Results

We present here the results of calculation of the matrix element on the resonant manifold

67#9 (3.7)

There are five topologically different configurations for the 68#10 interaction on (3.8) for arbitrary signs of wave vectors in 1D:

1. All wave vectors positive.
2. Positive 69#11 and 70#12, and one of the 71#13 negative.
3. Positive 69#11 and 70#12, and two of the 71#13 negative.
4. 69#11 70#12 with different signs, 71#13 positive.
5. 69#11 70#12 with different signs, and one of 71#13 negative

The results of this calculations is presented below, and summarized in the table at the end of the article. Some of the final expressions are naturally less symmetric than others, because the symmetry of expression reflects the symmetry of the wave-vector setup.

1. The answer is given in [1]
   

   $$=$$ 72#14 73#15 (3.8)

   $$=$$ 74#16 (3.9)

   
   

2. Choose 75#17 to be negative, and 76#18. One can parameterize the resonant manifold by
   

   77#19 (3.10)

   
   
   where 78#20, 79#21 with 80#22

   The result depends upon the sign of 81#23.

   • case when 82#24. In this case 83#25
     

     84#26 (3.11)

     
     

   • case of 85#27. Here 86#28
     

     87#29 (3.12)

     
     

3. zero
4. zero
5. chose 88#30 and 89#31 and positive 69#11, 90#32. Then for any positive 91#33 one can find 92#34 satisfying resonant condition (3.8) to be
   
   93#35
   
   
   
   
   94#36
   
   
   One can see from this parameterization, that 95#37. Choose 96#38. Then there are three variants of relations between 97#39. One of them 98#40 does not fix the relation between 99#41 and 100#42, so there are four different cases of relations between 101#43's for which the fifth order matrix element can be calculated.
   1. If 102#44 then 103#45 and
      
      104#46
      
      

   2. If 105#47 then 103#45 and
      
      106#48
      
      

   3. If 98#40 and 107#49 then
      
      108#50
      
      

   4. If 98#40 and 109#51, then
      
      110#52