Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation

Applied Mathematics and Computation 218 (2012) 7308–7321

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Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation Shou-Fu Tian a,b,⇑, Sheng-Wu Zhou a, Wu-You Jiang b, Hong-Qing Zhang b a b

Department of Mathematics, College of Science, China University of Mining and Technology, Xuzhou 221116, PR China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Keywords: Analytic solution Generalized Schrödinger equation Darboux transformation Supersymmetry Hopf–Cole transformation Transformed potential Hamiltonian

a b s t r a c t In this paper, analytically investigated is a generalized one-dimensional time-dependent Schrödinger equation. Using Darboux transformation operator technique, we construct the first-order Darboux transformation and the real-valued condition of transformed potential for the generalized Schrödinger equation. To prove the equivalence of the supersymmetry formalism and the Darboux transformation, we investigate the relationship among first-order Darboux transformation, supersymmetry and factorization of the corresponding effective mass Hamiltonian. Furthermore, the nth-order Darboux transformations are constructed by means of different method. Finally, by using Darboux transformation, some analytical solutions are generated in a recursive manner for some examples of the Schrödinger equation. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Among the many techniques used to study integrability and to obtain the multisoliton solutions for a given integrable model, the Darboux transformation has been widely used and it has established itself as an economic, convenient and efficient way of generating solutions. It is well known that the Darboux transformation [1] is one of the major tools for the analysis of physical systems and for finding new solvable systems. Using a linear differential operator, Darboux construct solutions of one ordinary differential equation in terms of another ordinary differential equation. It has been shown that the transformation method is useful in finding soliton solutions of the integrable systems [2–4] and constructing supersymmetric quantum mechanical systems [5–7]. Much later it was found that the Darboux transformation is equivalent to the supersymmetry formalism, see the reviews in [8,9]. An excellent survey of developments and some applications of the transformation method are given in [10]. After that, several generalizations of the stationary and the time-dependent Schrödinger equation were found that admitted particular Darboux transformations, all with a similar form and similar properties, such as their explicit form or their equivalence to factorization formalisms within the supersymmetry context. This large variety of applications explains the raising interest in solvable special cases of effective mass Schrödinger equations. Recently, several of such cases were found, particular combinations of potentials and step-like effective masses [11]. More general solvable cases were obtained by means of factorization methods [12] and via Lie algebraic approaches [13]. Darboux transformation is known as one of the most powerful methods for finding solvable Schrödinger equations with constant mass, in the context of which it is also called supersymmetric factorization method [14]. The Darboux transformation is

⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (S.-F. Tian). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.009

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usually constructed by means of the intertwining operator method, which has been used for the conventional Schrödinger equation [21], but can also be applied to more general operators [15,16]. In this paper, we consider a generalized one-dimensional Schrödinger equation

ih/t þ f /xx þ g/x  v / ¼ 0:

ð1:1Þ

If we take different coefficients, Eq. (1.1) can be transformed into the following well-known equations. 0

1 m Case 1. Taking f ¼  m ;g ¼ m 2 ; h ¼ 0; v ¼ e  VðxÞ, we have the effective mass Schrödinger equation and its associated Hamiltonian H in atomic units [17,18]

H/ðxÞ ¼ e/ðxÞ; where the energy function.

    d 1 d þ VðxÞ; H¼ dx mðxÞ dx

ð1:2Þ

e is a real constant, m stands for the effective mass, V denotes the potential and / is the wave

Case 2. Assuming that f ¼ 1; g ¼ 2iR; v ¼ V  iRx , we can obtain a Schrödinger equation with first-order derivatives [19]

i/t þ /xx þ 2iR/x þ ðiRx  VÞ/ ¼ 0;

ð1:3Þ

where R ¼ Rðx; tÞ is arbitrary, and V ¼ Vðx; tÞ denotes the potential. The Hamiltonian associated with Eq. (1.2) has the form of a three-dimensional Hamiltonian coupled to a magnetic field. 2

2

h  ; g ¼ h2 Case 3. Assuming that f ¼  2mðrÞ [20]



dm=dr ;h mðrÞ2

¼ 0 and

v ¼ E  U abca  VðrÞ, we have the following Schrödinger equation

2 2 2   h d / h dm=dr d/  þ VðrÞ þ U abca  E / ¼ 0; þ 2 2mðrÞ dr 2 mðrÞ2 dr 2

where U abca ¼  4m3hðaþ1Þ

h

ð1:4Þ

dm2 i 2 . ða þ c  aÞm ddrm 2 þ 2ða  ac  a  cÞ dr

Case 4. Taking f ¼  m1 and g ¼

1 m x

, we have the following Schrödinger equation

    1 @ x /  v / ¼ 0: ih/t þ @ x m

ð1:5Þ

The intertwining relations for Eq. (1.5) are investigated in Refs. [24,26]. Case 5. If f ¼ 1; g ¼ 0; h ¼ 1; v ¼ f ðx; tÞ þ gðx; tÞ, we can obtain the one-dimensional Schrödinger equation with a nonzero potential [25]

i/t ¼ /xx þ ½f ðx; tÞ þ gðx; tÞ;

ð1:6Þ

where f ðx; tÞ and gðx; tÞ are differentiable functions of their arguments.  h  2 i h  d h  ; g ¼  dx ; h ¼ 0 and Case 6. Taking f ¼  2mðxÞ 2mðxÞ

v ¼ E  VðxÞ, we have the Schrödinger equation associated with a

particle endowed with a positiondependent effective mass reads [11]

" !#   2 2 h d / d h d/ þ VðxÞ/ðxÞ ¼ E/ðxÞ;   2mðxÞ dx2 dx 2mðxÞ dx

ð1:7Þ

where mðxÞ stands for the particle’s effective mass, VðxÞ denotes the potential, /ðxÞ is the particle’s wave function, and E is the concomitant eigenenergy. The paper will be organized as follows: In Section 2, by using Darboux transformation operator technique, we obtain the first-order Darboux transformations and the transformed potentials for the linear, time-dependent Schrödinger equation in (1 + 1)-dimension. In Section 3, in order to the transformed potential is a real-valued function, we take into account the reality condition. In Section 4, to prove the equivalence of the supersymmetry formalism and the Darboux transformation, we investigate the relationship between first-order Darboux transformation, supersymmetry and factorization of the corresponding effective mass Hamiltonian. In Section 5, to further research, we consider the nth-order Darboux transformation. Finally, by using higher order Darboux transformation operator technique, the explicit multi-soliton solutions are generated in a recursive manner for some examples of Schrödinger equation. The propagation characteristic of one- and two-solitons are discussed.

