the critical in the KDV equation coefficients

How do you show the critical condition through the KDV equation coefficients? When we have the dispersion coefficient and the density rate, I could not formulate them in Matlab code

a = -42.37 - 26.69i;

b = -0.00165918 + 0.00613671i;

y2 = 1024.4;

c = -5.82 - 15.43i;

d = -0.0056625 + 0.0229537i;

Y = 0.1:0.00001:0.6;

A_real = zeros(size(Y));

for i = 1:length(Y)

A = (a + b - 2 * y2) / (c + d);

A_real(i) = real(A);

end

figure;

plot(Y, A_real, 'b-');

xlabel('Y');

ylabel('Real Part of A');

title('Plot of Real Part of A versus Y');

grid on;

Answer 1

[Shishir Reddy]

Hey Jazz

I understand that you would like to visualize the critical points through the Korteweg-de Vries (KDV) equation coefficients. For this, firstly the “critical condition” has to be defined in the context of the problem.

For instance, a critical condition might be when the real part of A (referring to your code) crosses a specific threshold or changes sign. For this critical condition, here’s how your code can be modified to plot the critical condition points.

critical_Y = [];
critical_A_real = [];

threshold = 0; % critical condition (Assuming zero crossing)

for i = 1:length(Y)
    A = (a + b_- 2 * y2) / (c + d);
    A_real = real(A);
    
    % Checking the critical condition
    if abs(A_real - threshold) < 1e-3 % Tolerance
        critical_Y = [critical_Y, Y(i)];
        critical_A_real = [critical_A_real, A_real];
    end
end

Finally, the critical points can be plotted as follows.

plot(critical_Y, critical_A_real, 'ro');

Ensure that the critical condition definition aligns with the theoretical understanding of the problem. Adjust the threshold and tolerance as needed to capture the relevant points.

I hope this helps.

[Jazz]

thank you