Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow

Shiping Zhong; Zehui Zhao; Xinjie Wan

doi:10.1515/math-2022-0600

Article Open Access

Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow

Shiping Zhong , Zehui Zhao and Xinjie Wan

Published/Copyright: July 3, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, the coupled matrix nonlinear Schrödinger (NLS) type equations are gauge equivalent to the equation of Schrödinger flow from R 1 to complex Grassmannian manifold G ˜ n , k = GL ( n , C ) ∕ GL ( k , C ) × GL ( n − k , C ) , which generalizes the correspondence between Schrödinger flow to the complex 2-sphere C S 2 ( 1 ) ↪ C 3 and the coupled Landau-Lifshitz (CLL) equation. This gives a geometric interpretation of the matrix generalization of the coupled NLS equation (i.e., CLL equation) via Schrödinger flow to the complex Grassmannian manifold G ˜ n , k . Finally, we explicit soliton solutions of the Schrödinger flow to the complex Grassmannian manifold G ˜ 2 , 1 .

Keywords: coupled matrix nonlinear Schrödinger-type equations; complex Grassmannian manifold; Schrödinger flow; gauge equivalence; soliton solutions

MSC 2010: 53C35; 53A04; 37K10; 35Q55; 35C08

1 Introduction

In this article, the matrix generalization of the second Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy (i.e., the coupled matrix nonlinear Schrödinger [NLS] type equations) (e.g., see [1–3]):

(1) ε Ψ 1 t − Ψ 1 x x + 2 Ψ 1 Ψ 2 Ψ 1 = 0 , ε Ψ 2 t + Ψ 2 x x − 2 Ψ 2 Ψ 1 Ψ 2 = 0 ,

where Ψ 1 = Ψ 1 ( x , t ) , Ψ 2 = Ψ 2 ( x , t ) are k × ( n − k ) and ( n − k ) × k complex matrix-valued functions and ε 2 = ± 1 . Note that the matrix form of the coupled matrix NLS equations (1) is

(2) u t = 1 ε 2 [ σ 3 , u x x ] − 1 2 ε 2 [ u , [ u , [ σ 3 , u ] ] ] ,

where

u = u ( x , t ) = 0 Ψ 1 Ψ 2 0 : R 2 → m ( see Secton 2 ) , σ 3 = ε 2 I k 0 0 − I n − k ,

which is called the G ˜ n , k -NLS equation (see Section 2), where [ ⋅ , ⋅ ] denotes Lie bracket. These symmetry reductions and special orders of Ψ 1 and Ψ 2 lead to several relevant cases (e.g., see [1–16] for more details on this fact) in the following:

Matrix Case 1 (Matrix-Nonlocal-types): The matrix nonlocal NLS equation (e.g., see [2]):

(3) − 1 Q t ( x , t ) − Q x x ( x , t ) + 2 ζ 2 Q ( x , t ) Q ∗ ( − x , t ) Q ( x , t ) = 0 , x , t ∈ R ,

is a reduction of equation (1) by Ψ 1 ( x , t ) = Q ( x , t ) and Ψ 2 ( x , t ) = ζ 2 Q ∗ ( − x , t ) , n = 2 k , ζ 2 = ± 1 , ε = − 1 , where the dagger indicates the complex conjugate transpose. In the special case where Q is a column vector, then equation (3) reduces to the vector nonlocal NLS equation (Vector-Nonlocal-types) (see [2]):

(4) Q − 1 , 1 ( n − 1 ) : − 1 Q t ( x , t ) − Q x x ( x , t ) + 2 ζ 2 Q ( x , t ) Q ∗ ( − x , t ) Q ( x , t ) = 0 , x , t ∈ R ,

namely, equation (4) is a reduction of equation (1) by Ψ 1 ( x , t ) = Q ( x , t ) = ( q 1 ( x , t ) , … , q n − 1 ( x , t ) ) , Ψ 2 ( x , t ) = ζ 2 Q ∗ ( − x , t ) , k = 1 , ζ 2 = ± 1 , ε = − 1 . And the natural generalization of equation (4) is a hierarchy of two-family-parameter { ( ε x j → , ε t j → ) ∣ ε x j → , ε t j → = ± 1 , j = 1 , 2 , … , n − 1 } equation (called Q ε x j → , ε t j → ( n − 1 ) hierarchy) (see [15]):

