In concentration field

The cubic amplitude equations:

Here, the quadratic term is from a resonance, k₁= -k₂ -k₃;
the cubic terms are from self- and cross-coupling.

It evolves three triplets.

To split the real and imaginary parts, let
One phase equation is

To seek stationary solutions: for h > 0 or h < 0, respectively, and then, h cos = p, a positive value. (h=0 allows A=-A, which kills hex)

Three moduli equations read.

Its stationary solutions:

• One trivial uniform solution,
• one stripes (supercritical),
• Two hexagonal solution (H+ and H-), and
• one mixed mode solution.

To check the stability of each branches, find SS first and then calculate the most positive eigenvalue.

1. Two hexs are

   The Jacobian matrix for hex is

   Eigenvalues are

   The hex bif is subcritical. Its limit point (the most left point which can be reached) is

   

   The three eigenvalues at the limit-point are

   

   • The lower branch (unstable always)exsists from mu1 to 0. The eigenvalues are

     

   • The higher branch exsists from mu1 to +infinity. It change stability at . The eigenvalues are

     

2. Stripes perpenticular to k₁ is . One eigenvalues switches sign at

   

   The overlapping range for g1 < g2 is .

   The reverse case stripes are always unstable. H+ is born stable, and always stable.

3. One mixed mode is

   

   which exsits when g2 > g1 for mu>mu₂^stp. At the starting point, lambda2=0.

   When the mixed mode meets the hex+, hex+ loses stability. At this special point, lambda1=lambda2=0.

   

   

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   Q & A

   1. Bifurcation to hexagons is supercritical or subcritical?
      It is subcritical.
   2. Bifurcation to stripes is supercritical or subcritical?
      It is supercritical in the cubic amplitude equations.
   3. Can the bifurcation to stripes turn to subcritical?
      Yes, In the quintic amplitude equations, both bifurcations to hexagons and stripes are subcritical.
   4. Is there any bistability between stripes and hexagons?
      Yes. the bistable region starts from mu₂^stp to mu₂^hex.
   5. Is it possible to have a gap between stripes and hexagons? If possible, what happens in the gap?
      Yes, a gap can be found in the quintic amplitude equation. In the gap neither stripes nor hexagons are unstable, instead, the mixed mode solution (we named them “beans”) becomes stable.
   6. Have you seen bistability between hexagons and beans?
      Yes. before hexagons loose their stability beans already become stable, which indicates a bistability between hexagons and beans.
   7. How do beans look like? Are they stable?
      This is a bean pattern (in color R=B=213,G=64 ), which can be stable in the quintic amplitude equations at least.

      

Square patterns selection