Modified KdV (mKdV) equation

introduction

• KdV equation

\[u_t+6uu_x+u_{xxx}=0\]

• mKdV equation

$$ u_t+6u^2u_x+u_{xxx}=0 $$

$N$-solution solution

• $N$-solution solution

$$ u(t,x)=-2\frac{\partial}{\partial x}\tan^{-1}[\frac{\Im \det (I+A)}{\Re \det (I+A)}] $$ where $I$ is the $N\times N$ matrix and $A$ denotes the $N\times N$ matrix with elements $$ A_{mn}=-\frac{d_n(t)}{\zeta_n+\zeta_m}\exp[i(\zeta_n+\zeta_m)x],\, m,n=1,2\cdots,N, $$ $\zeta_n=i\eta_n,\, \eta_n>0$ $d_n(t)=d_n(0)\exp (8i\zeta_n^3 t)$

• 1-soliton, $\zeta=i\eta,\, \eta>0$

$$ u(x)=-2\eta \operatorname{sech} (2\eta x) $$

• 2-soliton
• see
  • M. Wadati and K. Ohkuma, J. Phys. Soc. Japan. 51,2029 (1982).
  • M. Wadati, J. Phys. Soc. Japan.77, 074005 (2008)

periodic solution

• M. Wadati, J. Phys. Soc. Japan.77, 074005 (2008)
• M. Wadati, J. Phys. Soc.Japan.38, 673 (1975); ibid38,681 (1975).
• P.G. Kevrekidis, A. Khare, A. Saxena and G. Herring, J.Phys. A37, 10959 (2004).
• Z. Fu, S. Liu, S. Liu and Q. Zhao, Phys. Letts. A290,72 (2001).

expositions

• Ho, C.-L., and P. Roy. ‘mKdV Equation Approach to Zero Energy States of Graphene’. arXiv:1507.02649 [cond-Mat, Physics:math-Ph, Physics:nlin, Physics:quant-Ph], 9 July 2015. http://arxiv.org/abs/1507.02649.

articles

• Miguel A. Alejo, Claudio Muñoz, José M. Palacios, On the variational structure of breather solutions, arXiv:1309.0625 [math-ph], September 03 2013, http://arxiv.org/abs/1309.0625
• Germain, Pierre, Fabio Pusateri, and Frédéric Rousset. ‘Asymptotic Stability of Solitons for mKdV’. arXiv:1503.09143 [math], 31 March 2015. http://arxiv.org/abs/1503.09143.