(5) Q ε x j → , ε t j → ( n − 1 ) : − 1 Q t ( x , t ) − Q x x ( x , t ) + 2 ζ 2 Q ( x , t ) Q ∗ ( ε x j → x , ε t j → t ) Q ( x , t ) = 0 , x , t ∈ R ,

which is a reduction of equation (1) by Ψ 1 ( x , t ) = Q ( x , t ) = ( q 1 ( x , t ) , … , q n − 1 ( x , t ) ) , Ψ 2 ( x , t ) = ζ 2 Q ∗ ( ε x j → x , ε t j → t ) = ( q 1 ( ε x 1 → x , ε t 1 → t ) , … , q n − 1 ( ε x n − 1 → x , ε t n − 1 → t ) ) , k = 1 , ζ 2 = ± 1 , ε = − 1 .

Matrix Case 2 (Matrix-Local-types). The matrix NLS equation by Fordy and Kulish in [9] (or see [8,12,13]):

(6) − 1 Q t ( x , t ) − Q x x ( x , t ) + 2 ζ 2 Q ( x , t ) Q ∗ ( x , t ) Q ( x , t ) = 0 , x , t ∈ R ,

which is a reduction of equation (1) by Ψ 1 ( x , t ) = Q ( x , t ) , Ψ 2 ( x , t ) = ζ 2 Q ∗ ( x , t ) , n = 2 k , ζ 2 = ± 1 , ε = − 1 . In the special case where Q is a column vector, then equation (6) reduces to the vector local NLS equation (Vector-Local-types) (see [6,10,16]):

(7) Q 1 , 1 ( n − 1 ) : − 1 Q t ( x , t ) − Q x x ( x , t ) + 2 ζ 2 Q ( x , t ) Q ∗ ( x , t ) Q ( x , t ) = 0 ,

i.e., equation (7) is a reduction of equation (1) by Ψ 1 ( x , t ) = Q ( x , t ) = ( q 1 ( x , t ) , … , q n − 1 ( x , t ) ) , Ψ 2 ( x , t ) = ζ 2 Q ∗ ( x , t ) , k = 1 , ζ 2 = ± 1 , ε = − 1 . Though equations (1)–(7) have some dynamical properties, e.g., solitons, general N-solitons, multi-soliton solutions, reflectionless solutions, and rogue wave solutions. Hence, a quite relevant question arises: does there exist a unified geometric interpretation of equations (3)–(7) or equation (1)?

This question is motivated by the work of geometric characterization of equation (6); it is proved by Terng and Uhlenbeck [13] that equation (6) is gauge equivalent to the complex compact Grassmannian manifolds U ( n ) ∕ U ( k ) × U ( n − k ) , and by Terng and Thorbergsson in [17] for the other three classical Hermitian symmetric spaces. Chen [18] generalized their result to the cases of u ( k , n − k ) and G l ( n , R ) . This gives a unified geometric interpretation of the three typical second-order matrix NLS equations in the second matrix-AKNS hierarchies via Schrödinger flow. Along this route, in [19–22], the authors gave a complete description of the theory of vortex filament on symmetric algebras up to the second-order and third-order approximation from a purely geometric way. Recently, the first author showed that the coupled NLS equations, which are a reduction of equation (1) by Ψ 1 = φ 1 ( x , t ) , Ψ 2 = φ 2 ( x , t ) , k = 1 , n = 2 , ε = − 1 , are geometrically interpreted as the equation of Schrödinger flow from R 1 to the complex 2-sphere C S 2 ( 1 ) = { ( z 1 , z 2 , z 3 ) ∈ C 3 : z 1 2 + z 2 2 + z 3 2 = 1 } ↪ R 3 , 3 ≅ C 3 with the standard holomorphic metric, and is also gauge equivalent to the unconventional system of the coupled Landau-Lifshitz (CLL) equation (see [23,24]). CLL reads as the following evolution equation:

S t = − − 1 2 [ S , S x x ] ,

where S = S ( x , t ) is a 2 × 2 complex-matrix with S 2 = I 2 × 2 ( I 2 × 2 stands for the 2 × 2 unit matrix) and tr S = 0 . Some physical and geometrical properties of CLL are also discussed in [23,24].