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2. First-order Darboux transformation In this section, we consider the following generalized Schrödinger equation in (1 + 1)-dimension by using the Darboux transformation operator technique [18–30]

ih/t þ f /xx þ g/x  v / ¼ 0;

ð2:1Þ

where all involved functions f ; g; h and the potential V depend on the variables x and t. Note further that one of the linear, time-dependent Schrödinger equations in (1 + 1)-dimension is only a special case of Eq. (2.1), including above three kinds of special cases. This equation can be rewritten as

i/t þ H/ ¼ 0;

f h

g h

v

H ¼ @ xx þ @ x  :

ð2:2Þ

h

To relate the problem Eq. (2.1) to a problem of the same form, but for a different potential

^/ ^ ¼ 0; ^t þ H i/

f h

g h

v^

^ ¼ @ xx þ @ x  ; H

ð2:3Þ

h

^ – /. Introducing the transformation operator C yields into the following relation ^ – v , implying / we have v

^ Þ C: Cði@ t þ HÞ ¼ ði@ t þ H

ð2:4Þ

^ , respectively. The Eq. (2.4) and the operator C are called Darboux transformation operator for the Hamiltonians H and H operator C transforms any solution / of (2.1) into a new solution

^ tÞ ¼ C/ðx; tÞ: /ðx;

ð2:5Þ

Let Darboux transformation operator (the intertwiner) be the form of a linear, first-order differential operator

C ¼ A þ B@ x ;

ð2:6Þ

where the functions A ¼ Aðx; tÞ and B ¼ Bðx; tÞ are to be determined, such that C solves Eq. (2.4). In order to find A and B, we ^ into the Darboux transformation operator (2.4), and apply it to the soluinsert the explicit form of the Hamiltonians C and C tion / of Eq. (2.1), we can obtain



f h

g h

C i@ t þ @ xx þ @ x 

   f g v^ / ¼ i@ t þ @ xx þ @ x  C/: h h h h

v

Making linear independence of / and its partial derivatives, we collect their respective coefficients and equal them to zero, ^ from which we can obtain the following system about the functions A; B and v

  2f f B; Bx ¼ h h x

ð2:7aÞ

v f g A B ¼ ðv^  v Þ; iAt þ Axx þ Ax þ h h h h x  f g 2f g B B ¼ ðv^  v Þ: iBt þ Bxx þ Bx þ Ax  h h h h x h

ð2:7bÞ ð2:7cÞ

By virtue of Eq. (2.7a), we have

rffiffiffi   2Bx f f )B¼a ; ¼ ln h x h B

ð2:8Þ

where a ¼ aðtÞ is an arbitrary, purely time-dependent constant of integration. Eqs. (2.7b) and (2.7c) enable us to solve the ^ and the function A, respectively. For this, let us multiply Eq. (2.7b) with B and Eq. (2.7c) with A. Then, the leftpotential v hand sides of Eqs. (2.7b) and (2.7c) become the same and we can make them equal to each other

g v f g f g 2f iAt B þ Axx B þ Ax B þ B2  iABt  ABxx  ABx  AAx þ AB ¼ 0: h h h h h h x h x

ð2:9Þ

Taking into account Eq. (2.8), we have

     2 2Bx f Bxx 1 f 1 f ln ln ) þ : ¼ ln ¼ h x 2 h xx 4 h x B B In order to solve Eq. (2.9) with respect to A, a new auxiliary function P ¼ Pðx; tÞ is introduced by A ¼ BP. Eq. (2.9) is can conveniently be written in Burgers form for P only 2

2

ih Pt þ fhPxx  2fhPPx þ ðfx h  fhx þ ghÞPx  fx hP þ fhx P2 þ hg x P  hx gP þ hv x  hx v ¼ 0:

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This equation can be rewritten as 2

fP x fP gP v  þ  h h h h

iPt ¼

! ð2:10Þ

; x

which is a time-dependent Riccati equation that can be linearized and integrated by introducing the Hopf–Cole transformation

P¼

ux : u

ð2:11Þ

Assuming that u is twice continuously differentiable, satisfying uxt ¼ utx , we substitute Eq. (2.11) in Eq. (2.10) and obtain

  ut fu gu v i þ xx þ x  ¼ 0: u hu hu h x

ð2:12Þ

Integrating on both sides and multiply with u, we have

iut þ

fuxx gux uv þ  ¼ cu; h h h

ð2:13Þ

where c ¼ cðtÞ is a purely time-dependent constant of integration. Taking c ¼ 0, Eq. (2.13) is identical to the initial Eq. (2.1)

ihut þ fuxx þ gux  uv ¼ 0:

