2013, 18(5): 1361-1387. doi: 10.3934/dcdsb.2013.18.1361

Numerical study of blow-up in the Davey-Stewartson system

1. 

Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon Cedex, France, France

2. 

Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, MI 48109, United States

Received  December 2011 Revised  February 2013 Published  March 2013

Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow up. We numerically study the long time behavior and potential blow-up of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa solutions. It is shown in this way that both are unstable to blow-up and dispersion, and that blow-up in the Ozawa solution is generic.
Citation: Christian Klein, Benson Muite, Kristelle Roidot. Numerical study of blow-up in the Davey-Stewartson system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1361-1387. doi: 10.3934/dcdsb.2013.18.1361
References:
[1]

M. Ablowitz and R. Haberman, Nonlinear evolution equations in two and three dimensions,, Phys. Rev. Lett., 35 (1975), 1185.

[2]

M. J. Ablowitz and A. Fokas, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation,, Physics Letters A, 94 (1983), 67. doi: 10.1016/0375-9601(83)90208-6.

[3]

G. Agrawal, "Nonlinear Fiber Optics,", Academic Press, (2006).

[4]

V. Arkadiev, A. Pogrebkov and M. Polivanov, Inverse scattering transform method and soliton solutions for Davey-Stewartson II equation,, Physica D: Nonlinear Phenomena, 36 (1989), 189. doi: 10.1016/0167-2789(89)90258-3.

[5]

K. A. Bagrinovskii and S. Godunov, Difference schemes for multi-dimensional problems,, Dokl. Acad. Nauk., 115 (1957), 431.

[6]

C. Besse, N. Mauser and H. Stimming, Numerical study of the Davey-Stewartson system,, Math. Model. Numer. Anal., 38 (2004), 1035. doi: 10.1051/m2an:2004049.

[7]

M. Cross and P. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851.

[8]

A. Davey and K. Stewartson, On three-dimensional packets of surface waves,, Proc. R. Soc. Lond. A., 338 (1974), 101.

[9]

V. Djordjevic and L. Redekopp, On Two-dimensional Packets of Capillarity-Gravity Waves,, J. Fluid Mech., 79 (1977), 703.

[10]

A. Fokas and L. Sung, The Cauchy problem for the Kadomtsev-Petviashvili-I equation without the zero mass constraint,, Math. Proc. Camb. Philos. Soc., 125 (1999), 113. doi: 10.1017/S0305004198002850.

[11]

M. Forest and J. Lee, Geometry and modulation theory for the periodic nonlinear Schrödinger equation,, in, 2 (1986), 35. doi: 10.1007/978-1-4613-8689-6_3.

[12]

J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475.

[13]

W. Gropp, E. Lusk and A. Skjellum, "Using MPI,", MIT Press, (1999).

[14]

R. H. Hardin and F. D. Tappert, Applications of the Split-Step Fourier Method to the numerical Solution of nonlinear and variable Coefficient Wave Equations,, SIAM Rev., 15 (1973).

[15]

T. Kato, Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups,, in, (1978), 185.

[16]

C. Klein, Fourth-order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation,, Electronic Transactions on Numerical Analysis., 39 (2008), 116.

[17]

C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations,, SIAM J. Sci. Comp., 33 (2011), 3333. doi: 10.1137/100816663.

[18]

C. Klein and J.-C. Saut, Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations,, J. Nonl. Sci., 22 (2012), 763. doi: 10.1007/s00332-012-9127-4.

[19]

M. McConnell, A. Fokas and B. Pelloni, Localised coherent solutions of the DSI and DSII equations: A numerical study,, Mathematics and Computers in Simulation, 69 (2005), 424. doi: 10.1016/j.matcom.2005.03.007.

[20]

F. Merle and P. Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation,, Inventiones Mathematicae, 156 (2004), 565. doi: 10.1007/s00222-003-0346-z.

