2 added 3 characters in body
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We can not take the eps too small, because there is a stop because of the divergence of the solution. Empirically, I picked up eps = 0.005. Then there is a solution, but it is not like expected - it's a function similar to Bessel's function.

eps = 5*10^-3;end = 12;
a = 1.9123;
b = -28.9815;
d = 1;
A = 1.6;
gamma = 1;
chi = 3.5;
j[x_, r_] := 2 A^2 r x
g0[x_, r_] := 
 a + b/2 + 3 d/4 + A^2 (b + d) (x^2 + r^2) + 
  d A^4 ((x^2 + r^2)^2 + 4 x^2 r^2)
g1[x_, r_] := j[x, r] (b + 2 d) + 4 d A^4 r x (r^2 + x^2)
U[0][x_] := x/Sqrt[x^2 + 2]
q[x_] := U[0][x]^2
FNB2 = Interpolation[
   Table[{r, 
     NIntegrate[(q[x]*x*
        Exp[-A^2 (r^2 + x^2)] (g0[x, r] BesselI[0, j[x, r]] - 
          g1[x, r] BesselI[1, j[x, r]])), {x, 0, Infinity}]}, {r, eps,
      end, .005}]];
U[1] = NDSolveValue[{u''[r] + u'[r]/r - 2*u[r]u[r]/r^2 + u[r] - 
     chi*(u[r])^(45) - (2 Pi)^(3/2)/A u[r]*FNB2[r] == 0, 
   u'[eps] == 1.1, u[eps] == 0}, u, {r, eps, end}]

{Plot[U[1][r], {r, eps, end}, PlotRange -> All], 
 Plot[FNB2[r], {r, eps, end}, PlotRange -> All]}

fig1fig1

We can not take the eps too small, because there is a stop because of the divergence of the solution. Empirically, I picked up eps = 0.005. Then there is a solution, but it is not like expected - it's a function similar to Bessel's function.

eps = 5*10^-3;end = 12;
a = 1.9123;
b = -28.9815;
d = 1;
A = 1.6;
gamma = 1;
chi = 3.5;
j[x_, r_] := 2 A^2 r x
g0[x_, r_] := 
 a + b/2 + 3 d/4 + A^2 (b + d) (x^2 + r^2) + 
  d A^4 ((x^2 + r^2)^2 + 4 x^2 r^2)
g1[x_, r_] := j[x, r] (b + 2 d) + 4 d A^4 r x (r^2 + x^2)
U[0][x_] := x/Sqrt[x^2 + 2]
q[x_] := U[0][x]^2
FNB2 = Interpolation[
   Table[{r, 
     NIntegrate[(q[x]*x*
        Exp[-A^2 (r^2 + x^2)] (g0[x, r] BesselI[0, j[x, r]] - 
          g1[x, r] BesselI[1, j[x, r]])), {x, 0, Infinity}]}, {r, eps,
      end, .005}]];
U[1] = NDSolveValue[{u''[r] + u'[r]/r - 2*u[r]/r^2 + u[r] - 
     chi*(u[r])^(4) - (2 Pi)^(3/2)/A u[r]*FNB2[r] == 0, 
   u'[eps] == 1.1, u[eps] == 0}, u, {r, eps, end}]

{Plot[U[1][r], {r, eps, end}, PlotRange -> All], 
 Plot[FNB2[r], {r, eps, end}, PlotRange -> All]}

fig1

We can not take the eps too small, because there is a stop because of the divergence of the solution. Empirically, I picked up eps = 0.005. Then there is a solution, but it is not like expected - it's a function similar to Bessel's function.

eps = 5*10^-3;end = 12;
a = 1.9123;
b = -28.9815;
d = 1;
A = 1.6;
gamma = 1;
chi = 3.5;
j[x_, r_] := 2 A^2 r x
g0[x_, r_] := 
 a + b/2 + 3 d/4 + A^2 (b + d) (x^2 + r^2) + 
  d A^4 ((x^2 + r^2)^2 + 4 x^2 r^2)
g1[x_, r_] := j[x, r] (b + 2 d) + 4 d A^4 r x (r^2 + x^2)
U[0][x_] := x/Sqrt[x^2 + 2]
q[x_] := U[0][x]^2
FNB2 = Interpolation[
   Table[{r, 
     NIntegrate[(q[x]*x*
        Exp[-A^2 (r^2 + x^2)] (g0[x, r] BesselI[0, j[x, r]] - 
          g1[x, r] BesselI[1, j[x, r]])), {x, 0, Infinity}]}, {r, eps,
      end, .005}]];
U[1] = NDSolveValue[{u''[r] + u'[r]/r - u[r]/r^2 + u[r] - 
     chi*(u[r])^(5) - (2 Pi)^(3/2)/A u[r]*FNB2[r] == 0, 
   u'[eps] == 1.1, u[eps] == 0}, u, {r, eps, end}]

{Plot[U[1][r], {r, eps, end}, PlotRange -> All], 
 Plot[FNB2[r], {r, eps, end}, PlotRange -> All]}

fig1

1
source | link

We can not take the eps too small, because there is a stop because of the divergence of the solution. Empirically, I picked up eps = 0.005. Then there is a solution, but it is not like expected - it's a function similar to Bessel's function.

eps = 5*10^-3;end = 12;
a = 1.9123;
b = -28.9815;
d = 1;
A = 1.6;
gamma = 1;
chi = 3.5;
j[x_, r_] := 2 A^2 r x
g0[x_, r_] := 
 a + b/2 + 3 d/4 + A^2 (b + d) (x^2 + r^2) + 
  d A^4 ((x^2 + r^2)^2 + 4 x^2 r^2)
g1[x_, r_] := j[x, r] (b + 2 d) + 4 d A^4 r x (r^2 + x^2)
U[0][x_] := x/Sqrt[x^2 + 2]
q[x_] := U[0][x]^2
FNB2 = Interpolation[
   Table[{r, 
     NIntegrate[(q[x]*x*
        Exp[-A^2 (r^2 + x^2)] (g0[x, r] BesselI[0, j[x, r]] - 
          g1[x, r] BesselI[1, j[x, r]])), {x, 0, Infinity}]}, {r, eps,
      end, .005}]];
U[1] = NDSolveValue[{u''[r] + u'[r]/r - 2*u[r]/r^2 + u[r] - 
     chi*(u[r])^(4) - (2 Pi)^(3/2)/A u[r]*FNB2[r] == 0, 
   u'[eps] == 1.1, u[eps] == 0}, u, {r, eps, end}]

{Plot[U[1][r], {r, eps, end}, PlotRange -> All], 
 Plot[FNB2[r], {r, eps, end}, PlotRange -> All]}

fig1