I have four known functions, $u_i(x)$, and I suspect that all these four functions satisfy an operator, through a common force function, or at least a common force function type $(h(x)$, with some parameters, where each parameter could be adapted to the operator.
The "native" (unparametrized) operator is:
\begin{equation} \label{NLS_operator} P_i:=-\frac{1}{2}\frac{d^2}{dx^2}+|u_i(x)|^2 \end{equation}
$i=1,2,3,4$, the index of the respective known function. These functions are Fourier series ($L^2$ functions), and they are very long. See them at the bottom of this post. I would like to prepare a system of ODEs which are formed by the equation:
\begin{equation} P_{\beta_i} u_i(x)=h_i(x) \end{equation} $\beta$ is the parameter that we multiply to the operator (for simplicity) or we can consider it as a nonlinear term (function) which we add to the operator and $h(x)$ are the unknown force functions, $h$.
Here the force function $h$ should be in the form $$h(x)=Ae^{-\lambda(x-x_0)^2}.$$
Since I have four known $u(x)=\{u_1, u_2, u_3, u_4\}$, I should get:
$$P_{\beta} u_1(x)=h(x)\\ P_{\beta} u_1(x)=h(x)\\P_{\beta} u_3(x)=h(x)\\P_{\beta} u_4(x)=h(x)$$
With $$h(x)=Ae^{-\lambda(x-x_0)^2}$$
so we have three unknowns, $A$, $\lambda$ and $x_0$ plus the unknown parameter $\beta$. This makes 4 unknowns,and four linear equations.
Example 1: Let for example $u_1=e^{2it}$, $u_2=e^{4it}$, $u_3=e^{-3it}$ and $u_4=e^{-5it}$. Then we get the prepared inputs :
d[t_] = E^(2*I*t)
D2d[t_] = D[d[t],{t,2}]
The second
g[t_] = E^(4*I*t)
D2g[t_] = D[g[t],{t,2}]
the third:
b[t_] = E^(-3*I*t)
D2b[t_] = D[b[t],{t,2}]
and the fourth:
u[t_] = E^(-5*I*t)
D2u[t_] = D[u[t],{t,2}]
Then we arrange this such that we can solve for $\lambda$, $\beta$ , $x_0$ and $A$.
For this use Solve do:
Solve[{-1/2*D2d[t] + (Abs[d[t]])^2*d[t] ==
A*E^(-λ (t - B))^2/β,
-1/2*D2g[t] + (Abs[g[t]])^2*g[t] ==
A*E^(-λ (t - B))^2/β,
-1/2*D2b[t] + (Abs[b[t]])^2*b[t] ==
A*E^(-λ (t - B))^2/β,
-1/2*D2u[t] + (Abs[u[t]])^2*u[t] ==
A*E^(-λ (t - B))^2/β}, {B, A, β, λ}]
With this I get NO result.
