Multiple Plots over non-consecutive integer parameter values

Question

The following is a simple illustration of the difficulty I have encountered with multiple plots on the same diagram. The reason for trying to overcome (understand) the difficulty is more involved and is clarified after the illustration.

Consider two definitions of the same function as follows:

g[m_] = m E^(-m x^2)

gi[n_Integer] = n E^(-n x^2)

A plot of g[m] for several values of m simultaneously is possible with

Plot[g[m] /. m -> {1, 2, 3}, {x, -3, 3}, PlotRange -> All]

which results in

or ,

using Evaluate[g[m] /. m -> {1, 2, 3}, {x, -3, 3}] for the latter.

The same application of the Plot function, however, does not work with the second definition

Plot[gi[n] /. n -> {1, 2, 3}, {x, -3, 3}, PlotRange -> All]

Inserting Evaluate[g[m] /. m -> {1, 2, 3}, {x, -3, 3}] makes a difference and will make the plot lines distinct, however it still does not work with the second function gi[n], so I have left it out for simplicity.

The questions:

1. What causes the difference in plotting g[m_] and gi[n_Iinteger] as above?
2. How can Plot work with the second function definition gi[n_Integer], just as it does with the first g[m_]?

Clarification: I am aware of other methods for producing multiple plots in one diagram, such as constructing lists with Table and similar. The reason for pursuing the above approach for multiple plots with a list of values for a parameter, specified as in gi[n_Iinteger], is the possibility to extract non-consecutive terms from solutions of differential equations in the form of a sum. In the following example from the Mathematica Documentation

eqn = I  D[\[Psi][x, t], t] == -2 D[\[Psi][x, t], {x, 2}];
f[x_] := -350 + 155 x - 22 x^2 + x^3
sol = DSolve[{eqn, \[Psi][5, t] == 0, \[Psi][10, t] == 0, \[Psi][x, 2] == f[x]}, \[Psi], {x, t}]

the solution is a function as an infinite sum

,

from which a partial sum is extracted with

u[k_Integer] = \[Psi] /. Activate[First[sol] /. \[Infinity] -> k]

The above function is used to produce multiple plots with Table

...Table[Plot[{u[k][x, 2], f[x]}, {x, 5, 10}, ImageSize -> 200], {k, 4}]...

however, the plots are for consecutive values of k and not readily adaptable for extraction of individual terms and sums for non-consecutive k-values. Therefore, the summary question is

3. How can multiple plots be accomplished as illustrated for g[m_] above with a function specified as u[k_Integer]?

I shall be grateful for any help with these questions.

Clarification edit shifting emphasis to question 3. as the ultimate goal:

The first comment by Bob Hanlon suggested one successful approach to question 2., using Map (thank you Bob)! Unfortunately, I was unable to apply the same method to u[k_Integer] above - question 3. Consequently, I'd be grateful for responses related to the latter.

Answer 1

Clear["Global`*"]

eqn = I D[ψ[x, t], t] == -2 D[ψ[x, t], {x, 2}];
f[x_] := -350 + 155 x - 22 x^2 + x^3
sol = DSolve[{eqn, ψ[5, t] == 0, ψ[10, t] == 0, ψ[x, 2] == 
     f[x]}, ψ, {x, t}];

u[k_Integer?Positive] = ψ /. Activate[First[sol] /. ∞ -> k];

For k = {2, 4, 7}

Plot[Evaluate[u[#][x, 2] & /@ {2, 4, 7}], {x, 5, 10},
 PlotLegends -> Placed[
   LineLegend[{2, 4, 7}, LegendLabel -> k],
   {.7, .65}],
 Frame -> True,
 FrameLabel -> (Style[#, 14] & /@
    {x, HoldForm[u[k][x, 2]]})]

EDIT: For two parameters

u2[k1_Integer?Positive, k2_Integer?Positive] = ψ /. 
   Activate[First[sol] /. 
     {{K[1], 1, Infinity} :> {K[1], k1, k2}}];

kValues = {{1, 2}, {1, 4}, {3, 7}};

Plot[Evaluate[u2[#[[1]], #[[2]]][x, 2] & /@ kValues],
 {x, 5, 10},
 PlotLegends -> Placed[LineLegend[ToString /@ kValues,
    LegendLabel -> {Subscript[k, 1], Subscript[k, 2]}],
   {.2, .3}],
 Frame -> True,
 FrameLabel ->
  (Style[#, 14] & /@ {x, 
     HoldForm[u2[Subscript[k, 1], Subscript[k, 2]][x, 2]]})]

Answer 2

Consider: g[m] /. m -> {1, 2, 3} This will evaluate to: g[{1,2,3}] and further: {1, 2, 3} E^(-{1, 2, 3} x^2) and finally to: {E^-x^2, 2 E^(-2 x^2), 3 E^(-3 x^2)}

However, gi[n] /. n -> {1, 2, 3} evaluates only to: gi[{1, 2, 3}] and it can not evaluete further because {1,2,3}has a head: List and not "Integer".

If you want to use gi[n] /. n -> {1, 2, 3}, you must give gi the attribute "Listable"

Clear["Global`*"]
SetAttributes[gi, Listable];
gi[n_Integer] = n E^(-n x^2);
Plot[Evaluate[gi[n] /. n -> {1, 2, 3}], {x, -3, 3}, PlotRange -> All]

addendum

The third question is a bit more complex because you have 2 variables and the functions are complex. To make a simple example I will plot the Abs value and define u[k,t,x]:

SetAttributes[u, Listable]
u[k_Integer, t_, 
  x_] := (\[Psi] /. Activate[First[sol] /. \[Infinity] -> k] )[t, x]

Plot3D[Evaluate@Abs[u[{1, 2}, x, y]], {x, -1, 1}, {y, -1, 1}]

Answer 3

Give your function Attribute Listable.

g[m_] = m E^(-m x^2)

gi[n_Integer] = n E^(-n x^2)

Attributes[gi] = {Listable}

gi[n] /. n -> {2, 3, 4}  

gi[{2, 3, 4}]   (*  both yield   *)

(*   {2 E^(-2 x^2), 3 E^(-3 x^2), 4 E^(-4 x^2)}   *)

Many build in functions have this attribute.

Attributes[Sin]

If you don't do, or cancel it again, with Replace you get the whole list as argument of gi, which is not an Integer.

Attributes[gi] = {}

gi[n] /. n -> {2, 3, 4}

(*   gi[{2, 3, 4}]   *)

Without a restriction for argument like _Integer, function g is per se Listable g[{2, 3, 4}]. But

 Attributes[g]    (*   {}   *)

does not show it. Don't know why?