Modified KdV (mKdV) equation
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introduction
\[u_t+6uu_x+u_{xxx}=0\]
- 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.