Here is the actual question I am seeking an answer for
Let the given functions be $u_1=d(t),u_2=g(t),u_3=b(t), u_4=u(t)$ given below
d[t_] = 139.85 + (15.8404 + 4.76022 I) E^(-I t) + (15.8404 -
4.76022 I) E^(I t) + (4.64917 -
3.3024 I) E^(-2 I t) + (4.64917 +
3.3024 I) E^(2 I t) + (7.42191 -
0.300123 I) E^(-3 I t) + (7.42191 +
0.300123 I) E^(3 I t) + (0.340877 -
2.54665 I) E^(-4 I t) + (0.340877 +
2.54665 I) E^(4 I t) + (7.72422 +
6.71332 I) E^(-5 I t) + (7.72422 -
6.71332 I) E^(5 I t) + (3.16511 -
9.14479 I) E^(-6 I t) + (3.16511 +
9.14479 I) E^(6 I t) + (3.31502 +
2.18874 I) E^(-7 I t) + (3.31502 -
2.18874 I) E^(7 I t) + (9.31827 +
6.94538 I) E^(-8 I t) + (9.31827 -
6.94538 I) E^(8 I t) + (5.87173 +
18.8341 I) E^(-9 I t) + (5.87173 -
18.8341 I) E^(9 I t) + (8.75949 +
14.7107 I) E^(-10 I t) + (8.75949 -
14.7107 I) E^(10 I t) + (19.2903 +
7.78329 I) E^(-11 I t) + (19.2903 -
7.78329 I) E^(11 I t) - (3.39063 -
18.5502 I) E^(-12 I t) - (3.39063 +
18.5502 I) E^(12 I t) + (3.58427 +
18.7438 I) E^(-13 I t) + (3.58427 -
18.7438 I) E^(13 I t) + (2.66129 +
20.0781 I) E^(-14 I t) + (2.66129 -
20.0781 I) E^(14 I t) - (8.46335 -
9.96867 I) E^(-15 I t) - (8.46335 +
9.96867 I) E^(15 I t) - (8.50037 -
1.20377 I) E^(-16 I t) - (8.50037 +
1.20377 I) E^(16 I t) - (1.36102 -
16.9315 I) E^(-17 I t) - (1.36102 +
16.9315 I) E^(17 I t) - (5.78964 -
2.96094 I) E^(-18 I t) - (5.78964 +
2.96094 I) E^(18 I t) + (2.14681 +
8.29635 I) E^(-19 I t) + (2.14681 -
8.29635 I) E^(19 I t) - (3.91145 -
10.7712 I) E^(-20 I t) - (3.91145 + 10.7712 I) E^(20 I t);
g[t_] = 13.87 + (9.25477 + 1.51787 I) E^(-I t) + (9.25477 -
1.51787 I) E^(I t) + (1.08996 +
0.300406 I) E^(-2 I t) + (1.08996 -
0.300406 I) E^(2 I t) - (1.96638 +
1.26469 I) E^(-3 I t) - (1.96638 -
1.26469 I) E^(3 I t) + (0.218112 -
0.308038 I) E^(-4 I t) + (0.218112 +
0.308038 I) E^(4 I t) + (1.90783 +
1.17281 I) E^(-5 I t) + (1.90783 -
1.17281 I) E^(5 I t) + (1.09081 +
0.353231 I) E^(-6 I t) + (1.09081 -
0.353231 I) E^(6 I t) + (0.173557 -
1.10369 I) E^(-7 I t) + (0.173557 +
1.10369 I) E^(7 I t) + (0.224736 -
0.475649 I) E^(-8 I t) + (0.224736 +
0.475649 I) E^(8 I t) + (0.215461 +
0.922754 I) E^(-9 I t) + (0.215461 -
0.922754 I) E^(9 I t) + (0.127993 +
0.559526 I) E^(-10 I t) + (0.127993 -
0.559526 I) E^(10 I t) + (0.557846 -
0.59431 I) E^(-11 I t) + (0.557846 +
0.59431 I) E^(11 I t) + (0.801299 -
0.286539 I) E^(-12 I t) + (0.801299 +
0.286539 I) E^(12 I t) + (0.301617 +
0.825404 I) E^(-13 I t) + (0.301617 -
0.825404 I) E^(13 I t) + (0.115336 +
0.833234 I) E^(-14 I t) + (0.115336 -
0.833234 I) E^(14 I t) + (0.786769 +
