. Bony, Propagation et interaction des singularités pour les solutions des équations aux dérivées partielles non-linéaires, Proceedings of the International Congress of Mathematicians, 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1133–1147.  Google Scholar

[10]

J-M. Bony, Interaction des singularités pour les équations de Klein-Gordon non linéaires, Goulaouic-Meyer-Schwartz Seminar, Exp. No. 10, École Polytech., Palaiseau, 1984, 28 pp.  Google Scholar

[11]

J-M. Bony, Second microlocalization and propagation of singularities for semilinear hyperbolic equations, Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), Academic Press, Boston, MA, 1986, 11–49.  Google Scholar

[12]

J. Y. Chemin, Interaction de trois ondes dans les équations semi-linéaires strictement hyperboliques d'ordre 2, Communications in Partial Differential Equations, 12:11 (1987), 1203-1225.  doi: 10.1080/03605308708820525.  Google Scholar

[13]

X. Chen, M. Lassas, L. Oksanen and G. Paternain, Detection of Hermitian connections in wave equations with cubic nonlinearity, preprint, arXiv: 1902.05711. Google Scholar

[14]

J-M. Delort, Conormalité des ondes semi-linéaires le long des caustiques, Amer. J. Math., 113 (1991), 593-651.  doi: 10.2307/2374842.  Google Scholar

[15]

J. J. Duistermaat, Fourier integral operators, Series Progress in Mathematics, 130  Google Scholar

[16]

A. Fiorenza, M. R. Formica, T. Roskovec and F. Soudský, Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks, preprint, arXiv: 1812.04281. Google Scholar

[17]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232.  doi: 10.1016/0022-1236(90)90011-9.  Google Scholar

[18] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511721441.  Google Scholar
[19]

L. Holt, Singularities produced in conormal wave interactions, Trans. Amer. Math. Soc., 347 (1995), 289-315.  doi: 10.1090/S0002-9947-1995-1264146-3.  Google Scholar

[20]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume III, Springer Verlag, 1994.  Google Scholar

[21]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume IV, Springer Verlag, 1994.  Google Scholar

[22]

M. Joshi and A. Sá Barreto, The generation of semilinear singularities by a swallowtail caustic, Amer. J. Math., 120 (1998), 529-550.  doi: 10.1353/ajm.1998.0023.  Google Scholar

[23]

Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Inventiones Mathematicae, 212.3 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.  Google Scholar

[24]

G. Lebeau, Équations des ondes semi-linéaires. Ⅱ. Contrôle des singularités et caustiques nonlinéaires, Invent. Math., 95 (1989), 277-323.  doi: 10.1007/BF01393899.  Google Scholar

[25]

M. LassasG. Uhlmann and Y. Wang, Inverse problems for semilinear wave equations on Lorentzian manifolds, Comm. Math. Phys., 360 (2018), 555-609.  doi: 10.1007/s00220-018-3135-7.  Google Scholar

[26]

R. B. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.  Google Scholar

[27]

R. B. Melrose, Interaction of progressing waves through a nonlinear potential, Séminaire Équations aux Dérivées Partielles (Polytechnique), (1983-1984), Exp. No. 12, 1–13.  Google Scholar

[28]

R. B. Melrose, Semilinear waves with cusp singularities, Journées "Équations aux derivées partielles" (Saint Jean de Monts, 1987), Exp. No. X, École Polytech., Palaiseau, 1987, 10 pp.  Google Scholar

[29]

R. B. Melrose, Conormality, cusps and nonlinear interaction, Microlocal Analysis and Nonlinear Waves (Minneapolis, MN, 1988–1989), IMA Vol. Math. Appl., 30, Springer, New York, 1991,155–166. doi: 10.1007/978-1-4613-9136-4_11.  Google Scholar

[30]

R. B. Melrose and N. Ritter, Interaction of nonlinear progressing waves for semilinear wave equations, Ann. Math., 121 (1985), 187-213.  doi: 10.2307/1971196.  Google Scholar

[31]

R. B. Melrose and N. Ritter, Interaction of progressing waves for semilinear wave equations.Ⅱ, Ark. Mat., 25 (1987), 91-114.  doi: 10.1007/BF02384437.  Google Scholar

