Solution to the Transport Equation

transport=D[u[x,t],t]+D[u[x,t],x]==0

transoln=DSolve[{transport,u[x,0]==x*(1/Cosh[x])},u[x,t],{x,t}]

Manipulate[Plot[(t-x)*Sech[t-x],{x,0,100},PlotRange->{0,0.8}],{t,0,100}]

Soliton Solution to the KdV Equation

kdv=D[v[x,t],t]+D[v[x,t],{x,3}]-6*v[x,t]*D[v[x,t],x]==0

Manipulate[Plot[-(1/2)*Sech[(1/2)*(x-t+60)]^2,{x,-50,50},PlotRange->{-0.6,0.05}],{t,0,120,0.1}]

Tricomi Equation

tricomi=D[u[x,y],{x,2}]-x*D[u[x,y],{y,2}]==0

A Particular Solution to the Tricomi Equation (with parameters):

{K*6*y+G*12*x*y+F*(x^3+3*y^2)+H*(x^4+6*x*y^2),-K*3*x^2-G*(4*x^3+6*y^2)-F*3*x^2*y-H*(4*x^3*y+2*y^3)}

Manipulate[StreamDensityPlot[{K*6*y+G*12*x*y+F*(x^3+3*y^2)+H*(x^4+6*x*y^2),-K*3*x^2-G*(4*x^3+6*y^2)-F*3*x^2*y-H*(4*x^3*y+2*y^3)},{x,-10*S,20*S},{y,-20*S,20*S},VectorPoints->Fine,VectorStyle->Red, StreamPoints->Fine,StreamStyle->White,ColorFunction->"TemperatureMap"],{K,-100,100,1},{F,-100,100,1},{G,-100,100,1},{H,-100,100,1},{S,0.1,1}]