15 -3 -2 -1  0  1  2  3 -15 -10 -5  0  5  10  15 u n,m (x) x (0,1) (0,2), (2,0) (2,1), (0,3) -5 -4 -3 -2 -1  0  1  2  3  4 -15 -10 -5  0  5  10  15 u n,m (x) x (1,0) (1,1) (3,0), (1,2) FIG. 3: (Color online) Top panel: the displacement field u$_{nm}$ ( x ) for ( n, m ) = (0 , 1) , (0 , 2) and (0 , 3) in the strong corre- lation limit. Bottom panel: the same for ( n, m ) = (1 , 1) , (1 , 2) and (2 , 1). The thin solid lines represent the density profile. The large value of the displacement field for x ∼ 0 does not have a physical significance since the density is exponentially small in that region. where ψ$_{0}$ and φ$_{0}$ are the ground-state wave function in the center of mass and relative channel respectively. The interaction kernel K ( x, x ′ ) is given by We now have all the input that is necessary to set up and solve the fourth-order integro-differential equation (61). T$_{0}$ ( x ) = 2 m ∫ dy { [ $_{φ}$′ $_{0}$( x − y ) ψ$_{0}$ ( x + y 2 ) + 1 2 φ$_{0}$ ( x − y ) $_{ψ}$′ 0 ( x + y 2 )] 2 − 1 4 ∂ 2 x [ φ 2 $_{0}$( x − y ) ψ 2 0 ( x + y 2 )] } (110) K ( x, x ′ ) = − 2( x − x ′ ) 2 + a 2 [( x − x ′ ) 2 + a $^{2}$]5 / $_{2}$φ 2 $_{0}$( x − x ′ ) ψ 2 0 ( x + x ′ 2 ) . (111)