Solving constrained BKP hierarchy by boson–fermion correspondence

Solutions of ( k , m ) –constrained BKP hierarchy L k = ( L k ) ≥ 0 + ∑ i = 1 m ( q 1 , i ∂ − 1 ( q 2 , i ) x − q 2 , i ∂ − 1 ( q 1 , i ) x ) with k ∈ Z ≥ 0 odd are investigated by boson–fermion correspondence. Firstly, ( k , m ) –constrained BKP hierarchy is converted into the fermionic form Ω k ( τ 0 ⊗ τ 0 ) = 1 2 ∑ i = 1 m ( τ 2 , i ⊗ τ 1 , i − τ 1 , i ⊗ τ 2 , i ) . Then we determine the conditions for the fermionic field β , which keeps the fermionic form of constrained BKP hierarchy after the transformations τ α → β τ α for α = 0 , { 1 , i } and { 2 , i } . Finally based upon this, the corresponding solutions are given by boson–fermion correspondence. • Darboux transformations of Constrained BKP hierarchy are treated in the fermionic forms. • Avoid the choosing the special group elements in vacuum expectation value of the BKP tau functions. • Provide an effective method to solve (k,m)-constrained BKP hierarchy for any k and m.