0.0749826 I) E^(-15 I t) + (0.786769 -
0.0749826 I) E^(15 I t) + (1.03796 -
0.0574653 I) E^(-16 I t) + (1.03796 +
0.0574653 I) E^(16 I t) + (0.302552 +
0.238826 I) E^(-17 I t) + (0.302552 -
0.238826 I) E^(17 I t) + (0.032363 +
0.135126 I) E^(-18 I t) + (0.032363 -
0.135126 I) E^(18 I t) + (0.952115 -
0.0508812 I) E^(-19 I t) + (0.952115 +
0.0508812 I) E^(19 I t) + (1.59932 +
0.164806 I) E^(-20 I t) + (1.59932 - 0.164806 I) E^(20 I t);
b[t_] = 22.2253 + (6.05951 + 1.26288 I) E^(-I t) + (6.05951 -
1.26288 I) E^(I t) - (2.28516 +
0.718175 I) E^(-2 I t) - (2.28516 -
0.718175 I) E^(2 I t) + (3.48496 +
1.91013 I) E^(-3 I t) + (3.48496 -
1.91013 I) E^(3 I t) + (1.02469 -
1.28294 I) E^(-4 I t) + (1.02469 +
1.28294 I) E^(4 I t) + (1.18499 +
0.102604 I) E^(-5 I t) + (1.18499 -
0.102604 I) E^(5 I t) + (1.05572 +
0.532558 I) E^(-6 I t) + (1.05572 -
0.532558 I) E^(6 I t) + (1.43148 -
1.00592 I) E^(-7 I t) + (1.43148 +
1.00592 I) E^(7 I t) + (1.14346 +
1.19477 I) E^(-8 I t) + (1.14346 -
1.19477 I) E^(8 I t) + (1.43917 +
0.795927 I) E^(-9 I t) + (1.43917 -
0.795927 I) E^(9 I t) + (1.91234 +
2.80903 I) E^(-10 I t) + (1.91234 -
2.80903 I) E^(10 I t) + (3.70871 +
2.25491 I) E^(-11 I t) + (3.70871 -
2.25491 I) E^(11 I t) + (3.20857 +
0.708758 I) E^(-12 I t) + (3.20857 -
0.708758 I) E^(12 I t) - (0.407118 -
0.0128365 I) E^(-13 I t) - (0.407118 +
0.0128365 I) E^(13 I t) + (0.747326 +
0.854073 I) E^(-14 I t) + (0.747326 -
0.854073 I) E^(14 I t) - (1.65033 -
1.69294 I) E^(-15 I t) - (1.65033 +
1.69294 I) E^(15 I t) - (1.70632 -
1.04809 I) E^(-16 I t) - (1.70632 +
1.04809 I) E^(16 I t) + (1.05899 +
1.95888 I) E^(-17 I t) + (1.05899 -
1.95888 I) E^(17 I t) + (1.36781 -
0.334041 I) E^(-18 I t) + (1.36781 +
0.334041 I) E^(18 I t) + (1.98706 -
0.0544715 I) E^(-19 I t) + (1.98706 +
0.0544715 I) E^(19 I t) + (0.890154 +
0.772225 I) E^(-20 I t) + (0.890154 - 0.772225 I) E^(20 I t);
u[t_] = 93.2259 + (9.87447 + 1.23806 I) E^(-I t) + (9.87447 -
1.23806 I) E^(I t) - (7.28392 +
0.41437 I) E^(-2 I t) - (7.28392 -
0.41437 I) E^(2 I t) + (12.8921 -
2.58731 I) E^(-3 I t) + (12.8921 +
2.58731 I) E^(3 I t) - (2.22197 -
2.18978 I) E^(-4 I t) - (2.22197 +
2.18978 I) E^(4 I t) + (11.0601 -
0.575458 I) E^(-5 I t) + (11.0601 +
0.575458 I) E^(5 I t) - (1.22048 -
2.68096 I) E^(-6 I t) - (1.22048 +
2.68096 I) E^(6 I t) + (6.47151 +
1.21344 I) E^(-7 I t) + (6.47151 -
1.21344 I) E^(7 I t) + (0.260353 +
4.33147 I) E^(-8 I t) + (0.260353 -
4.33147 I) E^(8 I t) + (0.61904 +
3.37889 I) E^(-9 I t) + (0.61904 -
3.37889 I) E^(9 I t) - (0.585492 -
2.98757 I) E^(-10 I t) - (0.585492 +
