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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

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 $n\leq2$.
Citation: Christian Klein, Ralf Peter. Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1689-1717. doi: 10.3934/dcdsb.2014.19.1689
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C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations,, ETNA, 29 (): 116.   Google Scholar

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SIAM J. Sci. Comput., 33 (2011), 3333-3356. doi: 10.1137/100816663.  Google Scholar

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show all references

References:
[1]

Phys. Lett. A, 94 (1983), 67-70. doi: 10.1016/0375-9601(83)90208-6.  Google Scholar

[2]

Inverse Problems, 10 (1994), 505-519. doi: 10.1088/0266-5611/10/3/001.  Google Scholar

[3]

Appl. Num. Maths., 10 (1992), 335-355. doi: 10.1016/0168-9274(92)90049-J.  Google Scholar

[4]

Phil. Trans. R. Soc. Lond. A, 351 (1995), 107-164. doi: 10.1098/rsta.1995.0027.  Google Scholar

[5]

J. Comp. Phys., 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995.  Google Scholar

[6]

Ann. Inst. Henri Poincaré, Anal. Non Lineaire, 14 (1997), 211-236. doi: 10.1016/S0294-1449(97)80145-X.  Google Scholar

[7]

SIAM J. Math. Anal., 28 (1997), 1064-1085. doi: 10.1137/S0036141096297662.  Google Scholar

[8]

Differ. Integral Equ., 11 (1998), 679-723.  Google Scholar

[9]

Sov. Phys. JETP, 62 (1985), 146-152. Google Scholar

[10]

Math. Proc. Camb. Phil. Soc., 125 (1999), 113-138. doi: 10.1017/S0305004198002850.  Google Scholar

[11]

Discrete Contin. Dyn. Syst., 21 (2008), 835-882. doi: 10.3934/dcds.2008.21.835.  Google Scholar

[12]

Nonlinearity, 23 (2010), 237-275. doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[13]

SIAM J. Numer. Anal., 43 (2005), 1069-1090. Google Scholar

[14]

Sov. Phys. Dokl., 15 (1970), 539-541. Google Scholar

[15]

SIAM J. Sci. Comput., 26 (2005), 1214-1233. doi: 10.1137/S1064827502410633.  Google Scholar

[16]

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

[17]

SIAM Journal on Scientific Computing, 33 (2011), 3333-3356. doi: 10.1137/100816663.  Google Scholar

[18]

J. Nonl. Sci., 22 (2012), 763-811. doi: 10.1007/s00332-012-9127-4.  Google Scholar

[19]

J. Nonl. Sci., 17 (2007), 429-470. doi: 10.1007/s00332-007-9001-y.  Google Scholar

[20]

C. Klein, Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations,, ETNA, 29 (): 116.   Google Scholar

[21]

SIAM J. Sci. Comput., 33 (2011), 3333-3356. doi: 10.1137/100816663.  Google Scholar

[22]

SIAM Journal of Optimization, 9 (1999), 112-147. doi: 10.1137/S1052623496303470.  Google Scholar

[23]

Trans. Amer. Math. Soc., 353 (2001), 191-208. doi: 10.1090/S0002-9947-00-02465-X.  Google Scholar

[24]

Phys. Lett. A, 63 (1977), 205-206. doi: 10.1016/0375-9601(77)90875-1.  Google Scholar

[25]

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

[26]

SIAM J. Math. Anal., 39 (2007), 627-641. doi: 10.1137/060654256.  Google Scholar

[27]

PhD thesis, Oxford University, 2007. Google Scholar

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