-
Previous Article
Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system
- DCDS-B Home
- This Issue
-
Next Article
Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations
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 |
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. |
[1] |
Mengxian Lv, Jianghao Hao. General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021058 |
[2] |
Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11 |
[3] |
Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 |
[4] |
Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 |
[5] |
Christian Klein, Benson Muite, Kristelle Roidot. Numerical study of blow-up in the Davey-Stewartson system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1361-1387. doi: 10.3934/dcdsb.2013.18.1361 |
[6] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
[7] |
Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 |
[8] |
Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 |
[9] |
Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881 |
[10] |
Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449 |
[11] |
Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828 |
[12] |
Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835 |
[13] |
Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 |
[14] |
Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations and Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 |
[15] |
Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126 |
[16] |
Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585 |
[17] |
Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 |
[18] |
Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733 |
[19] |
John A. D. Appleby, Denis D. Patterson. Blow-up and superexponential growth in superlinear Volterra equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3993-4017. doi: 10.3934/dcds.2018174 |
[20] |
Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]