November  2013, 33(11&12): 4811-4840. doi: 10.3934/dcds.2013.33.4811

Singularity formation and blowup of complex-valued solutions of the modified KdV equation

1. 

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States

2. 

Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539 Laboratoire Analyse, Géométrie et Applications, 99 avenue J.B. Clément - 93430 Villetaneuse, France, France

Received  November 2011 Revised  August 2012 Published  May 2013

The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are investigated. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time.
Citation: Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811
References:
[1]

J. Bi, Novel solutions of MKdV-equation with the modified Bäcklund transformation,, J. Shanghai Univ., 8 (2004), 286.  doi: 10.1007/s11741-004-0065-8.  Google Scholar

[2]

B. Birnir, An example of blow-up for the complex KdV-equation and existence beyond the blow-up,, SIAM J. Appl. Math., 47 (1987), 710.  doi: 10.1137/0147049.  Google Scholar

[3]

J. L. Bona, J. Cohen and G. Wang, Global well posedness for a system of KdV-type equations with coupled quadratic nonlinearities,, to appear in the Nagoya Mathematical Journal., ().   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,, Philos. Trans. Royal Soc. London, 351 (1995), 107.  doi: 10.1098/rsta.1995.0027.  Google Scholar

[5]

J. L. Bona and Z. Grujiç, Spatial analyticity properties of nonlinear waves,, Math. Models Methods Appl. Sci., 13 (2003), 345.  doi: 10.1142/S0218202503002532.  Google Scholar

[6]

J. L. Bona, Z. Grujiç and H. Kalisch, Algebraic lower bounds for the uniform radius of spatrial analyticity for the generalized Korteweg-de Vries equation,, Ann. Inst. Henri Poincaré, 22 (2005), 783.  doi: 10.1016/j.anihpc.2004.12.004.  Google Scholar

[7]

J. L. Bona and J.-C. Saut, Dispersive blowup of generalized Korteweg-de Vries equations,, J. Differential Equations, 103 (1993), 3.  doi: 10.1006/jdeq.1993.1040.  Google Scholar

[8]

J. L. Bona and F. B. Weissler, Blow-up of spatially periodic complex-valued solutions of nonlinear dispersive equations,, Indiana Univ. Math. J., 50 (2001), 759.  doi: 10.1512/iumj.2001.50.1865.  Google Scholar

[9]

J. L. Bona and F. B. Weissler, Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV,, Nonlinearity, 22 (2009), 311.  doi: 10.1088/0951-7715/22/2/005.  Google Scholar

[10]

J. L. Bona and F. B. Weissler, Finite time blowup of spatially periodic, complex-vlaued solutions of the Kortweg-de Vries equation,, in preparation., ().   Google Scholar

[11]

G. Bowtell and A. E. G. Stuart, A particle representation of the Kortweg-de Vries soliton,, J. Math. Phys., 24 (1983), 969.  doi: 10.1063/1.525786.  Google Scholar

[12]

A. C. Bryan and A. E. G. Stuart, On the dynamics of soliton interactions for the Korteweg-de Vries equation,, Chaos, 2 (1992), 487.  doi: 10.1016/0960-0779(92)90024-H.  Google Scholar

[13]

Z. Grujiç and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions,, Diff. Integral Eqns., 15 (2002), 1325.   Google Scholar

[14]

M. D. Kruskal, The Korteweg-de Vries equation and related evolution equations,, Nonlinear Wave Motion (Lectures in Applied Mathematics 15 ) (ed. A. C. Newell), 15 ) (): 61.   Google Scholar

[15]

Y.-C. Li, Simple explicit formulae for finite time blowup solutions to the complex KdV equation,, Chaos, 39 (2009), 369.   Google Scholar

[16]

Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation,, Ann. of Math. (2), 155 (2002), 235.  doi: 10.2307/3062156.  Google Scholar

[17]

F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation,, J. American Math. Soc., 14 (2001), 555.  doi: 10.1090/S0894-0347-01-00369-1.  Google Scholar

[18]

W. R. Thickstun, A system of particles equivalent to solitons,, J. Math. Anal. Appl., 55 (1976), 335.  doi: 10.1016/0022-247X(76)90164-5.  Google Scholar

[19]

