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This is a corrected version of pdeProb2.mw, in which we examine the 1-D classical burgers equation, and find an asymptotic steady state in the solution fields u, v which is not reached by a solution via numerical simulation.
NOTE: When generating and displaying PLOTS AT HIGH RESOLUTION, do not use p1 := plot(bla, etc); i.e. do not end with a semicolon. Instead, end with a colon viz p1 := plot(bla, etc): which sends the result to p1 instead of generating an excess memory use message. Then create the plot by executing p1; i.e. end the assigned p1 with a semicolon to display the graphics result.
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We load the MAPLE Physics package from the MapleCloud, in order to support solutions using pdsolve().
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Start of definition of problem:
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![[CanonicalCoordinates, ChangeSymmetry, CharacteristicQ, CharacteristicQInvariants, ConservedCurrentTest, ConservedCurrents, ConsistencyTest, D_Dx, DeterminingPDE, Eta_k, Euler, FromJet, FunctionFieldSolutions, InfinitesimalGenerator, Infinitesimals, IntegratingFactorTest, IntegratingFactors, InvariantEquation, InvariantSolutions, InvariantTransformation, Invariants, Laplace, Library, PDEplot, PolynomialSolutions, ReducedForm, SimilaritySolutions, SimilarityTransformation, Solve, SymmetryCommutator, SymmetryGauge, SymmetrySolutions, SymmetryTest, SymmetryTransformation, TWSolutions, ToJet, ToMissingDependentVariable, build, casesplit, charstrip, dchange, dcoeffs, declare, diff_table, difforder, dpolyform, dsubs, mapde, separability, splitstrip, splitsys, undeclare]](/view.aspx?sf=228741_question/8ff8165fdd5e2c1c7de14b55e7f07027.gif)
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Start of definition of problem:
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with(PDETools); with(CodeTools);with(plots);
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![[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot, display, dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d, inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d, listdensityplot, listplot, listplot3d, loglogplot, logplot, matrixplot, multiple, odeplot, pareto, plotcompare, pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d, polyhedra_supported, polyhedraplot, rootlocus, semilogplot, setcolors, setoptions, setoptions3d, shadebetween, spacecurve, sparsematrixplot, surfdata, textplot, textplot3d, tubeplot]](/view.aspx?sf=228741_question/542ae583c6be2d03267b1b7de027b863.gif)
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Two 1-D coupled Burgers equations - semiclassical case O(1), O(  ) : retain O(1) only for u(x,t) and O(1), O(  ) for v(x,t):
In the quantum case, there are two coupled quantum Burgers equations, which each include the quantum potential terms. As in the classical case above, we apply constant external forces  and  . Our aim is to display the profiles of  and  as strings on  space.
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#hBar := 'hBar': m := 'm':Fu := 'Fu': Fv := 'Fv': # define constants
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hBar:= 1:m:= 1:Fu:= 0.2:Fv:= 0.1: # set constant values - same as above ...consider reducing
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Notice that we set 
At O(  ) the real quantum potential term is zero, leaving the classical expression:
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pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;
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As in the classical case above, the temporal and spatial derivative are each of order 1; so only one initial condition and one boundary condition are required for this part of the semiclassical equations.
On the otherhand, the imaginary quantum potential equation for v(x,t) has only O(  ) terms so together the pair of equations for  are semiclassical:
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pdev := diff(v(x,t),t)+u(x,t)/m*(diff(v(x,t),x))-hBar*(diff(u(x,t),x$2))/(2*m)+v(x,t)*(diff(u(x,t),x))/m = Fv;
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By inspection of the derivatives in above two equations we now set up the ICs and BCs for  and  Note that the above second order spatial derivative requires a 1st order derivative boundary condition as defined below.
The quantum initial and boundary conditions are similar to the classical case, but also comprise additional boundary condition terms for  and for  , notably a 1st derivative reflective BC term for  .
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ICu:={u(x,0) = 0.1*sin(2*Pi*x)};# initial conditions for PDE pdeu
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ICv:={v(x,0) = 0.2*sin(Pi*x)};# initial conditions for PDE pdev
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BCu := {u(0,t) = 0.5*(1-cos(2*Pi*t)),D[1](u)(1,t) = 0}; # boundary conditions for PDE pdeu: note the reflective derivative term D[1](u)
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)(1, t) = 0}](/view.aspx?sf=228741_question/b020787a7beaf15c7121f802dabc3386.gif)
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BCv := {v(0,t) = 0.5*sin(2*Pi*t), v(1,t)=-0.5*sin(2*Pi*t)}; # boundary conditions for PDE pdev
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)(1, t) = 0}](/view.aspx?sf=228741_question/27ae0383019cc9889c96189b0680b11f.gif)
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This set of equations and conditions can now be solved numerically.
The above IC and BC are both  at  and thus consistent.
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pdu := pdsolve({pdeu,pdev},{IC,BC},numeric, time = t,range = 0..1,spacestep = 1/66,timestep = .1);
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Here is the 3D plot of u(x,t):
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T := 3; p1 := pdu:-plot3d(u,t=0..T,numpoints = 2000,x=0.0..2, shading = zhue,orientation=[-146,54,0],scaling = constrained, title = print("Figure 1",u(x, t), numeric));
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