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August  2014, 19(6): 1689-1717. doi: 10.3934/dcdsb.2014.19.1689

Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations

Christian Klein 1, and  Ralf Peter 1,

1. 

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

Received  December 2013 Revised  March 2014 Published  June 2014

We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we use a dynamic rescaling to identify the type of the singularity. We present the first discussion of the observed blow-up scenarios. We show that the blow-up in solutions to the $L_{2}$ critical generalized Kadomtsev-Petviashvili I case is similar to what is known for the $L_{2}$ critical generalized Korteweg-de Vries equation. No blow-up is observed for solutions to the generalized Kadomtsev-Petviashvili II equations for $n\leq2$.
Keywords: blow-up, numerical approaches, Generalized Kadomtsev-Petviasvili equations, dynamic rescaling..
Mathematics Subject Classification: Primary: 35Q53, 35B44; Secondary: 35B65, 65M2.
Citation: Christian Klein, Ralf Peter. Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1689-1717. doi: 10.3934/dcdsb.2014.19.1689
References:
[1]

M. Ablowitz and A. Fokas, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation, Phys. Lett. A, 94 (1983), 67-70. doi: 10.1016/0375-9601(83)90208-6.

[2]

M. Boiti, F. Pempinelli and A. K. Pogrebkov, Solutions of the KPI equation with smooth initial data, Inverse Problems, 10 (1994), 505-519. doi: 10.1088/0266-5611/10/3/001.

[3]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries-Burgers equation, Appl. Num. Maths., 10 (1992), 335-355. doi: 10.1016/0168-9274(92)90049-J.

[4]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Phil. Trans. R. Soc. Lond. A, 351 (1995), 107-164. doi: 10.1098/rsta.1995.0027.

[5]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys., 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.

[6]

A. de Bouard and J.-C. Saut, Solitary waves of the generalized KP equations, Ann. Inst. Henri Poincaré, Anal. Non Lineaire, 14 (1997), 211-236. doi: 10.1016/S0294-1449(97)80145-X.

[7]

A. de Bouard and J.-C. Saut, Symmetry and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085. doi: 10.1137/S0036141096297662.

[8]

D. B. Dix and W. R. McKinney, Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equations, Differ. Integral Equ., 11 (1998), 679-723.

[9]

G. Falkovitch and S. Turitsyn, Stability of magnetoelastic solitons and self-focusing of sound in antiferromagnet, Sov. Phys. JETP, 62 (1985), 146-152.

[10]

A. Fokas and L. Sung, The Cauchy problem for the KP I equation without the zero mass constraint, Math. Proc. Camb. Phil. Soc., 125 (1999), 113-138. doi: 10.1017/S0305004198002850.

[11]

P. Gravejat, Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations, Discrete Contin. Dyn. Syst., 21 (2008), 835-882. doi: 10.3934/dcds.2008.21.835.

[12]

P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.

[13]

M. Hochbruck and A. Ostermann, Exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), 1069-1090.

[14]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539-541.

[15]

A.-K. Kassam and L. N. Trefethen, Fourth order time-stepping for stiff pdes, SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.

[16]

C. Klein and R. Peter, Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations, Preprint available at: arXiv:1307.0603.

[17]

C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations, SIAM Journal on Scientific Computing, 33 (2011), 3333-3356. doi: 10.1137/100816663.

[18]

C. Klein and J. Saut, Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations, J. Nonl. Sci., 22 (2012), 763-811. doi: 10.1007/s00332-012-9127-4.

[19]

C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation, J. Nonl. Sci., 17 (2007), 429-470. doi: 10.1007/s00332-007-9001-y.

[20]

C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations, ETNA, 29 (2007/08), 116-135.

[21]

C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewardson equations, SIAM J. Sci. Comput., 33 (2011), 3333-3356. doi: 10.1137/100816663.

[22]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9 (1999), 112-147. doi: 10.1137/S1052623496303470.

[23]

Y. Liu, Blow-up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc., 353 (2001), 191-208. doi: 10.1090/S0002-9947-00-02465-X.

[24]

S. Manakov, V. Zakharov, L. Bordag and V. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A, 63 (1977), 205-206. doi: 10.1016/0375-9601(77)90875-1.

