Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation
aa r X i v : . [ n li n . S I] A ug Constructing Solvable Models of Vector Non-linear Schr ¨ o dinger Equation withBalanced Loss and Gain via Non-unitary transformation Pijush K. Ghosh ∗ Department of Physics, Siksha-Bhavana, Visva-Bharati University, Santiniketan, PIN 731 235, India. (Dated: August 28, 2020)We consider vector Non-linear Schr¨ o dinger Equation(NLSE) with balanced loss-gain(BLG), linearcoupling(LC) and a general form of cubic nonlinearity. We use a non-unitary transformation toshow that the system can be exactly mapped to the same equation without the BLG and LC,and with a modified time-modulated nonlinear interaction. The nonlinear term remains invariant,while BLG and LC are removed completely, for the special case of a pseudo-unitary transformation.The mapping is generic and may be used to construct exactly solvable autonomous as well as non-autonomous vector NLSE with BLG. We present an exactly solvable two-component vector NLSEwith BLG which exhibits power-oscillation. The NLSE finds application in many diverse branchesof modern science, including optics[1], Bose-Einsteincondensation[2], plasma physics[3], gravity waves[4] and α -helix protein dynamics[5]. The NLSE is integrableand admits soliton solutions[1]. Several generalizationsof NLSE have been considered over the years to describeand model various emerging physical phenomenon[2, 6–10]. With the emergence of Parity-Time( PT ) symmet-ric theory[11] and its growing relevance in optics andother areas, the NLSE with BLG is an active area ofresearch[12–16]. Investigations on NLSE with BLG aremostly based on numerical and approximate methods.Integrable and/or exactly solvable NLSE with BLG arestill elusive.Within this background, we present a generic methodto investigate such systems analytically leading to exactsolutions. The BLG and LC terms are removed via a non-unitary transformation with its effect manifested in thetime-dependence of the nonlinear term. Even the nonlin-ear term is not changed due to the transformation if itsgenerator is pseudo-unitary[17]. The mapping is used toconstruct exactly solvable NLSE models with BLG.The vector NLSE is introduced in terms of a N -component complex scalar field Ψ and its hermitian ad-joint Ψ † as, i (cid:18) I ∂∂t + iA (cid:19) Ψ = − ∂ Ψ ∂x − δ (cid:0) Ψ † M Ψ (cid:1) Ψ (1)where I is the N × N identity matrix and δ is a realparameter. The N × N non-hermitian matrix A is de-composed in terms of two hermitian matrices B and C as A = B + iC with the condition that C is a traceless diag-onal matrix. The loss-gain terms in Eq. (1) are describedby the term − iC Ψ, while the LC among different field-components are governed by B Ψ. The N × N hermitianand non-singular matrix M does not depend on complexscalar fields and Eq. (1) describes a coupled cubic non-linear Schr¨ o dinger equation with BLG. The space-time ∗ [email protected] modulation of the nonlinear strengths may be incorpo-rated via explicit space-time dependence of M .The NLSE in Eq. (1) may be obtained from the La-grangian density, L = i (cid:2) Ψ † M ( D Ψ) − ( D o Ψ) † M Ψ (cid:3) − ∂ Ψ † ∂x M ∂ Ψ ∂x + δ (cid:0) Ψ † M Ψ (cid:1) + Ψ † F Ψ (2)where the operator D := I ∂∂t + iA has formal resem-blance with the temporal component of covariant deriva-tive with non-hermitian gauge potential A and the anti-hermitian matrix F := (cid:0) A † M − M A (cid:1) . The hermitianadjoint of Eq. (1) does not describe the equation satis-fied by Ψ † for F = 0, rather it describes the equationobeyed by Ψ † of a system whose Lagrangian density is L ∗ , i.e. complex conjugate of L . This is because L iscomplex for F = 0. The equation satisfied by Ψ † hasto be derived by using the Euler-Lagrange equation for L . The conjugate momenta corresponding to Ψ and Ψ † are Π Ψ = i Ψ † M and Π Ψ † = − iM Ψ, respectively. TheHamiltonian density H corresponding to L has the form, H = ∂ Ψ † ∂x M ∂ Ψ ∂x − δ (cid:0) Ψ † M Ψ (cid:1) + Ψ † M A