[32]

R. B. Melrose and A. Sá Barreto, Non-linear interaction of a cusp and a plane, Comm. Partial Differential Equations, 20 (1995), 961-1032.  doi: 10.1080/03605309508821121.  Google Scholar

[33]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.  doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[34]

A. Piriou, Calcul symbolique non linéaire pour une onde conormale simple, Ann. Inst. Fourier (Grenoble), 38 (1988), 173-187.  doi: 10.5802/aif.1153.  Google Scholar

[35]

J. Rauch and M. Reed, Singularities produced by the nonlinear interaction of three progressing waves; Examples, Comm. Partial Differential Equations, 7 (1982), 1117-1133.  doi: 10.1080/03605308208820246.  Google Scholar

[36]

J. Rauch and M. Reed, Non-linear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J., 49 (1982), 379-475.   Google Scholar

[37]

J. Rauch and M. Reed, Classical conormal solutions of semilinear systems, Comm. Partial Differential Equations, 13 (1988), 1297-1335.  doi: 10.1080/03605308808820577.  Google Scholar

[38]

A. Sá Barreto, Interactions of conormal waves for fully semilinear wave equations, J. Funct. Anal., 89 (1990), 233-273.  doi: 10.1016/0022-1236(90)90094-2.  Google Scholar

[39]

A. Sá Barreto, Second microlocal ellipticity and propagation of conormality for semilinear wave equations, J. Funct. Anal., 102 (1991), 47-71.  doi: 10.1016/0022-1236(91)90135-R.  Google Scholar

[40]

A. Sá Barreto, Evolution of semilinear waves with swallowtail singularities, Duke Math. J., 75 (1994), 645-710.  doi: 10.1215/S0012-7094-94-07520-0.  Google Scholar

[41]

A. Sá Barreto, G. Uhlmann and Y. Wang, Inverse scattering for critical semilinear wave equations, preprint, arXiv: 2003.03822. Google Scholar

[42]

A. Sá Barreto and Y. Wang, Singularities generated by the triple interaction of semilinear conormal waves, preprint, arXiv: 1809.09253. Google Scholar

[43]

E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc., 87 (1958), 159-172.  doi: 10.1090/S0002-9947-1958-0092943-6.  Google Scholar

[44]

G. Uhlmann and Y. Wang, Determination of space-time structures from gravitational perturbations, Comm. Pure Appl. Math., 73 (2020), 1315-1367.  doi: 10.1002/cpa.21882.  Google Scholar

Figure 1.  The interaction of three conormal plane waves in two space dimensions. The only possible singularities created by the triple interaction appear on the surface of the light cone. Fig.3, Fig.5 and Fig.4 below illustrate the higher dimensional cases
Figure 2.  A swallowtail singularity formed on the light cone emanating from a point in two space dimensions. This can be due to the existence of conjugate points of the geodesic flow in the case $ P = D_t^2-\Delta_g, $ $ g $ a Riemannian metric in $ {{\mathbb{R}}^{2}}. $ The solution to (2.4) would remain conormal to $ {\mathcal{Q}} $ away from the caustic, but other singularities could be generated by the caustic. This figure resembles one after equation 5.1.24 in Duistermaat's book [15]
Figure 3.  The dotted line represents an expanding cylindrical wave, generated by the interaction of three plane waves given by (3.1) in $ {{\mathbb{R}}^{4}}, $ viewed by an observer in $ {\mathbb{R}}^3 $ as time increases. The speed in which the radius of the wave expands is equal to one
Figure 4.  Singularities produced by the intersection of three plane waves (3.2). An observer in $ {\mathbb{R}}^3 $ sees a conic shaped wave
Figure 5.  The dotted line shows the surface (3.4) as $ (x_1,x_2,x_3) $ vary for $ t $ fixed. Unlike the wave formed by the interaction of three plane waves considered above, which is an infinite cylinder, three spherical waves intersect along a bounded curve for fixed time. The level sets of this surface for $ \{x_3 = c\} $ are circles centered on the line $ \{x_1 = a, x_2 = b\}. $
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