[25]

Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation I: Dynamics near the solition, Preprint available at: arXiv:1204.4625.

[26]

L. Molinet, J. C. Saut and N. Tzvetkov, Remarks on the mass constraint for KP type equations, SIAM J. Math. Anal., 39 (2007), 627-641. doi: 10.1137/060654256.

[27]

T. Schmelzer, The Fast Evaluation of Matrix Functions for Exponential Integrators, PhD thesis, Oxford University, 2007.

show all references

References:
[1]

M. Ablowitz and A. Fokas, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation, Phys. Lett. A, 94 (1983), 67-70. doi: 10.1016/0375-9601(83)90208-6.

[2]

M. Boiti, F. Pempinelli and A. K. Pogrebkov, Solutions of the KPI equation with smooth initial data, Inverse Problems, 10 (1994), 505-519. doi: 10.1088/0266-5611/10/3/001.

[3]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries-Burgers equation, Appl. Num. Maths., 10 (1992), 335-355. doi: 10.1016/0168-9274(92)90049-J.

[4]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Phil. Trans. R. Soc. Lond. A, 351 (1995), 107-164. doi: 10.1098/rsta.1995.0027.

[5]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys., 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.

[6]

A. de Bouard and J.-C. Saut, Solitary waves of the generalized KP equations, Ann. Inst. Henri Poincaré, Anal. Non Lineaire, 14 (1997), 211-236. doi: 10.1016/S0294-1449(97)80145-X.

[7]

A. de Bouard and J.-C. Saut, Symmetry and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085. doi: 10.1137/S0036141096297662.

[8]

D. B. Dix and W. R. McKinney, Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equations, Differ. Integral Equ., 11 (1998), 679-723.

[9]

G. Falkovitch and S. Turitsyn, Stability of magnetoelastic solitons and self-focusing of sound in antiferromagnet, Sov. Phys. JETP, 62 (1985), 146-152.

[10]

A. Fokas and L. Sung, The Cauchy problem for the KP I equation without the zero mass constraint, Math. Proc. Camb. Phil. Soc., 125 (1999), 113-138. doi: 10.1017/S0305004198002850.

[11]

P. Gravejat, Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations, Discrete Contin. Dyn. Syst., 21 (2008), 835-882. doi: 10.3934/dcds.2008.21.835.

[12]

P. Guyenne, D. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.

[13]

M. Hochbruck and A. Ostermann, Exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), 1069-1090.

[14]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539-541.

[15]

A.-K. Kassam and L. N. Trefethen, Fourth order time-stepping for stiff pdes, SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.

[16]

C. Klein and R. Peter, Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations, Preprint available at: arXiv:1307.0603.

[17]

C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations, SIAM Journal on Scientific Computing, 33 (2011), 3333-3356. doi: 10.1137/100816663.

[18]

C. Klein and J. Saut, Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations, J. Nonl. Sci., 22 (2012), 763-811. doi: 10.1007/s00332-012-9127-4.

[19]

C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation, J. Nonl. Sci., 17 (2007), 429-470. doi: 10.1007/s00332-007-9001-y.

[20]

C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations, ETNA, 29 (2007/08), 116-135.

[21]

C. Klein and K. Roidot, Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewardson equations, SIAM J. Sci. Comput., 33 (2011), 3333-3356. doi: 10.1137/100816663.

[22]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9 (1999), 112-147. doi: 10.1137/S1052623496303470.

[23]

Y. Liu, Blow-up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc., 353 (2001), 191-208. doi: 10.1090/S0002-9947-00-02465-X.

[24]

S. Manakov, V. Zakharov, L. Bordag and V. Matveev, Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A, 63 (1977), 205-206. doi: 10.1016/0375-9601(77)90875-1.

[25]

Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation I: Dynamics near the solition, Preprint available at: arXiv:1204.4625.

[26]

L. Molinet, J. C. Saut and N. Tzvetkov, Remarks on the mass constraint for KP type equations, SIAM J. Math. Anal., 39 (2007), 627-641. doi: 10.1137/060654256.

[27]

T. Schmelzer, The Fast Evaluation of Matrix Functions for Exponential Integrators, PhD thesis, Oxford University, 2007.

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