[21]

K. Nishinari, K. Abe and J. Satsuma, A new-type of soliton behavior of the Davey-Stewartson equations in a plasma system,, Teoret. Mat. Fiz., 99 (1994), 487. doi: 10.1007/BF01017062.

[22]

K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an electrostaic Ion wave in a magnetized plasma,, Phys. Plasmas, 1 (1994), 2559.

[23]

T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems,, Proc. R. Soc. Lond. A., 436 (1992), 345. doi: 10.1098/rspa.1992.0022.

[24]

D. Pelinovsky and C. Sulem, Eigenfunctions and eigenvalues for a scalar Riemann-Hilbert problem associated to inverse scattering,, Commun. Math. Phys., 208 (2000), 713. doi: 10.1007/s002200050008.

[25]

D. Pelinovsky and C. Sulem, Spectral decomposition for the Dirac system associated to the DSII equation,, Inv. Prob., 16 (2000), 59. doi: 10.1088/0266-5611/16/1/306.

[26]

P. Stinis, Numerical computation of solutions of the critical nonlinear Schrödinger equation after the singularity,, Multiscale Model. Simul., 10 (2012), 48. doi: 10.1137/110831222.

[27]

G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.

[28]

C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation,", 139, 139 (1999).

[29]

L.-Y. Sung, Long-time decay of the solutions of the Davey-Stewartson II equations,, J. Nonlinear Sci., 5 (1995), 433.

[30]

T. Tao, Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation,, Analysis and PDE, 2 (2009), 61. doi: 10.2140/apde.2009.2.61.

[31]

F. Tappert, Numerical solutions of the Korteweg-de vries equation and its generalizations by the split-step fourier method,, Lectures in Applied Mathematics, 15 (1974), 215.

[32]

H. Trotter, On the product of semi-groups of operators,, Proceedings of the American Mathematical Society, 10 (1959), 545.

[33]

P. White and J. Weideman, Numerical simulation of solitons and dromions in the Davey-Stewartson system,, Math. Comput. Simul., 37 (1994), 469. doi: 10.1016/0378-4754(94)00032-8.

[34]

, href=, ().

[35]

H. Yoshida, Construction of higher order symplectic integrators,, Physics Letters A, 150 (1990), 262. doi: 10.1016/0375-9601(90)90092-3.

show all references

References:
[1]

M. Ablowitz and R. Haberman, Nonlinear evolution equations in two and three dimensions,, Phys. Rev. Lett., 35 (1975), 1185.

[2]

M. J. Ablowitz and A. Fokas, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation,, Physics Letters A, 94 (1983), 67. doi: 10.1016/0375-9601(83)90208-6.

[3]

G. Agrawal, "Nonlinear Fiber Optics,", Academic Press, (2006).

[4]

V. Arkadiev, A. Pogrebkov and M. Polivanov, Inverse scattering transform method and soliton solutions for Davey-Stewartson II equation,, Physica D: Nonlinear Phenomena, 36 (1989), 189. doi: 10.1016/0167-2789(89)90258-3.

[5]

K. A. Bagrinovskii and S. Godunov, Difference schemes for multi-dimensional problems,, Dokl. Acad. Nauk., 115 (1957), 431.

[6]

C. Besse, N. Mauser and H. Stimming, Numerical study of the Davey-Stewartson system,, Math. Model. Numer. Anal., 38 (2004), 1035. doi: 10.1051/m2an:2004049.

[7]

M. Cross and P. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851.

[8]

A. Davey and K. Stewartson, On three-dimensional packets of surface waves,, Proc. R. Soc. Lond. A., 338 (1974), 101.

[9]

V. Djordjevic and L. Redekopp, On Two-dimensional Packets of Capillarity-Gravity Waves,, J. Fluid Mech., 79 (1977), 703.

[10]

A. Fokas and L. Sung, The Cauchy problem for the Kadomtsev-Petviashvili-I equation without the zero mass constraint,, Math. Proc. Camb. Philos. Soc., 125 (1999), 113. doi: 10.1017/S0305004198002850.

