\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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 $ - oo \lt x \lt oo $, exponentially decreasing to zero as $|x| \to oo$, that blow up in finite time.

    Mathematics Subject Classification: 35Q53, 35Q51, 37K40, 35A21, 35B44.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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. WangGlobal 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. WeisslerFinite 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. KruskalThe Korteweg-de Vries equation and related evolution equations, Nonlinear Wave Motion (Lectures in Applied Mathematics 15 ) (ed. A. C. Newell), (Providence, RI: American Mathematical Society), pp. 61-83.

    [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(66) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return