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

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

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

• 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.