ΨThe Hamiltonian is real-valued for F = 0.We use a non-unitary transformation relating Ψ witha N -component complex scalar field Φ as follows,Ψ( t, x ) = U ( t )Φ( t, x ) , U ( t ) = e − iAt (3)which when substituted in Eq. (1) results in the equation, i ∂ Φ ∂t = − ∂ Φ ∂x − δ (cid:0) Φ † G Φ (cid:1) Φ , G = U † M U (4)The time-dependent non-unitary transformation removesthe loss-gain and the LC terms by modifying the nonlin-ear interaction. The nonlinear term remains unchangeddue to the transformation if and only if G = M , i.e. U ispseudo-unitary[17] with respect to M or equivalently A is M -pseudo-hermitian, U † M U = M ⇔ A † = M AM − (5)The pseudo-hermiticity of A can also be derived by ex-panding G ( t ) in powers of t with the identification of F = M , G ( t ) = ∞ X n =0 ( it ) n n ! F n , F n +1 = A † F n − F n A The condition G = M leads to F = 0, i.e. A is M -pseudo-hermitian. The first important result is that Eq.(1) with M -pseudo-hermitian A can be mapped to thesame equation without the loss-gain and the LC terms asgiven in Eq.(4). Further, if the transformed equation (4)is exactly solvable, solutions for Eq. (1) can be obtainedby using the pseudo-unitary transformation. The actionand Hamiltonian of the system is real-valued for a M -pseudo-hermitian A . The inclusion of more generalizedcubic nonlinear interaction in Eq. (1) may be achieved byreplacing Ψ † M Ψ with K , where K is a N × N hermitianmatrix with elements [ K ] ij = Ψ † H ij Ψ and H ij are N constant hermitian matrices of dimension N × N . Thematrix K can be re-expressed as [ K ] ij = Φ † (cid:0) U † H ij U (cid:1) Φ.If U is pseudo-unitary with respect to each matrix H ij ,then [ K ] ij = Φ † H ij Φ remains form invariant. The BLGand LC are removed by the pseudo-unitary transforma-tion without changing the nonlinear term. The Eq. (1)may or may not admit a Hamiltonian for a generic K .The second important result concerns the case forwhich A is neither hermitian nor M -pseudo-hermitianor equivalently U is neither unitary nor pseudo-unitary.The matrix M for a non-unitary U may be chosen tobe time-dependent such that Eq. (1) is necessarily non-autonomous, while Eq. (4) is autonomous. We choosethe matrix M in terms of real parameters α j as, M ( t ) = N − X j =0 α j (cid:2) U † ( − t ) λ j U ( − t ) (cid:3) (6)where the constant matrices λ j denote a suitable basis forexpanding M and G with λ being the identity matrix.The matrix G for the choice of M in Eq. (6) has the form G = P j α j λ j and Eq. (4) reduces to integrable Manakovsystem[6] of coupled vector NLSE for G = α λ , whichmay be realized by choosing all α j = 0 except α . The so-lution of the non-autonomous equation (1) may be foundfrom the solution of Eq. (4) by using the non-unitarytransformation. Various integrable and/or solvable gen-eralizations of Manakov systems are known[7–10]. Theparameters α j may be chosen appropriately to find solv-able non-autonomous system with BLG and LC corre-sponding to these known solvable models. There is anuseful duality relation between M and G . The matrix M ( t ) in Eq. (6) is time-dependent for a constant G . If M is chosen as time-independent M = P j α j λ j , then G becomes time-dependent G ( t ) = M ( − t ) where M ( t ) isgiven by Eq. (6).We present an example of a two-component NLSE to elucidate the general results by choosing, G = X j =0 α j σ j (7)where σ is the 2 × σ , σ , σ denotethe Pauli matrices with σ being diagonal. The terms inΦ † G Φ in Eq. (4) has standard physical interpretation.In particular, terms containing α and α are related toself-phase modulation and the cross-phase modulation,while terms which include α , α describe the effect offour-wave mixing. Eq. (4) for the above choice of G is integrable for any values of the real parameters α j [9].The celebrated Manakov system[6] of two coupled NLSEis obtained for α = 0 , b = 0, while α = 0 , b = 0 corre-spond to Zakharov-Schulman system[7].The non-hermitian matrix A in Eq. (1) is chosen as, A = β σ + + β σ + i Γ σ (8)The real constants β , linearly couple two components ofΨ, while Γ measures the loss-gain strength. The matrix A is M -pseudo-hermitian for the conditions, α = 0 , η ≡ Γ α + β α − β α = 0 (9)for which G = M is time-independent, M = X j =0 α j σ j + 1 β (Γ α + β α ) σ . (10)The NLSE in Eq. (1) with δ = 1 is solvable for A and M given by Eqs. (8) and (10), respectively. The non-unitaryoperator U connecting Ψ and Φ may be expressed as a 2 × θ ≡ p β + β − Γ , U = σ cos( θt ) − iAθ sin( θt ) for θ = 0 (11)The parameter θ becomes purely imaginary for Γ >β + β and the periodic functions change to the corre-sponding hyperbolic functions. Consequently, a boundedsolution for Φ will correspond to an unbounded solu-tion for Ψ in the long time behaviour. This is true for θ = 0 also for which U has a linear time-dependence.The loss-gain parameter is restricted within the range − p β + β < Γ < p β + β so that time-dependenceof U is periodic.The solution of Eq. (4) for δ = 1 and G = M given byEq. (10) has the expression[9],Φ = r C κW sech [ κ ( x − vt )] e i ( vx − ωt ) (12)where ω = v − κ , C = W † M W and W is an arbitrarytwo-component constant complex vector. The constants v, ω, κ correspond to the velocity, frequency and momen-tum, respectively for the one soliton solution Φ. Thepower P = Ψ † Ψ = Φ † { U † ( t ) U ( t ) } Φ for the loss-gain sys-tem oscillates with time, P = 2 κ W † W | C | sech [ κ ( x − vt )] N ( t ) N ( t ) = 1 + N sin ( θt ) + N sin(2 θt ) (13)where N = θ (Γ + β C + β C C ), N = Γ C θC and C j = W † σ j W, j = 0 , , ,
3. The power-oscillation vanishesfor no loss-gain in the system, i.e. Γ = 0. Thecondition N ( t ) ≥ ∀ t may be implemented in sev-eral ways by choosing the integration constants and pa-rameters appropriately. For example, the constant N vanishes if the two components of the complex vector W are chosen as W e iφ j , j = 1 , β ≡ β + iβ = p β + β e iφ , φ ≡ tan − ( β β ) andfixing φ = φ + φ + (2 n + 1) π , n ∈ Z , the constant N = θ becomes semi-positive definite. The amplitudeand time-period of oscillation grows as Γ is increased andapproaches β . The loss-gain parameter may be used asa controlling parameter for power-oscillation.The matrix M ( t ) corresponding to G and A given inEqs. (7) and (8), respectively, may be evaluated by usingEq. (6) and substituting λ j = σ j . This leads to theexpression, M ( t ) = σ (cid:20) α + 2Γ ηθ sin ( θt ) − Γ α θ sin(2 θt ) (cid:21) + σ (cid:20) α − β ηθ sin ( θt ) + β α θ sin(2 θt ) (cid:21) + σ (cid:20) α + 2 β ηθ sin ( θt ) − β α θ sin(2 θt ) (cid:21) + σ h α cos(2 θt ) − ηθ sin(2 θt ) i (14)and the limit η = 0 , α = 0 reproduces the result M = G .The NLSE in Eq. (1) with the above M ( t ) and A in eq.(8) is exactly solvable. The expressions for Ψ and P maybe obtained directly from the solutions for the case of M -pseudo-hermitian A by evaluating the factor C withoutthe condition (9). This is an overall multiplication factorand does not affect the physical behaviour of the system.The constant C can be chosen to be positive-definite for α ≥ | α | ≥ φ = φ + cos − α − α p α + α ! + (2 n + 1) π φ = tan − ( α α ) and n ∈ Z . The condition C > G = M can be obtained by using Eq. (9) inthe above equation.To conclude, we have presented a generic method toremove loss-gain and LC terms from a vector NLSE bya time-dependent non-unitary transformation which im-parts time-dependence to the nonlinear term. Further,if the generator of the transformation is pseudo-unitary,the non-linear term remains unchanged even though theloss-gain and LC terms are completely removed. The method is applicable to any vector NLSE with cubic non-linearity that is subjected to BLG and LC, and useful toconstruct solvable models. We have constructed an ex-actly solvable two-component NLSE with BLG and LCthat exhibits power-oscillation. Exactly solvable modelsof NLSE with more than two components and subjectedto BLG may also be constructed[18].The results can be trivially generalized to higher spa-tial dimensions and/or by including a space-time depen-dent inhomogenous term λ V ( x, t )Ψ. The time mod-ulated gain-loss strength and LC can be implementedby replacing the non-hermitian matrix A with ˜ A ( t ) = R dtA ( t ) in the definition of U ( t ) and for all subsequentsteps. Investigations along these directions could be car-ried out by using the method prescribed in this articleto explore a wide variety of physically interesting modelswith BLG. ACKNOWLEDGMENTS
This work is supported by a grant (
SERB Ref. No.MTR/2018/001036 ) from the Science & EngineeringResearch Board(SERB), Department of Science & Tech-nology, Govt. of India under the
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