On the other hand, by the loop group factorization method, Terng and Uhlenbeck [25] first constructed Bäcklund transformations for the Zakharov-Shabat (ZS)-AKNS sl( n , C )-hierarchy. Afterward, using this method, many hierarchies (such as SU(1,1)-hierarchy, vmKdV-hierarchy, A ( 1 ) n − 1 -KdV-hierarchy, Gelfand-Dickey-hierarchy, B ˆ n ( 1 ) -hierarchy and A ˆ 2 n ( 2 ) -KdV-hierarchy) of Bäcklund and Darboux transformations are obtained (see [26–32]).

This article is organized as follows. Section 2 gives preliminary about the symmetric Lie algebras G l ( n , C ) and the second-order matrix-AKNS hierarchy. In Section 3, we show that the coupled matrix NLS-type equations are gauge equivalent to the equation of Schrödinger flow from R 1 to complex Grassmannian manifold G ˜ n , k , which generalizes the correspondence between Schrödinger flow to the complex 2-sphere C S 2 ( 1 ) ↪ C 3 and the CLL equations. Section 4 explicits the soliton solutions of the Schrödinger flow to the complex Grassmannian manifold G ˜ 2 , 1 .

2 Symmetric Lie algebras and the second-order matrix-AKNS hierarchy

In this section, we recall some fundamental facts about the complex general linear Lie algebra G l ( n , C ) ( n ≥ 2 ) with index k ( 1 ≤ k < n ): the space of all n × n complex matrices and the second-order matrix-AKNS hierarchy.

First, we recall the concept of symmetric Lie algebra g . The so-called symmetric Lie algebra g is a Lie algebra that has a decomposition as a vector space sum: g = k ⊕ m satisfying the (bracket) symmetric conditions: [ k , k ] ⊂ k , [ m , m ] ⊂ k and [ k , m ] ⊂ m (see [9,33,34]). In such a symmetric Lie algebra, there is an element denoted by σ 3 in k such that k = Kernel ( ad σ 3 ) = { χ ∈ g ∣ [ χ , σ 3 ] = 0 } . A homogeneous space is a manifold M with a transitive action of a Lie group G. Equivalently, it is a manifold of the form G ∕ K , where G is a Lie group and K is a closed subgroup of G .

Now, we recall some results of the complex general linear Lie algebra G l ( n , C ) ( n ≥ 2 ) with index k ( 1 ≤ k < n ).

Lemma 1

The complex general linear Lie algebra G l ( n , C ) = k ⊕ m is a symmetric Lie algebra, where

k = Kernel ( ad σ 3 ) = A k × k 0 0 B ( n − k ) × ( n − k ) ∈ G l ( n , C )

and

m = 0 U k × ( n − k ) V ( n − k ) × k 0 ∈ G l ( n , C ) ∀ U k × ( n − k ) and V ( n − k ) × k .

And the adjoint obit space

G ˜ n , k = { E − 1 σ 3 E ∣ ∀ E ∈ GL ( n , C ) }

is a homogeneous symmetric space, where

(8) σ 3 = ε 2 I k 0 0 − I n − k , ε 2 = ± 1 .

Proof

Let us define a (left) operation of the complex general linear Lie group GL ( n , C ) on G ˜ n , k by

Φ : GL ( n , C ) × G ˜ n , k → G ˜ n , k , ( X , γ ) ↦ Φ ( X , γ ) = X ∘ γ = X γ X − 1 ,

since

Φ ( X , γ ) = X ∘ γ = X γ X − 1 = X E − 1 σ 3 E X − 1 = X E − 1 σ 3 ( X E − 1 ) − 1 .