J. Wu and J.-M. Yuan, The complex KdV equation with or without dissipation,, Discrete Cont. Dynamical Sys., 5 (2005), 489.  doi: 10.3934/dcdsb.2005.5.489.  Google Scholar

show all references

References:
[1]

J. Bi, Novel solutions of MKdV-equation with the modified Bäcklund transformation,, J. Shanghai Univ., 8 (2004), 286.  doi: 10.1007/s11741-004-0065-8.  Google Scholar

[2]

B. Birnir, An example of blow-up for the complex KdV-equation and existence beyond the blow-up,, SIAM J. Appl. Math., 47 (1987), 710.  doi: 10.1137/0147049.  Google Scholar

[3]

J. L. Bona, J. Cohen and G. Wang, Global well posedness for a system of KdV-type equations with coupled quadratic nonlinearities,, to appear in the Nagoya Mathematical Journal., ().   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,, Philos. Trans. Royal Soc. London, 351 (1995), 107.  doi: 10.1098/rsta.1995.0027.  Google Scholar

[5]

J. L. Bona and Z. Grujiç, Spatial analyticity properties of nonlinear waves,, Math. Models Methods Appl. Sci., 13 (2003), 345.  doi: 10.1142/S0218202503002532.  Google Scholar

[6]

J. L. Bona, Z. Grujiç and H. Kalisch, Algebraic lower bounds for the uniform radius of spatrial analyticity for the generalized Korteweg-de Vries equation,, Ann. Inst. Henri Poincaré, 22 (2005), 783.  doi: 10.1016/j.anihpc.2004.12.004.  Google Scholar

[7]

J. L. Bona and J.-C. Saut, Dispersive blowup of generalized Korteweg-de Vries equations,, J. Differential Equations, 103 (1993), 3.  doi: 10.1006/jdeq.1993.1040.  Google Scholar

[8]

J. L. Bona and F. B. Weissler, Blow-up of spatially periodic complex-valued solutions of nonlinear dispersive equations,, Indiana Univ. Math. J., 50 (2001), 759.  doi: 10.1512/iumj.2001.50.1865.  Google Scholar

[9]

J. L. Bona and F. B. Weissler, Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV,, Nonlinearity, 22 (2009), 311.  doi: 10.1088/0951-7715/22/2/005.  Google Scholar

[10]

J. L. Bona and F. B. Weissler, Finite time blowup of spatially periodic, complex-vlaued solutions of the Kortweg-de Vries equation,, in preparation., ().   Google Scholar

[11]

G. Bowtell and A. E. G. Stuart, A particle representation of the Kortweg-de Vries soliton,, J. Math. Phys., 24 (1983), 969.  doi: 10.1063/1.525786.  Google Scholar

[12]

A. C. Bryan and A. E. G. Stuart, On the dynamics of soliton interactions for the Korteweg-de Vries equation,, Chaos, 2 (1992), 487.  doi: 10.1016/0960-0779(92)90024-H.  Google Scholar

[13]

Z. Grujiç and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions,, Diff. Integral Eqns., 15 (2002), 1325.   Google Scholar

[14]

M. D. Kruskal, The Korteweg-de Vries equation and related evolution equations,, Nonlinear Wave Motion (Lectures in Applied Mathematics 15 ) (ed. A. C. Newell), 15 ) (): 61.   Google Scholar

[15]

Y.-C. Li, Simple explicit formulae for finite time blowup solutions to the complex KdV equation,, Chaos, 39 (2009), 369.   Google Scholar

[16]

Y. Martel and F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation,, Ann. of Math. (2), 155 (2002), 235.  doi: 10.2307/3062156.  Google Scholar

[17]

F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation,, J. American Math. Soc., 14 (2001), 555.  doi: 10.1090/S0894-0347-01-00369-1.  Google Scholar

[18]

W. R. Thickstun, A system of particles equivalent to solitons,, J. Math. Anal. Appl., 55 (1976), 335.  doi: 10.1016/0022-247X(76)90164-5.  Google Scholar

[19]

J. Wu and J.-M. Yuan, The complex KdV equation with or without dissipation,, Discrete Cont. Dynamical Sys., 5 (2005), 489.  doi: 10.3934/dcdsb.2005.5.489.  Google Scholar

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