ð2:14Þ

However, making c to zero is not a restriction, since solutions of Eq. (2.13) with c – 0 and c ¼ 0 differ from each other only by a purely time-dependent factor, which cancels out in Eq. (2.11). For a given u, one can then obtain the function P by virtue of Eq. (2.11), which in turn solves A by virtue of A ¼ BP and Eq. (2.8), that is

rffiffiffi f ðln uÞx : h

A ¼ a

ð2:15Þ

^ by considering Eq. (2.7c) for v ^ By means of Eq. (2.15), one can obtain the equation for the new potential v

v^ ¼ v þ

fBxx 2fAx gBx ghx hBt þ þ þ g þi : B qffiffiB B h qxffiffi B

ð2:16Þ

By virtue of A ¼ a hf ðln uÞx and B ¼ a hf , we obtain the explicit form of the transformed potential (2.16). Furthermore, we ^ from Eqs. (2.5) and (2.6), respectively can construct the operator C and the transformed solution /

pffiffiffiffiffi v^ ¼ v þ a fh

sffiffiffi rffiffiffi! " rffiffiffi! rffiffiffi! rffiffiffi!# pffiffiffiffiffi f ux f h f hx f þ 2 fh  þg þ g  g x þ ih ln a ; h f h h u h h xx

x

x

ð2:17Þ

t

rffiffiffi rffiffiffi rffiffiffi rffiffiffi f f f f ðP þ @ x Þ ¼ a ðln uÞx þ a @x ¼ a @ x  ðln uÞx ; h h h h rffiffiffi f ^ / ¼ C/ ¼ a @ x  ðln uÞx /: h

C¼a

ð2:18Þ

ð2:19Þ

^ The ^ and the corresponding solution /. The function u defines the Darboux transformation operator C, the new potential v new potential depends not only on the potential v, but also on the additional potentials f ; g and h.

3. Reality condition of the first-order Darboux transformation ^ in Eq. (2.17) is real valued, if Suppose that the coefficients f ; g; h and v in Eq. (2.1) are all real valued. Then the function v the reality condition

pffiffiffiffiffi Imðv^ Þ ¼ ImðaÞ fh

(" rffiffiffi! rffiffiffi! rffiffiffi!# ) pffiffiffiffiffi f f f þ 2 fh ImðPÞ þ hRe ln a ; h h h xx

x

ð3:1Þ

t

is fulfilled. Because the function a can be complex and appears within a logarithm in Eq. (2.17), we can write a as

a ¼ a1 expðia2 Þ; where a1 ¼ a1 ðtÞ ¼ AbsðaÞ and a2 ¼ a2 ðtÞ ¼ ArgðaÞ. We have

ImðaÞ ¼ a1 sin a2 ;

ð3:2Þ

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(" Re

qffiffi 3 qffiffi 3 2 qffiffi 2 qffiffi f f f f rffiffiffi!# )   a a ð a expði a Þ þ i a expði a Þ a Þ a expði a Þ t 1t 2 1 2 2t 1 2 h h h h 6 7 6 7 f t7 t7 6 6 qffiffi qffiffi ¼ Re4 ln a 5 ¼ Re4 5 h f f a a expðia Þ t

1

h

2

h

sffiffiffi!# " rffiffiffi! a1t f h  ln ¼ ln a1 : ¼ h f a1 t

ð3:3Þ

t

Substituting Eqs. (3.2) and (3.3) into Eq. (3.1), we obtain the corresponding equation

Imðv^ Þ ¼ a1 sin a2

pffiffiffiffiffi fh

sffiffiffi!# " rffiffiffi! rffiffiffi! pffiffiffiffiffi f f h þ 2 fh ImðPÞ þ h ln a1 : h h f xx

x

ð3:4Þ

t

^ Þ ¼ 0 and P ¼  uux , then the transformed potential v ^ is real valued, that is If the expression Imðv

sffiffiffi!# " rffiffiffi! rffiffiffi! pffiffiffiffiffi f ux f h þ 2 fh Imð Þ þ h ln a1 h h f u xx x t sffiffiffi!# " rffiffiffi! rffiffiffi! pffiffiffiffiffi pffiffiffiffiffi f f h ¼ a1 sin a2 fh þ 2 fh Imððln uÞx Þ þ h ln a1 h h f xx x t sffiffiffi!# ( " rffiffiffi! rffiffiffi) p ffiffiffiffi ffi pffiffiffiffiffi  f f h ¼ a1 sin a2 fh  i fh ðln uÞx  ðln u Þx þ h ln a1 h h f xx x t sffiffiffi!# " " rffiffiffi! rffiffiffi#  pffiffiffiffiffi p ffiffiffiffi ffi f u f h ¼ a1 sin a2 fh  i fh ln  þ h ln a1 ¼ 0: h u x h f

pffiffiffiffiffi a1 sin a2 fh

xx

x

ð3:5Þ

t

This is the reality condition for our transformed potential Eq. (2.17). If f ; g and h are constants, condition (3.5) becomes

sffiffiffi!# " " rffiffiffi# pffiffiffiffiffi  u f h i fh ln   h ln a1 ¼ 0: u x h f x

t

This reality condition can be rewritten as

ðln a1 Þt ¼ i

f  u ln  : h u xx

ð3:6Þ

This is fulfilled if the right-hand side does not depend on x, that is if

ln

u u

xxx

¼ 0:

ð3:7Þ

Now suppose that the reality condition (3.5) is fulfilled. We substitute into the transformed potential Eq. (2.17), which then takes the following form

v^ ¼ v

pffiffiffiffiffi þ a1 cos a2 fh

¼ v þ a1 cos a2

pffiffiffiffiffi fh

¼ v þ a1 cos a2

pffiffiffiffiffi fh

¼ v þ a1 cos a2

pffiffiffiffiffi fh

sffiffiffi rffiffiffi! (" " rffiffiffi! rffiffiffi# rffiffiffi!# ) g  u f pffiffiffiffiffi f h f f x þ 2 fh Re  þg h þ hIm ln a h h f h h x h u xx x x t sffiffiffi rffiffiffi! " rffiffiffi! rffiffiffi# g pffiffiffiffiffi   f f h f  2 fh Re ðln uÞx þg h þ ha2t h h f h h x xx x x sffiffiffi rffiffiffi! ( rffiffiffi! rffiffiffi) g pffiffiffiffiffi   f f h f   fh ðln uÞx þ ðln u Þx þg h þ ha2t h h f h h x xx x x sffiffiffi rffiffiffi! " rffiffiffi! rffiffiffi# g f pffiffiffiffiffi  f h f  fh ln juj2 þg h þ ha2t : x h h f h h x xx

x

ð3:8Þ

x

This is clearly real valued and compatible with the conventional case, where f ; g and h are constants, we get for the transformed potential Eq. (3.8)

v^ ¼ v  f



ln juj2

xx

þ ha2t :

These reality conditions also holds for the higher order Darboux transformations.

