Newton trajectories for the tilted Frenkel–Kontorova model
by    W.Quapp   and J.M.Bofill
in:    Molecular Physics,117:9-12,1541-1558 (2019)
DOI: 10.1080/00268976.2019.1576930
To link to this article:
https://doi.org/10.1080/00268976.2019.1576930

(* Figure 1 ::  2D PES for 2 atoms in a linear 'chain' *) 
(* Ansatz of a 2D PES *)

PES[x_,y_]=0.5*(((y-4./3.*Pi -x))^2)+2.-Cos[x]-Cos[y]

conk=ContourPlot[PES[x,y],{x,-1.0,6.2},{y,2.3,9.3},Contours->{1.0,1.5,2.0,2.5,3,3.5,4,4.5,5,5.5,6,6.5,7,7.5,8,8.5,9,9.5,10},  PlotPoints->79,ContourShading->False,
ContourStyle->Thickness[0.0015],FrameStyle->Directive[20],PlotPoints->79,FrameLabel->{"x1","x2"}]

(*  Determine the Newton trajectory to direction (1,1) *)

p1[x_,y_]=D[PES[x,y],x];
p3[x_,y_]=D[PES[x,y],y];
a13[x_,y_]= p1[x,y]-p3[x,y];
congr13=ContourPlot[a13[x,y],{x,-1.0,6.2},{y,2.3,9.3},
ContourShading->False,PlotPoints->55,Contours->{0.0},ContourStyle->{Thickness[0.004]}]

(* The flat line in the harmonic potential of the FK model  *)

PES1[x_,y_]=0.5* (((y-4./3.*Pi -x))^2)
con1line=ContourPlot[PES1[x,y],{x,-1.0,5.95},{y,2.3,9.3},Contours->{0.000999},  ContourShading->False,PlotPoints->79,ContourStyle->{Dashed,Thickness[0.005]},FrameStyle->Directive[20],PlotPoints->49,FrameLabel->{"x1","x2"}]

(* Make some symbols for the Figure *)

xx=Show[Graphics[{Text[Style["x\*SubscriptBox[\(\),\(1\)]",Large],{4.341,1.1}],
Text[Style["x\*SubscriptBox[\(\),\(2\)]",Large],{8.512,1.1}],Text[Style["Min",Large],{0.50,6.24}],
Text[Style["SP",Large],{4.750,6.98}],
PointSize[Large],Black,Point[{4.688649,7.87772}],
PointSize[Large],Black,Point[{0.718187,5.564998}]},
  FormatType->StandardForm,PlotRange->{{1.0,3.1},{1.0,3.1}}]];

Fig1=Show[conk,xx,congr13,con1line]

Export["FKmodelx1x2PES.pdf",Fig1,ImageResolution->Automatic,
ImageSize->Automatic]