# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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 (Enlgish Edition) (2004), 286-288. doi: 10.1007/s11741-004-0065-8. [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), 710-725. doi: 10.1137/0147049. [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., (). [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, Ser. A, 351 (1995), 107-164. doi: 10.1098/rsta.1995.0027. [5] J. L. Bona and Z. Grujiç, Spatial analyticity properties of nonlinear waves, Math. Models Methods Appl. Sci., 13 (2003), 345-360. doi: 10.1142/S0218202503002532. [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, Anal. Non Linéaire, 22 (2005), 783-797. doi: 10.1016/j.anihpc.2004.12.004. [7] J. L. Bona and J.-C. Saut, Dispersive blowup of generalized Korteweg-de Vries equations, J. Differential Equations, 103 (1993), 3-57. doi: 10.1006/jdeq.1993.1040. [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), 759-782. doi: 10.1512/iumj.2001.50.1865. [9] J. L. Bona and F. B. Weissler, Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV, Nonlinearity, 22 (2009), 311-349. doi: 10.1088/0951-7715/22/2/005. [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., (). [11] G. Bowtell and A. E. G. Stuart, A particle representation of the Kortweg-de Vries soliton, J. Math. Phys., 24 (1983), 969-981. doi: 10.1063/1.525786. [12] A. C. Bryan and A. E. G. Stuart, On the dynamics of soliton interactions for the Korteweg-de Vries equation, Chaos, Solitons & Fractals, 2 (1992), 487-491. doi: 10.1016/0960-0779(92)90024-H. [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), 1325-1334. [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. [15] Y.-C. Li, Simple explicit formulae for finite time blowup solutions to the complex KdV equation, Chaos, Solitons & Fractals, 39 (2009), 369-372. [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), 235-280. doi: 10.2307/3062156. [17] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. American Math. Soc., 14 (2001), 555-578. doi: 10.1090/S0894-0347-01-00369-1. [18] W. R. Thickstun, A system of particles equivalent to solitons, J. Math. Anal. Appl., 55 (1976), 335-346. doi: 10.1016/0022-247X(76)90164-5. [19] J. Wu and J.-M. Yuan, The complex KdV equation with or without dissipation, Discrete Cont. Dynamical Sys., Ser. B, 5 (2005), 489-512. doi: 10.3934/dcdsb.2005.5.489.

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##### References:
 [1] J. Bi, Novel solutions of MKdV-equation with the modified Bäcklund transformation, J. Shanghai Univ., 8 (Enlgish Edition) (2004), 286-288. doi: 10.1007/s11741-004-0065-8. [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), 710-725. doi: 10.1137/0147049. [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., (). [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, Ser. A, 351 (1995), 107-164. doi: 10.1098/rsta.1995.0027. [5] J. L. Bona and Z. Grujiç, Spatial analyticity properties of nonlinear waves, Math. Models Methods Appl. Sci., 13 (2003), 345-360. doi: 10.1142/S0218202503002532. [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, Anal. Non Linéaire, 22 (2005), 783-797. doi: 10.1016/j.anihpc.2004.12.004. [7] J. L. Bona and J.-C. Saut, Dispersive blowup of generalized Korteweg-de Vries equations, J. Differential Equations, 103 (1993), 3-57. doi: 10.1006/jdeq.1993.1040. [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), 759-782. doi: 10.1512/iumj.2001.50.1865. [9] J. L. Bona and F. B. Weissler, Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV, Nonlinearity, 22 (2009), 311-349. doi: 10.1088/0951-7715/22/2/005. [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., (). [11] G. Bowtell and A. E. G. Stuart, A particle representation of the Kortweg-de Vries soliton, J. Math. Phys., 24 (1983), 969-981. doi: 10.1063/1.525786. [12] A. C. Bryan and A. E. G. Stuart, On the dynamics of soliton interactions for the Korteweg-de Vries equation, Chaos, Solitons & Fractals, 2 (1992), 487-491. doi: 10.1016/0960-0779(92)90024-H. [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), 1325-1334. [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. [15] Y.-C. Li, Simple explicit formulae for finite time blowup solutions to the complex KdV equation, Chaos, Solitons & Fractals, 39 (2009), 369-372. [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), 235-280. doi: 10.2307/3062156. [17] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. American Math. Soc., 14 (2001), 555-578. doi: 10.1090/S0894-0347-01-00369-1. [18] W. R. Thickstun, A system of particles equivalent to solitons, J. Math. Anal. Appl., 55 (1976), 335-346. doi: 10.1016/0022-247X(76)90164-5. [19] J. Wu and J.-M. Yuan, The complex KdV equation with or without dissipation, Discrete Cont. Dynamical Sys., Ser. B, 5 (2005), 489-512. doi: 10.3934/dcdsb.2005.5.489.
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