ð3:9Þ

S.-F. Tian et al. / Applied Mathematics and Computation 218 (2012) 7308–7321

7313

4. Supersymmetry and equivalence with Darboux transformation In what follows we will prove that the formalism of supersymmetry for our generalized Schrödinger equation is equivalent to the Darboux transformation by using Refs. [12,14,18,24,26,28]. Suppose the Schrödinger operator i@ t  H is selfadjoint

ði@ t þ HÞ ¼ i@ t þ H: Taking the operation of conjugation on Darboux transformation (2.4), we obtain

^ Þ; ði@ t þ HÞC ¼ C ði@ t þ H

ð4:1Þ



where the operator C adjoint to C, as given in Eq. (2.18), is determined as

C ¼ a

rffiffiffi   f 1 f : @ x þ P  ln h 2 h x

ð4:2Þ

Eqs. (2.2) and (2.3) can then be rewritten as one single matrix equation of the form





i@ t

0

0

i@ t

 þ

H 0

0 ^ H

  / ^ ¼ 0: /

ð4:3Þ

^ Þ and U ¼ ð/; /Þ ^ T , the above Schrödinger (4.3) can be written as Assuming that H ¼ diagðH; H

½i@ t þ HU ¼ 0:

ð4:4Þ 

Two supercharge operators L; L are defined as follows

 L¼

0

0

C 0

 ;

L ¼



0 C 0

0

 ;

ð4:5Þ

where C and C are the operators given by Eqs. (2.18) and (4.2), respectively. One can show that the Hamiltonian H satisfies the following system

8   > < fL; Lg ¼ fL ; L g ¼ 0; ½L; i@ t þ H ¼ ½i@ t þ H; L ¼ 0; > : ½i@ t þ H; L  ¼ ½L ; i@ t þ H;

ð4:6Þ

where fM; Ng ¼ MN þ NM and ½M; N ¼ MN  NM, named anticommutator and commutator, respectively. The first equation of Eq. (4.6) is trivially fulfilled since the matrices in Eq. (4.5) are nilpotent. The second and third equations of Eq. (4.6) are precisely our Darboux transformations Eqs. (2.4) and (4.1). Considering the complementing relations of the supersymmetric b ¼ LL and consider the relations of algebra: the anticommutators fL; L g and fL ; Lg, we obtain the operators R ¼ L L and R ^ them with our Hamiltonians H and H. By virtue of Eqs. (2.18), (4.2), and some algebraic transformations, we have

      f f f f R ¼ C C ¼ jaj2  @ xx  @x  P þ ðjPj2  Px Þ ; h h h x h  x     f f f 1 fx fx 3fh fh b ¼ CC ¼ jaj2  @ xx  R @ x þ ðjPj2 þ P x Þ  þ  3x þ xx2 : h h x h 2 h x 2fh 2h 2h

ð4:7aÞ ð4:7bÞ

Compared with Eqs. (4.7a) and (4.7b), we have

b  R ¼ jaj2 R

rffiffiffi" rffiffiffi! rffiffiffi! # f f f  : þ2 P h h h xx

ð4:8Þ

x

b of our matrix (4.4). The supercharges L; L , and the symmetry operator R We consider a symmetry operator R ¼ diagðR; RÞ generate the simplest superalgebra as

8 2  2 > < L ¼ ðL Þ ¼ 0; ½L; R ¼ ½L ; R ¼ 0; > : R ¼ fL; L g ¼ fL ; Lg:

ð4:9Þ

Compared with the relations of Eqs. (4.6) and (4.9), the first equations in Eqs. (4.6) and (4.9) are coincide, but their Darboux b that is transformations are different. They are standard for the operators R and R,

b C  CR ¼ 0; R

b  RC ¼ 0; C R

ð4:10Þ

b  ¼ ðLL Þ ¼ LL ¼ R. b But the Darboux transformations for the operators H and H ^ are where R ¼ ðL LÞ ¼ L L ¼ R and R nonstandard. From Eqs. (2.4) and (4.1), we have 







 



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^  HC ¼ iC  iB @ xt : C H t

^ C ¼ iCt  iB@ xt ; CH  H

ð4:11Þ

^ are nonstandard. Considering the third equation of (4.9) and comThe Darboux transformations for the operators H and H paring the Darboux transformations for the elements of H and R, we can obtain the relationship between operators R and H as follows. Taking  ¼ diagðc1 ; c2 Þ, where ci ¼ ci ðx; tÞ ði ¼ 1; 2Þ to be determined, we have

 R¼

R 0

0 b R

 ¼

!

H þ c1

0

0

^ þc H 2

¼ H þ  :

ð4:12Þ

Taking jaj ¼ 1, we can obtain the connection between operators R and H

H ¼ R þ c1 ¼ C C þ c1 ;

ð4:13aÞ

b þ c ¼ CC þ c ; ^ ¼ R H 2 2

ð4:13bÞ

which are equivalent to

H ¼ fL ; Lg þ  :

ð4:14Þ

v in the form

To determine function c1 , we can express

v

ut f g ¼ i þ ðjPj2  P x Þ  P  c; h h u h

ð4:15Þ

by using Eqs. (2.11) and (2.13). where c ¼ cðtÞ is an arbitrary function that appeared as a constant of integration in Eq. (2.13). Substituting Eq. (4.7a) into (4.13a) and comparing to Eqs. (2.2) and (4.15), we can obtain

c1 ¼ i

ut þ c: u

ð4:16Þ

^ =h in the form To determine function c2 , we can express v

" rffiffiffi rffiffiffi! rffiffiffi! rffiffiffi!# g f f f g f f 2 þ pffiffiffiffiffi   c1 þ i ln a ; ¼ ðjPj þ Px Þ þ a h h h h h h h x fh

v^

xx

x

ð4:17Þ

t

by using Eqs. (2.17) and (4.15). Substituting Eq. (4.7b) into Eq. (4.13b) and comparing to Eq. (2.2) and Eq. (4.15), we can obtain