It is obvious to see that an operation satisfies the following:

I n × n ∘ γ = I n × n γ I n × n − 1 = γ , ∀ γ ∈ G ˜ n , k , ( X Y ) ∘ γ = ( X Y ) γ ( X Y ) − 1 = X ( Y γ Y − 1 ) X − 1 = X ∘ ( Y ∘ γ ) , ∀ X , Y ∈ GL ( n , C ) ,

and the action is transitive. In fact, ∀ γ 1 = E 1 − 1 σ 3 E 1 , γ 2 = E 2 − 1 σ 3 E 2 ∈ G ˜ n , k , then ∃ X = E 2 − 1 E 1 ∈ GL ( n , C ) , s.t.,

X ∘ γ 1 = E 2 − 1 E 1 γ 1 ( E 2 − 1 E 1 ) − 1 = E 2 − 1 E 1 E 1 − 1 σ 3 E 1 E 1 − 1 E 2 = E 2 − 1 σ 3 E 2 = γ 2 .

Moreover, the isotropy group at the point σ 3 ∈ G ˜ n , k is

G σ 3 = { X ∈ GL ( n , C ) ∣ X ∘ σ 3 = σ 3 } = { X ∈ GL ( n , C ) ∣ X σ 3 = σ 3 X } = A k × k 0 0 B ( n − k ) × ( n − k ) ∈ GL ( n , C ) ≔ K .

Hence, G ˜ n , k is a homogeneous space of the group GL ( n , C ) ; in fact, the map

G ∕ K → G ˜ n , k , [ x ] ↦ X ∘ σ 3

is diffeomorphism and the K-principle bundle K → GL ( n , C ) → G ˜ n , k . The Lie algebra G l ( n , C ) ( n ≥ 2 ) with index k ( 1 ≤ k < n ) of GL ( n , C ) decomposes as G l ( n , C ) = k ⊕ m , where

k = A k × k 0 0 B ( n − k ) × ( n − k ) ∈ G l ( n , C )

is the Lie algebra of K with the property: [ k , k ] ⊂ k and

m = 0 U k × ( n − k ) V ( n − k ) × k 0 ∈ G l ( n , C ) ∀ U k × ( n − k ) and V ( n − k ) × k .

It is easy to verify that the symmetric connections fulfill

[ k , m ] ⊂ m , [ m , m ] ⊂ k .

Therefore, G l ( n , C ) = k ⊕ m is a symmetric algebra, and hence, G ˜ n , k is a homogeneous symmetric space.□

Now, we recall the concept of ( J 2 = ± 1 )-Kähler manifold ( M , J , g ) (such as see [35]). A manifold will be called to have a J 2 = ± 1 -structure if an almost complex ( J 2 = − 1 ) or almost product ( J 2 = 1 ) structure J is an isometry and ∇ J = 0 , where ∇ denotes the Levi-Civita connection of g . It is also said that ( M , J , g ) is a ( J 2 = ± 1)-Kähler manifold.

Lemma 2

The adjoint obit space G ˜ n , k is a J 2 = ± 1 -Kähler manifold with tensor J γ = [ γ , ⋅ ] at a point γ ∈ G ˜ n , k .

Proof

First, we consider the tangent space of a point γ = E − 1 σ 3 E on G ˜ n , k . We can decompose the element P of G l ( n , C ) of the form P = diag(P) + P ˜ , where P ˜ = off-diag(P) ∈ m is defined to be the off-diagonal part of P with respect to the decomposition G l ( n , C ) = k ⊕ m . So,

σ ( t ) = exp ( − t diag(P) − t P ˜ ) σ 3 exp ( t diag(P) + t P ˜ )

is a curve on G ˜ n , k passing the point σ 3 . By taking its derivation, we can obtain the tangent space of the point σ 3 of G ˜ n , k :

σ 3 P ˜ − P ˜ σ 3 + σ 3 diag(P) − diag(P) σ 3 = [ σ 3 , P ˜ ] ,

which is a matrix of form P ˜ , because diag(P) is communicative with σ 3 . So the tangent space T γ G ˜ n , k at γ consisted of

T γ G ˜ n , k = { E − 1 [ σ 3 , P ] E ∣ ∀ P ∈ G l ( n , C ) } = { E − 1 [ σ 3 , P ˜ ] E ∣ ∀ P ˜ ∈ m } .