[11]

M. Forest and J. Lee, Geometry and modulation theory for the periodic nonlinear Schrödinger equation,, in, 2 (1986), 35. doi: 10.1007/978-1-4613-8689-6_3.

[12]

J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475.

[13]

W. Gropp, E. Lusk and A. Skjellum, "Using MPI,", MIT Press, (1999).

[14]

R. H. Hardin and F. D. Tappert, Applications of the Split-Step Fourier Method to the numerical Solution of nonlinear and variable Coefficient Wave Equations,, SIAM Rev., 15 (1973).

[15]

T. Kato, Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups,, in, (1978), 185.

[16]

C. Klein, Fourth-order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation,, Electronic Transactions on Numerical Analysis., 39 (2008), 116.

[17]

C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations,, SIAM J. Sci. Comp., 33 (2011), 3333. doi: 10.1137/100816663.

[18]

C. Klein and J.-C. Saut, Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations,, J. Nonl. Sci., 22 (2012), 763. doi: 10.1007/s00332-012-9127-4.

[19]

M. McConnell, A. Fokas and B. Pelloni, Localised coherent solutions of the DSI and DSII equations: A numerical study,, Mathematics and Computers in Simulation, 69 (2005), 424. doi: 10.1016/j.matcom.2005.03.007.

[20]

F. Merle and P. Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation,, Inventiones Mathematicae, 156 (2004), 565. doi: 10.1007/s00222-003-0346-z.

[21]

K. Nishinari, K. Abe and J. Satsuma, A new-type of soliton behavior of the Davey-Stewartson equations in a plasma system,, Teoret. Mat. Fiz., 99 (1994), 487. doi: 10.1007/BF01017062.

[22]

K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an electrostaic Ion wave in a magnetized plasma,, Phys. Plasmas, 1 (1994), 2559.

[23]

T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems,, Proc. R. Soc. Lond. A., 436 (1992), 345. doi: 10.1098/rspa.1992.0022.

[24]

D. Pelinovsky and C. Sulem, Eigenfunctions and eigenvalues for a scalar Riemann-Hilbert problem associated to inverse scattering,, Commun. Math. Phys., 208 (2000), 713. doi: 10.1007/s002200050008.

[25]

D. Pelinovsky and C. Sulem, Spectral decomposition for the Dirac system associated to the DSII equation,, Inv. Prob., 16 (2000), 59. doi: 10.1088/0266-5611/16/1/306.

[26]

P. Stinis, Numerical computation of solutions of the critical nonlinear Schrödinger equation after the singularity,, Multiscale Model. Simul., 10 (2012), 48. doi: 10.1137/110831222.

[27]

G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.

[28]

C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation,", 139, 139 (1999).

[29]

L.-Y. Sung, Long-time decay of the solutions of the Davey-Stewartson II equations,, J. Nonlinear Sci., 5 (1995), 433.

[30]

T. Tao, Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation,, Analysis and PDE, 2 (2009), 61. doi: 10.2140/apde.2009.2.61.

[31]

F. Tappert, Numerical solutions of the Korteweg-de vries equation and its generalizations by the split-step fourier method,, Lectures in Applied Mathematics, 15 (1974), 215.

[32]

H. Trotter, On the product of semi-groups of operators,, Proceedings of the American Mathematical Society, 10 (1959), 545.

[33]

P. White and J. Weideman, Numerical simulation of solitons and dromions in the Davey-Stewartson system,, Math. Comput. Simul., 37 (1994), 469. doi: 10.1016/0378-4754(94)00032-8.

[34]

, href=, ().

[35]

H. Yoshida, Construction of higher order symplectic integrators,, Physics Letters A, 150 (1990), 262. doi: 10.1016/0375-9601(90)90092-3.

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