h

 pffiffiffi i

c2 ¼ c1  i ln a f

t

¼ c1  i

Bt : B

ð4:18Þ

^ can be factorized as Eqs. (4.13) or Eq. (4.14), supplementing the supersymmetric relaFrom above, two Hamiltonians H and H tions Eq. (4.6) for the generalized time-dependent Hamiltonian. The fact that the factorization matrix  depends on u ¼ uðx; tÞ and B ¼ Bðx; tÞ is nonstandard. Particularly, once the potentials f ; g and h do not depend on time,  can be written as  ¼ c1 E , where E is the identity matrix. Furthermore, if uðx; tÞ ¼ nðxÞgðtÞ, then c1 becomes independent of the spatial variable. If the initial Hamiltonian H is hold for the factorized form Eq. (4.13a), one can obtain its supersymmetric partner in a factorized form too. Indeed, multiplying Eq. (4.13a) from the left by C and taking into account the first Darboux transformation from Eq. (4.11), we have

^ C/ þ iCt /  iB/ : CH/ ¼ CðC C þ c1 Þ/ ¼ H xt

ð4:19Þ

^ in terms of C and C , we have Expressing H

Cc1 / ¼ BðP þ @ x Þc1 / ¼ Bðc1 Þx / þ c1 C/: By virtue of Eq. (4.19), i/t ¼ H/ and

i uut

ð4:20Þ

¼ H, we obtain

 ut iB/xt þ Bðc1 Þx / ¼ B i/t  / ¼ 0 u x

 Bt ut Bt iCt / ¼ iBt ðP þ @ x Þ/ þ iBðP þ @ x Þt / ¼ i C/ þ i@ t  i / ¼ i C/: B u x B

ð4:21Þ ð4:22Þ

By means of Eqs. (4.19)–(4.22), we have

^ ¼ CC þ c  i H 1

Bt ¼ CC þ c2 ; B

ð4:23Þ

which is the same as Eq. (4.13b). ^ . Hamiltonians Eqs. (4.13a) From above, we have the explicit forms of the supersymmetric partner Hamiltonians H and H ^ are given by Eqs. (4.15) and and (4.13b) are the same as their definitions Eqs. (2.2) and (2.3), respectively, if jaj2 ¼ 1 and v ; v (4.17). Finally, considering the difference of the Hamiltonians Eqs. (4.13a) and (4.13b) yields the potential difference Eq. (2.17). Hence, we obtain the equivalence of the Darboux transformation and the supersymmetry formalism.

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5. Higher order Darboux transformation In principle, we would have to go back to an Darboux transformation operator (2.4) with generalized Hamiltonians Eqs. (2.2), (2.3) and solve it for an nth-order differential operator C, which would need large and involved calculations. Therefore, we are here to make use of other methods. With the conventional Schrödinger Darboux transformation and its well-known Darboux transformation operator, we will derive nth-order Darboux transformations for the generalized Schrödinger Eq. (2.1) by means of a point transformation. We will now construct the point transformation as follows (based on the method of undetermined coefficient)

/ðx; tÞ ¼ expðNðx; tÞÞwðnðx; tÞ; gðtÞÞ;  R with Nðx; tÞ ¼  i 2fu

x

ð5:1Þ

 pffiffiffiffiffi R qhffiffi hut dx, where g0 ¼ ddtg ; g must not depend on x, in order to preserve lineardx and nðx; tÞ ¼ g0 g f þ þ uxx 2f

2ux

ity of the equation. Using the point transformation (5.1), we can construct a connection between the conventional Schrödinger equation and its generalized counterpart Eq. (2.1). Substituting Eq. (5.1) into Eq. (2.1), we have

iwg þ wnn þ

1 ðihNt þ Nxx þ f N2x þ g Nx  v Þw ¼ 0; hg0

ð5:2Þ

where g remains arbitrary. Note that Eq. (5.2) is of the Schrödinger form, such that the Darboux transformation becomes applicable. As was shown in Ref.[9,21,24,26,28], there is an Darboux operator Cn of order n, satisfying the relation of Darboux transformation

^ 0 ÞCn ; Cn ði@ t þ H0 Þ ¼ ði@ t þ H

ð5:3aÞ

1 H0 ¼ @ nn þ 0 ðihNt þ Nxx þ f N2x þ g Nx  v Þ; hg ^ 0 ¼ @ nn þ 1 ðihNt þ Nxx þ f N2 þ g Nx  v^ Þ; H x hg0 W nþ1;ðwi Þ ðÞ Cn ðÞ ¼ a0 ; i ¼ 1; 2; . . . ; n; W n;ðwi Þ

ð5:3bÞ ð5:3cÞ ð5:3dÞ

where w1 ; w2 ; . . . ; wn are the solutions of the Schrödinger equation (5.2), such that the family (w1 ; w2 ; . . ., wn ; wÞ is linearly independent. a0 ¼ a0 ðtÞ is arbitrary, W nþ1;ðwi Þ and W n;ðwi Þ stand for the Wronskians of the families ðw1 ; w2 ; . . . ; wn Þ and ðw1 ; w2 ; . . . ; wn Þ, respectively. Substituting Eqs. (5.3b)–(5.3d) into Eq. (5.3a), we can obtain the transformed potential function v^ as

v^ ¼ v  2hg0



ln W n;ðwi Þ

 nn

þ ihg0 ðln aÞt :

ð5:4Þ

Considering the inverse point transformation Eq. (5.1) to Eq. (5.3a), we have

/ ¼ expðNÞw;

ð5:5aÞ

wj ¼ expðNÞ/j ;

j ¼ 1; 2; . . . ; n:

ð5:5bÞ

where /1 ; /2 ; . . . ; /n are solutions of the generalized Schrödinger Eq. (2.1). Eq. (5.3a) can be rewritten in terms of the variables x and t. From above, we have