Hence, ∀ X = E − 1 [ σ 3 , P ] E ∈ T γ G ˜ n , k , P ∈ m . Let us define an operation J γ at γ by

(9) J γ = [ γ , ⋅ ] : T γ G ˜ n , k → T γ G ˜ n , k , X ↦ J γ ( X ) = [ γ , X ] .

Therefore, ∀ X = E − 1 [ σ 3 , P ˜ ] E ∈ T γ G ˜ n , k ; it is a direct verification that

J γ ( X ) = [ γ , X ] = γ X − X γ = E − 1 σ 3 E E − 1 [ σ 3 , P ˜ ] E − E − 1 [ σ 3 , P ˜ ] E E − 1 σ 3 E = E − 1 σ 3 [ σ 3 , P ˜ ] E − E − 1 [ σ 3 , P ˜ ] σ 3 E = ε 2 2 E − 1 P ˜ E − ε 2 2 E − 1 ( − P ˜ ) E = ε 2 E − 1 P ˜ E , J γ 2 ( X ) = [ γ , [ γ , X ] ] = [ γ , ε 2 E − 1 P ˜ E ] = E − 1 σ 3 E ε 2 E − 1 P ˜ E − ε 2 E − 1 P ˜ E E − 1 σ 3 E = ε 2 E − 1 σ 3 P ˜ E − ε 2 E − 1 P ˜ σ 3 E = ε 2 E − 1 [ σ 3 , P ˜ ] E = ε 2 X = ± X .

This shows that the J 2 = ± 1 -Kähler structure of G ˜ n , k is given by (9). In fact, we can define bi-invariant metric on G ˜ n , k :

⟨ ⋅ , ⋅ ⟩ γ : T γ G ˜ n , k → T γ G ˜ n , k , ( X , Y ) ↦ ⟨ X , Y ⟩ γ = − tr ( X Y ) .

It is a direct verification that ∀ X , Y , Z ∈ T γ G ˜ n , k , we have

□ d ω γ ( X , Y , Z ) = X ( ω γ ( Y , Z ) ) − Y ( ω γ ( X , Z ) ) + Z ( ω γ ( X , Y ) ) − ω γ ( [ X , Y ] , Z ) + ω γ ( [ X , Z ] , Y ) − ω γ ( [ Y , Z ] , X ) = ⟨ ∇ X ( J γ Y ) , Z ⟩ γ + ⟨ J γ Y , ∇ X Z ⟩ γ − ⟨ ∇ Y ( J γ X ) , Z ⟩ γ − ⟨ J γ X , ∇ Y Z ⟩ γ + ⟨ ∇ Z ( J γ X ) , Y ⟩ γ + ⟨ J γ X , ∇ Z Y ⟩ γ − ⟨ J γ [ X , Y ] , Z ⟩ γ + ⟨ J γ [ X , Z ] , Y ⟩ γ − ⟨ J γ [ Y , Z ] , X ⟩ γ = 0 .

From Lemma 1, the symmetric space G ˜ n , k is simply written as G ˜ n , k = GL ( n , C ) ∕ GL ( k , C ) × GL ( n − k , C ) and is called the complex Grassmannian manifold.

Next, let us recall the three typical classes of the Hermitian symmetric Lie algebras G l ( n , C ) with index k ( 1 ≤ k < n ) having three types.

The first subclass of symmetric Lie algebras G l ( n , C ) consists of Hermitian symmetric Lie algebras u ( n ) ⊂ G l ( n , C ) ( n ≥ 2 ) with index k ( 1 ≤ k < n ) of compact type. In fact, for any given 1 ≤ k < n , u ( n ) is decomposable as u ( n ) = k 1 ⊕ m 1 satisfy the symmetric conditions, where

k 1 = Kernel ( ad σ 3 ) = A k × k 0 0 B ( n − k ) × ( n − k ) ∈ u ( n )

and

m 1 = 0 U k × ( n − k ) − U ( n − k ) × k ∗ 0 ∈ u ( n ) ,

where U ( n − k ) × k ∗ stands for the transposed conjugate matrix of U k × ( n − k ) , σ 3 is given by (8), and ε 2 = − 1 .

