# American Institute of Mathematical Sciences

July  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.   Google Scholar [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.  Google Scholar [3] G. Agrawal, "Nonlinear Fiber Optics,", Academic Press, (2006).   Google Scholar [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.  Google Scholar [5] K. A. Bagrinovskii and S. Godunov, Difference schemes for multi-dimensional problems,, Dokl. Acad. Nauk., 115 (1957), 431.   Google Scholar [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.  Google Scholar [7] M. Cross and P. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851.   Google Scholar [8] A. Davey and K. Stewartson, On three-dimensional packets of surface waves,, Proc. R. Soc. Lond. A., 338 (1974), 101.   Google Scholar [9] V. Djordjevic and L. Redekopp, On Two-dimensional Packets of Capillarity-Gravity Waves,, J. Fluid Mech., 79 (1977), 703.   Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475.   Google Scholar [13] W. Gropp, E. Lusk and A. Skjellum, "Using MPI,", MIT Press, (1999).   Google Scholar [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).   Google Scholar [15] T. Kato, Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups,, in, (1978), 185.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an electrostaic Ion wave in a magnetized plasma,, Phys. Plasmas, 1 (1994), 2559.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.   Google Scholar [28] C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation,", 139, 139 (1999).   Google Scholar [29] L.-Y. Sung, Long-time decay of the solutions of the Davey-Stewartson II equations,, J. Nonlinear Sci., 5 (1995), 433.   Google Scholar [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.  Google Scholar [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.   Google Scholar [32] H. Trotter, On the product of semi-groups of operators,, Proceedings of the American Mathematical Society, 10 (1959), 545.   Google Scholar [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.  Google Scholar [34] , href=, ().   Google Scholar [35] H. Yoshida, Construction of higher order symplectic integrators,, Physics Letters A, 150 (1990), 262.  doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

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##### References:
 [1] M. Ablowitz and R. Haberman, Nonlinear evolution equations in two and three dimensions,, Phys. Rev. Lett., 35 (1975), 1185.   Google Scholar [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.  Google Scholar [3] G. Agrawal, "Nonlinear Fiber Optics,", Academic Press, (2006).   Google Scholar [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.  Google Scholar [5] K. A. Bagrinovskii and S. Godunov, Difference schemes for multi-dimensional problems,, Dokl. Acad. Nauk., 115 (1957), 431.   Google Scholar [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.  Google Scholar [7] M. Cross and P. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851.   Google Scholar [8] A. Davey and K. Stewartson, On three-dimensional packets of surface waves,, Proc. R. Soc. Lond. A., 338 (1974), 101.   Google Scholar [9] V. Djordjevic and L. Redekopp, On Two-dimensional Packets of Capillarity-Gravity Waves,, J. Fluid Mech., 79 (1977), 703.   Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems,, Nonlinearity, 3 (1990), 475.   Google Scholar [13] W. Gropp, E. Lusk and A. Skjellum, "Using MPI,", MIT Press, (1999).   Google Scholar [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).   Google Scholar [15] T. Kato, Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups,, in, (1978), 185.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] K. Nishinari, K. Abe and J. Satsuma, Multidimensional behavior of an electrostaic Ion wave in a magnetized plasma,, Phys. Plasmas, 1 (1994), 2559.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.   Google Scholar [28] C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation,", 139, 139 (1999).   Google Scholar [29] L.-Y. Sung, Long-time decay of the solutions of the Davey-Stewartson II equations,, J. Nonlinear Sci., 5 (1995), 433.   Google Scholar [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.  Google Scholar [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.   Google Scholar [32] H. Trotter, On the product of semi-groups of operators,, Proceedings of the American Mathematical Society, 10 (1959), 545.   Google Scholar [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.  Google Scholar [34] , href=, ().   Google Scholar [35] H. Yoshida, Construction of higher order symplectic integrators,, Physics Letters A, 150 (1990), 262.  doi: 10.1016/0375-9601(90)90092-3.  Google Scholar
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