• 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
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
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.  doi: 10.1016/0375-9601(83)90208-6.  Google Scholar

[2]

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

[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.  doi: 10.1016/0168-9274(92)90049-J.  Google Scholar

[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.  doi: 10.1098/rsta.1995.0027.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1137/S0036141096297662.  Google Scholar

[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.   Google Scholar

[9]

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

[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.  doi: 10.1017/S0305004198002850.  Google Scholar

[11]

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

[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.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[13]

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

[14]

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

[15]

A.-K. Kassam and L. N. Trefethen, Fourth order time-stepping for stiff pdes,, SIAM J. Sci. Comput., 26 (2005), 1214.  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]

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

[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.  doi: 10.1007/s00332-012-9127-4.  Google Scholar

[19]

C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation,, J. Nonl. Sci., 17 (2007), 429.  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]

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

[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.  doi: 10.1137/S1052623496303470.  Google Scholar

[23]

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

[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.  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]

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

[27]

T. Schmelzer, The Fast Evaluation of Matrix Functions for Exponential Integrators,, PhD thesis, (2007).   Google Scholar

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.  doi: 10.1016/0375-9601(83)90208-6.  Google Scholar

[2]

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

[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.  doi: 10.1016/0168-9274(92)90049-J.  Google Scholar

[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.  doi: 10.1098/rsta.1995.0027.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1137/S0036141096297662.  Google Scholar

[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.   Google Scholar

[9]

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

[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.  doi: 10.1017/S0305004198002850.  Google Scholar

[11]

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

[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.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[13]

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

[14]

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

[15]

A.-K. Kassam and L. N. Trefethen, Fourth order time-stepping for stiff pdes,, SIAM J. Sci. Comput., 26 (2005), 1214.  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]

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

[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.  doi: 10.1007/s00332-012-9127-4.  Google Scholar

[19]

C. Klein, C. Sparber and P. Markowich, Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation,, J. Nonl. Sci., 17 (2007), 429.  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]

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

[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.  doi: 10.1137/S1052623496303470.  Google Scholar

[23]

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

[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.  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]

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

[27]

T. Schmelzer, The Fast Evaluation of Matrix Functions for Exponential Integrators,, PhD thesis, (2007).   Google Scholar

[1]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[2]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[3]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[4]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[5]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[6]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[7]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[8]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[9]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[10]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[11]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[12]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[13]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[14]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322

[15]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[16]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[17]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

[18]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[19]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[20]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]