Procedure Input Summary:
   - Parameter varied: `b`
   - Values used     : [.3, .6, .9, 1.2]
   - Mathematical role: linear damping coefficient (affects dissipation: -ab)
   - Fixed parameters: a = 1, c = 1, d = .2, epsilon = 1
 

 Procedure Input Summary:
   - Parameter varied: `c`
   - Values used     : [.3, .6, .9, 1.2]
   - Mathematical role: dispersion parameter (affects nonlocal wave spreading)
   - Fixed parameters: a = 1, b = 1, d = .2, epsilon = 1
 

 Procedure Input Summary:
   - Parameter varied: `d`
   - Values used     : [.1, .2, .3, .4]
   - Mathematical role: material damping (viscous dissipation, scaled with a²)
   - Fixed parameters: a = 1, b = 1, c = 1, epsilon = 1
 

 Procedure Input Summary:
   - Parameter varied: `epsilon`
   - Values used     : [.5, 1.0, 1.5, 2.0]
   - Mathematical role: nonlinear interaction strength (affects higher-order terms)
   - Fixed parameters: a = 1, b = 1, c = 1, d = .2,
 

Step 1: Declare l and b as free variables for the 3D plot.
Step 2: Set fixed values for remaining parameters.
Step 3: Define the 3D gain function G(l,b) with fixed a,c,d and variable l,b.
Step 4: Create a 3D surface plot of G(l,b).
Step 5: Display the 3D surface plot.

Step 1: Define the wave number variable l and parameter range for plotting.
Step 2: Define constants and gain parameters used in the equation.
Step 3: Define gain function G(l,a) using corrected version with l instead of k.

Step 4: Create individual plots for a = 1, 0.5, 0.3 over wave number l.
Step 5: Overlay all plots together with proper labels, legend, and styling.
Step 6: Display the final combined plot.

Step 1: Declare symbolic variables used in the gain expression.
Step 2: Define the discriminant D, which arises from the stability analysis of a nonlinear wave.
         This expression comes from the square root part of a complex frequency solution w.
Step 3: Define the gain function G(k,a).
         Gain is defined as the imaginary part of the square root of D.
         If Discr is negative, the square root becomes imaginary , this corresponds to instability.
Step 4: Physically, G(k,a) > 0 implies that perturbations with wave number k grow exponentially.
         This is the signature of modulation instability in nonlinear wave systems.
         The region where G > 0 is the unstable region in the (k,a)-parameter space.
Step 5: Generate a 3D plot over k ∈ [-3, 3] and a ∈ [0.1, 2] to visualize the instability region.
Step 6: Display the gain surface. Regions with high gain indicate fast-growing perturbations.

Step 1: Define the variables: wave number l and parameter a.
Step 2: Define the MI gain function G(l,a) = 2 * Im(sqrt(l^4 - 2*a*l^2)).
Step 3: Create a 3D plot over a range of l and a.
Step 4: Display the 3D plot.

Step 1: Define the wave number variable l and parameter range for plotting.
Step 2: Define values for the parameter a: a = 1, 0.5, and 0.3.
Step 3: Define the growth rate function G(l) = 2 * Im(sqrt(l^4 - 2*a*l^2)).
Step 4: Create individual plots for each value of a.
Step 5: Overlay all plots together with proper labels, legend, and styling.
Step 6: Display the final combined plot.