2.98757 I) E^(10 I t) + (1.95162 +
8.81749 I) E^(-11 I t) + (1.95162 -
8.81749 I) E^(11 I t) + (0.534066 +
11.0168 I) E^(-12 I t) + (0.534066 -
11.0168 I) E^(12 I t) - (1.45227 -
11.017 I) E^(-13 I t) - (1.45227 +
11.017 I) E^(13 I t) - (0.987428 -
6.69947 I) E^(-14 I t) - (0.987428 +
6.69947 I) E^(14 I t) - (4.88124 -
6.62883 I) E^(-15 I t) - (4.88124 +
6.62883 I) E^(15 I t) - (2.56445 -
4.71859 I) E^(-16 I t) - (2.56445 +
4.71859 I) E^(16 I t) - (4.16251 -
5.0285 I) E^(-17 I t) - (4.16251 +
5.0285 I) E^(17 I t) - (3.24296 -
2.11604 I) E^(-18 I t) - (3.24296 +
2.11604 I) E^(18 I t) - (4.22329 -
0.335417 I) E^(-19 I t) - (4.22329 +
0.335417 I) E^(19 I t) + (0.506471 -
0.205546 I) E^(-20 I t) + (0.506471 + 0.205546 I) E^(20 I t);
Then we declare the second differential operator:
D2u[t_] = D[u[t], {t, 2}];
D2b[t_] = D[b[t], {t, 2}];
D2g[t_] = D[g[t], {t, 2}];
D2d[t_] = D[d[t], {t, 2}];
and we prepare the system of algebraic equations:
sys=[{-1/2*D2d[t] + (Abs[d[t]])^2*d[t] ==
A*E^(-λ*I* (t - B))^2/β,
-1/2*D2g[t] + (Abs[g[t]])^2*g[t] ==
A*E^(-λ*I* (t - B))^2/β,
-1/2*D2b[t] + (Abs[b[t]])^2*b[t] ==
A*E^(-λ *I*(t - B))^2/β,
-1/2*D2u[t] + (Abs[u[t]])^2*u[t] ==
A*E^(-λ *I*(t - B))^2/β}, {B, A, β, λ}]
How can I solve this system?
NSolve Does not work, Reduce gives "This system cannot be solved with the methods available to Reduce.",
and Solve seems not to converge..
An idea is to dicretize the functions $u$ into a set of vectors, and then use these discretized functions to identify the force functions $h$. But this I am not sure may work?
With Alex's suggestion, we obtain:
D2u[t_] = D[u[t], {t, 2}];
D2y[t_] = D[y[t], {t, 2}];
D2g[t_] = D[g[t], {t, 2}];
D2d[t_] = D[d[t], {t, 2}];
Du[t_] = D[u[t], {t, 1}];
Dy[t_] = D[y[t], {t, 1}];
Dg[t_] = D[g[t], {t, 1}];
Dd[t_] = D[d[t], {t, 1}];
NSolve[{-1/2*D2d[t] + (Abs[d[t]])^2*d[t] + I*u[t]*Du[t] ==
A*E^(-λ *I*(t - B))^2/β,
-1/2*D2g[t] + (Abs[g[t]])^2*g[t] + I*g[t]*Dg[t] ==
A*E^(-λ*I* (t - B))^2/β,
-1/2*D2y[t] + (Abs[y[t]])^2*y[t] + I*y[t]*Dy[t] ==
A*E^(-λ *I*(t - B))^2/β,
-1/2*D2u[t] + (Abs[u[t]])^2*u[t] + I*d[t]*Dd[t] ==
A*E^(-λ *I*(t - B))^2/β},
{B, A, β, λ}]
However, there are no changes currently, NSolve is running for hours.
A*E^(-\[Lambda] (t - B))^2/\[Beta]set equal to four different expressions. There would certainly be no solutions for{B, A, \[Beta], \[Lambda]}as the variables, except possibly for special values oft(probably not, since it reduces to three equations in one unknown). -- You should change thee^toE^in the question. Copying nonworking is discouraging. $\endgroup$