 14nðnþ1Þ 1 expððn þ 1ÞNðx; tÞÞW nþ1;ð/i Þ ð/Þ; nx  14nðn1Þ 1 W n;ðwi Þ ðwÞ ¼ expðnNðx; tÞÞW n;ð/i Þ ð/Þ; nx

W nþ1;ðwi Þ ðwÞ ¼

ð5:6aÞ ð5:6bÞ

where w ¼ wðnðx; tÞ; gðx; tÞÞ and / ¼ /ðx; tÞ. From above, using Eqs. (5.1), (5.3d), (5.6a) and (5.6b), we obtain nth-order Darboux transformations for the generalized Schrödinger Eq. (2.1)

Cn ð/Þ ¼ Cn ðexpðNðx; tÞÞwðnðx; tÞ; gðtÞÞÞ ¼ expðNðx; tÞÞCn ðwðnðx; tÞ; gðtÞÞÞ ¼ a0 expðNðx; tÞÞ rffiffiffi!n W nþ1;ð/i Þ ð/Þ f ¼a ; W n;ð/i Þ ð/Þ h

W nþ1;ðwi Þ ðwÞ W n;ðwi Þ ðwÞ ð5:8Þ

0  where a ¼ paffiffiffi n is the arbitrary function with t-dependent. By virtue of Eqs. (5.4) and (5.6b), we have

g0

v^ ¼ v þ 2nf where Nðx; tÞ ¼ 

R

Nxx þ  i 2fu

(rffiffiffi" !# )  nðn1Þ     2 pffiffiffiffiffi f f f  2 fh W þ ihg0 ðln aÞt ; ln ln n;ð/j Þ h t h hg0 2

Nx

x

hut g uxx x þ2f þ2u

x

x

  4 dx and g0 ¼ aa0 n is an arbitrary function with t-dependent.

ð5:9Þ

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6. Some examples of application for Schrödinger equation In this section, we will present two simple applications of the generalized Schrödinger equation. Example 1. Assuming that f ¼ 1; g ¼ 0 are hold, we have a generalized Schrödinger equation [22]

ih/t þ /xx  v / ¼ 0:

ð6:1Þ

Equivalently, this equation can be seen as a Schrödinger equation with linearly energy-dependent potential. We consider a specific case of Eq. (6.1)

h¼

ki p i x and b

v ¼ li pxi  q2 ; i ¼ 1; 2; . . . ;

ð6:2Þ

where ki P 0; li P 0, p and q are real and positive constants, and b ¼ bðtÞ is an arbitrary function. By virtue of Eq. (6.2), we have a particular solution of Eq. (6.1)

/ ¼ sinðqxÞ expði

Z

ð6:3Þ

bdtÞ:

To perform the first-order Darboux transformation Eq. (2.19) on this function, we take an auxiliary solution u the function Eq. (6.1)

 Z  u ¼ cosðqxÞ exp i bdt :

ð6:4Þ

From above, the two solutions / and u are linearly independent, as required for the Darboux transformation. Substituting f ¼ 1; g ¼ 0, Eqs. (6.2), (6.3), (6.4) into Eqs. (2.17) and (2.19), we can obtain the following results (see Fig. 1)

rffiffiffi  R  sffiffiffiffiffiffiffiffiffiffi f ux aq exp i bdt b ^ ; / ¼ C1 ð/Þ ¼ a /  / ¼ cosðqxÞ h x u ki pxi

ð6:5Þ

sffiffiffi rffiffiffi! " rffiffiffi! rffiffiffi! rffiffiffi!# pffiffiffiffiffi f ux f h f hx f þ 2 fh  þg þ g  g x þ ih ln a h f h h u h h xx x x t sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi! sffiffiffiffiffiffiffiffiffiffi!# "  ki p i b ki p i b ki p i b x x tanðqxÞ x ln a ¼ li pxi  q2 þ a þ 2q þi ; b ki pxi xx b ki pxi b ki pxi pffiffiffiffiffi fh

v^ ¼ v þ a

x

ð6:6Þ

t

where a ¼ aðtÞ; b ¼ are arbitrary functions. h bðtÞ  q ffiffiffiffiffiffiffit-dependent ffi i 1 If we take kbi p xi ln a k bpxi ¼ 0, that is a ¼ b2 , we have the imaginary part of the transformed potential t

i

sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi!  ki p i b ki p i b i 2 x x tanðqxÞ v^ ¼ li px  q þ a þ 2q : b ki pxi xx b ki pxi

ð6:7Þ

x

Using Eqs. (5.8) and (5.9), we can obtain a sequence of new nth-order Darboux transformation and transformed potential for the generalized Schrödinger equation Eq. (6.1).

1e+16 5e+15

1e+28

0

0

-5e+15

-1e+28 100

-1e+16 -1

50

-0.5

x

0

0

-50

0.5 1

-100

(a)

100

-1

50

-0.5

t

x

0

0

-50

0.5 1

t

-100

(b)

^ and (b) transformed potential v ^ with i ¼ 2; l2 ¼ i; k2 ¼ 3; p ¼ 5i; q ¼ 7; a ¼ sinðtÞ; b ¼ cosðtÞ. Fig. 1. (a) the transformed solution /

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S.-F. Tian et al. / Applied Mathematics and Computation 218 (2012) 7308–7321 1 Example 2. Taking f ¼ 2m ; g ¼  mmx ; h ¼ 1, we have a position-dependent mass Schrödinger equation [23]

i/t þ

1 mx /  v / ¼ 0; /  2m xx 2m2 x

ð6:8Þ

where m ¼ mðx; tÞ stands for the nonconstant mass,

v ¼ v ðx; tÞ is the potential.

