Differential substitutions and symmetries of hyperbolic equations

Details Differential substitutions and symmetries of hyperbolic equations Differential substitutions and symmetries of hyperbolic equations

1995-09-15

arxiv.org/abs/solv-int/9509006v2

Nonlinear Sciences

Pattern Formation and Solitons

8 pages, AmSTeX

Scientific paper

There are considered differential substitutions of the form $v=P(x,u,u_{x})$ for which there exists a differential operator $H=\sum^{k}_{i=0} \alpha_{i} D^{i}_{x}$ such that the differential substitution maps the equation $u_{t}=H[s(x,P,D_{x}(P),...,D^{k}_{x}(P))]$ into an evolution equation for any function $s$ and any nonnegative integer $k$. All differential substitutions of the form $v=P(x,u,u_{x})$ known to the author have this property. For example, the well-known Miura transformation $v=u_{x}-u^{2}$ maps any equation of the form $$u_{t}=(D^{2}_{x}+2uD_{x}+2u_{x}) [s(x,u_{x}-u^{2},D_{x}(u_{x}-u^{2}),...,D^{k}_{x}(u_{x}-u^{2}))]$$ into the equation $$v_{t}=(D^{3}_{x}+4vD_{x}+2v_{x})[s(x,v,{{\partial v}\over{\partial x }},...,{{\partial^{k} v}\over{\partial x^{k}}})].$$ The complete classification of such differential substitutions is given. An infinite set of the pairwise nonequivalent differential substitutions with the property mentioned above is constructed. Moreover, a general result about symmetries and invariant functions of hyperbolic equations is obtained.

Affiliated with

Also associated with

No associations

Rating

Differential substitutions and symmetries of hyperbolic equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Differential substitutions and symmetries of hyperbolic equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Differential substitutions and symmetries of hyperbolic equations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-88693