Case (I). From above, we will now take into account a specific case of Eq. (6.8)

m¼

c sinð2sxÞ

v¼

;

2s2 þ hc sinð2sxÞ ; c sinð2sxÞ

ð6:9Þ

where s is the real and positive constant, and h ¼ hðtÞ and c ¼ cðtÞ are arbitrary functions. By means of Eq. (6.8), we obtain a particular solution with Eq. (6.9)

 Z  / ¼ tanðsxÞ exp i cdt :

ð6:10Þ

To perform the first-order Darboux transformation Eq. (2.19) on this function, we take an auxiliary solution u the function Eq. (6.8)

 Z  u ¼ cotðsxÞ exp i cdt :

ð6:11Þ

Similarly, the two solutions / and u are linearly independent, as required for the Darboux transformation. Substituting 1 f ¼ 2m ; g ¼  mmx ; h ¼ 1, Eqs. (6.9) and (6.10) into Eqs. (2.17) and (2.19), we can obtain the following results (see Fig. 2)

qffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi  Z   2as sinð2sxÞ f u 2c x ^ ¼ C1 ð/Þ ¼ a /x  / ¼ / c dt ; exp i h u cos2 ðsxÞ

ð6:12Þ

sffiffiffi rffiffiffi! " rffiffiffi! rffiffiffi! rffiffiffi!# pffiffiffiffiffi f ux f h f hx f v^ ¼ v þ 2 fh  þg þ g  g x þ ih ln a h f h h u h h xx x x t  0  2 2 2 2 2 a s sin ð2sxÞ 4s þ ð a  1Þs cos ð2sxÞ þ 2sð1  sÞ cosð2sxÞ a c0 ; ¼ h þ 2scsc2 ð2sxÞ    þi 2c sinð2sxÞ c a 2c pffiffiffiffiffi þ a fh

ð6:13Þ

where a ¼ aðtÞ; c ¼ cðtÞ; h ¼ hðtÞ, are t-dependent arbitrary functions and a0 ¼ dadtðtÞ ; c0 ¼ dcdtðtÞ 0 0 If we take aa  2cc ¼ 0, that is c ¼ a2 , we have the imaginary part of the transformed potential

v^ ¼ h þ 2s csc2 ð2sxÞ 

as2 sin2 ð2sxÞ 4s2 þ ða  1Þs2 cos2 ð2sxÞ þ 2sð1  sÞ cosð2sxÞ  : 2c sinð2sxÞ c

ð6:14Þ

By means of Eq. (6.9), we have a sequence of new nth-order Darboux transformations and transformed potential for the generalized Schrödinger Eq. (6.8).

1000

1e+31

0

5e+30 -20

-1000 -15

4 2

x

-10

0

-5

-2 -4

0

(a)

t

0

-20

4 2 0

-15

x -10

-2 -5

t

0

(b)

^ and (b) transformed potential v ^ with s ¼ 3; a ¼ tanhðtÞ; c ¼ cothðtÞ; h ¼ tanðtÞ. Fig. 2. (a) the transformed solution /

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40 0.2

20

0.1

0

0

-20 -40

-0.1 2

2 3

1 4

5

x

0 -1

6

2

2

1

3

4

t

0 x

5

6

-2

7

(a)

-1 7

t

-2

(b)

^ 1 and (b) is real part of transformed potential v ^ 1 with s ¼ 3; a ¼ tanhðtÞ; c ¼ tanðtÞ; h ¼ cothðtÞ. Fig. 3. (a) is real part of one-soliton solution /

Case (II). From above, we will now consider a specific case of Eq. (6.8)

v ¼cþ

m ¼ h sec2 ðsxÞcsc2 ðsxÞ;

3s2 2 sin ðsxÞ cos2 ðsxÞ; 2h

ð6:15Þ

where s is a real and positive constant, and h ¼ hðtÞ and c ¼ cðtÞ are arbitrary functions. With the help of Eq. (6.8), we have a particular solution of Eq. (6.15)

 Z  / ¼ ðsecðsxÞ þ x secðsxÞÞ exp i cdt :

ð6:16Þ

0.8

0.0003 0.00025

0.6

0.0002 0.4

0.00015 0.0001

0.2

5e-05 -10

-8

-6

-4

-2 0

2

-5e-05

x

4

6

8

10

-10

-8

-6

-4

-2 0

2

x

4

6

8

10

4

6

8

10

-0.2

(a)

(b) 0.3

0.015

0.25 0.2

0.01

0.15 0.1

0.005

0.05 -10

-8

-6

-4

-2 0 2

(c)

2

x

4

6

8

10

-10

-8

-6

-4

-2

0

-0.05

2

x

(d)

Fig. 4. Four kinds of one-soliton solutions (6.18) with different t. The related physical quantities are s ¼ 3; a ¼ tanhðtÞ; c ¼ tanðtÞ; h ¼ cothðtÞ and (a) t ¼ 0:1 (b) t ¼ 1 (c) t ¼ 10 (d) t ¼ 100.

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To perform the first-order Darboux transformation Eq. (2.19) on this function, we take an auxiliary solution u of Eq. (6.8)

 Z  u ¼ x secðsxÞ exp i cdt :

ð6:17Þ

Similarly, the two solutions / and u are linearly independent, as required for the Darboux transformation. Substituting mx 1 f ¼ 2m ; g ¼  2m 2 ; h ¼ 1 and Eqs. (6.8), (6.15)–(6.17) into Eqs. (2.17) and (2.19), we can obtain the following one-soliton solution

pffiffiffi ^ 1 ¼ C1 ð/Þ ¼ a f /  ux / ¼  a sinðsxÞ / x 2x u

v^ 1 ¼ v þ

pffiffiffipffiffiffi pffiffiffi ux pffiffiffi f þg f f þ2 f  xx u x 3

¼cþ

rffiffiffi Z 2 expði cdtÞ; h

ð6:18Þ

sffiffiffi h  pffiffiffi i 1pffiffiffi f  g x þ i ln a f x t f

2

s cosðsxÞ sin ðsxÞ sin ðsxÞ cos2 ðsxÞ a0 þ þi ; 2 hx hx a

ð6:19Þ

where a ¼ aðtÞ; c ¼ cðtÞ; h ¼ hðtÞ, are t-dependent arbitrary function and a0 ¼ dadtðtÞ ; h0 ¼ dhðtÞ . The graph of these one-soliton dt ^ 1 and transformed potential v ^ 1 is plotted in Fig. 3 for one choice of the parameters. solution / ^ 1 and transformed potential v ^ 1 are singular for some values of x and t. In particular for x ¼ 0 and The one-soliton solution / ^ 1 and v ^ 1 are 0. This means that x ¼ 0; t ¼ 0 are the t ¼ 0, the equality h ¼ cothðtÞ ¼ 0 is hold. Then the denominators of the / singularities of the solution (6.18) and (6.19). Fig. 4 shows four kinds of soliton profile structures with different time variables. Although solitons in Fig. 4 a,b,c and d hold larger time gaps, they both can propagate stably for long distances by the results of numerical simulations of linear pulses.

0.4

1

0.2

0

0

-1

-0.2 4

2

4

2

2

3 x

0

4

-2

5 6

2

3 x

t

0

4

-2

5

-4

-4

6

(a)

t

(b)

4 4 2

2

t 0

t 0 -2

-2

-4 -4 2

3

4

x

(c)

5

6

2

3

4

x

5

6

(d)

^ 2 and symmetric transformed potential v ^ 2 for the position-dependent mass Schrödinger equation with parameters: Fig. 5. A two-soliton solution / ^2. s ¼ 3; a ¼ sinðtÞ; c ¼ tanhðtÞ; h ¼ cothðtÞ. This figure shows that the two-soliton wave is periodic in two directions. (a) Perspective view of the wave / ^ 2 , with contour plot shown. The bright hexagons are crests and the dark hexagons are ^ 2 (c) Overhead view of the wave / (b) Perspective view of the wave v ^ 2 , with contour plot shown. The bright lines are crests and the dark lines are troughs. troughs. (d) Overhead view of the wave v

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^ 2 and new transformed potential v ^2 Similarly, with the use of Eqs. (6.18) and (6.19), we can obtain the two-soliton solution /

  Z  a sinð2sxÞ 2sx2 cos2 ðsxÞ þ 2sx cos2 ðsxÞ  x sinð2sxÞ  12 sinð2sxÞ  sx2  sx ^ /2 ¼  exp i cdt ; 2 4hx ð1 þ xÞ cosðsxÞ

v^ 2 ¼

x2 c þ 2xc þ c ð1 þ xÞ þ

 12 s2

2 2

3

þ

3

3s sin ðsxÞ cosðsxÞ þ 52 xs2 cos2 ðsxÞ  2xs2  12 x2 s2 cos4 ðsxÞ þ 2xs sin ðsxÞ cosðsxÞ hð1 þ xÞ 2

4

2

2

cos ðsxÞ sin ðsxÞ þ 2xs sin ðsxÞ hð1 þ xÞ

2

þ

2

3

4 sin ðsxÞ cos2 ðsxÞ þ s sin ðsxÞ cosðsxÞ 2

hxð1 þ xÞ

2

þ

sin ðsxÞ cos2 ðsxÞ hx2 ð1 þ xÞ

2

þi

2a0

a

;

^ 2 and transformed potential v ^ 2 is plotted in Fig. 5 for one choice of the parameters. The graph of these two-soliton solution / ^ 2 and transformed potential v ^ 2 are singular for some values of x and t. In particular for The two-soliton solution / x ¼ 0; 1; kp=3 þ p=6ðk 2 ZÞ and t ¼ 0, the equalities h ¼ cothðtÞ ¼ 0; cosðsxÞ ¼ 0; ð1 þ xÞ ¼ 0 are hold. Then the denomina^ 2 and v ^2 ^ 2 are 0. This means that x ¼ 0; 1; kp=3 þ p=6ðk 2 ZÞ and t ¼ 0 are the singularities of the solution / tors of the / ^2. and / With the help of nth-order Darboux transformation [see Eq. (5.8) and Eq. (5.9)], we can obtain

pffiffiffiffiffiffi!n sinð2sxÞ 2h W nþ1;ð/i Þ ð/Þ ^ /n ¼ Cn ð/Þ ¼ a ; W n;ð/i Þ 4h 39 " # !nðn1Þ pffiffiffiffiffiffi 8 pffiffiffiffiffiffi 2 2 2 2 = sin ð2xtÞ Nx sin2 ð2xtÞ sinð2sxÞ 2h
where /1 ; /1 ; . . . ; /n are solutions of the generalized Schrödinger Eq. (6.8), g0 ¼ 0 1 RB B and Nðx; tÞ ¼  B @

cx secðsxÞ

sin2 ðsxÞ cosðsxÞþx sin3 ðsxÞ þs tanðsxÞ exp h

 R i



xs2 exp

cdt þ

a0 4n a

x

is an arbitrary function with t-dependent

C  R C Cdx. i cdt A

2½1þsx tanðsxÞ

From above, we can obtain a set of solutions of Eq. (6.1)

n o ^ 1 ; v^ 1 Þ; ð/ ^ 2 ; v^ 2 Þ; . . . ; ð/ ^ n ; v^ n Þ; . . . : ð/; v Þ; ð/

7. Conclusions and discussions Under the Darboux transformation operator technique, in this paper, we investigate the first-order Darboux transformations and the transformed potential for the linear, time-dependent generalized Schrödinger equation (1.1) in (1 + 1)-dimension. We take into account the real-valued function condition of the transformed potential. We further research on nth-order Darboux transformation and on its connection with and factorization of the corresponding effective mass Hamiltonian. And we have proved the equivalence of the supersymmetry formalism and the Darboux transformation. Finally, the results are applied into some examples of Schrödinger equation and obtained the corresponding transformed solutions and transformed potentials that are given out analytically and graphically. Acknowledgments The authors are grateful to referees’s and editor’s comments. The work is partially supported by the Doctoral Academic Freshman Award of Ministry of Education of China under the grant 0213–812002, Doctaral Fund of Ministry of Education of China under the grant 20100041120037, Natural Sciences Foundation of China under the grant 11026165, 50909017 and the Fundamental Research Funds for the Central Universities under grant JGK101677 and DUT11SX03. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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