The second subclass of symmetric Lie algebras G l ( n , C ) consists of Hermitian symmetric Lie algebras u ( k , n − k ) ⊂ G l ( n , C ) with index k ( 1 ≤ k < n ) of noncompact type. In this case, we see that u ( k , n − k ) is decomposable as u ( k , n − k ) = k 2 ⊕ m 2 satisfy the symmetric conditions, where

k 2 = Kernel ( ad σ 3 ) = A k × k 0 0 B ( n − k ) × ( n − k ) ∈ u ( k , n − k )

and

m 2 = 0 U k × ( n − k ) U ( n − k ) × k ∗ 0 ∈ u ( k , n − k ) ,

where σ 3 is given by (8), and ε 2 = − 1 .

The third subclass of symmetric Lie algebras G l ( n , C ) consists of para-Hermitian symmetric Lie algebras G l ( n , R ) ⊂ G l ( n , C ) ( n ≥ 2 ) with index k ( 1 ≤ k < n ). In this case, for any given 1 ≤ k < n , we see that G l ( n , R ) = k 3 ⊕ m 3 is a symmetric Lie algebra, where

k 3 = Kernel ( ad σ 3 ) = A k × k 0 0 B ( n − k ) × ( n − k ) ∈ G l ( n , R )

and

m 3 = 0 U k × ( n − k ) + U ( n − k ) × k − 0 ∈ G l ( n , R ) ∀ U k × ( n − k ) + and U ( n − k ) × k − ,

where σ 3 is given by (8), and ε 2 = 1 .

Finally, we briefly review the second matrix-AKNS hierarchy on a symmetric Lie algebra. It is well known that the coupled matrix NLS equations are equivalent to the compatibility condition of a Lax pair for the potential matrix

u = 0 Ψ 1 ( x , t ) Ψ 2 ( x , t ) 0 ∈ m ,

where the first equation in the Lax pair is the so-called matrix ZS or AKNS system (see [1]). Specifically, the coupled matrix NLS equations (1) admit the Lax pair

(10) E x = ( − σ 3 λ + u ) E , E t = ( σ 3 λ 2 − u λ + ε 2 P − 1 ( u ) ) E ,

where E : R 2 × C → GL ( n , C ) and

P − 1 ( u ) = 2 σ 3 ( u x − u 2 ) = ε − Ψ 1 ( x , t ) Ψ 2 ( x , t ) Ψ 1 x ( x , t ) − Ψ 2 x ( x , t ) Ψ 2 ( x , t ) Ψ 1 ( x , t ) .

We call E : R 2 → GL ( n , C ) a frame of the solution u of the G ˜ n , k -NLS equation (2) with E ( 0 , 0 , λ ) = I n × n . Hence, rewrite equation (10) as:

(11) ε 2 u t = P − 1 ( u ) x + [ P − 1 ( u ) , u ] .

3 Schrödinger flow into the complex Grassmannian manifold G ˜ n , k

It is well known that the equation of Schrödinger flow from a Riemannian manifold ( M , g ) to a J 2 = ± 1 -Kähler manifold ( N , J , h ) , where J satisfies J 2 = ± 1 and compatible to the metric h , is given by the following Hamiltonian gradient flow:

(12) u t = J u τ ( u ) ,

where τ ( u ) is the tension field of map u : M → N ([13,36–38]).

Theorem 1

The equation (12) of Schrödinger flow from R 1 to complex Grassmannian manifold G ˜ n , k is

(13) γ t = ε 2 [ γ , γ x x ] .

Proof

Let γ = E − 1 σ 3 E ∈ G ˜ n , k be the equation of Schrödinger flow from R 1 to complex Grassmannian manifold G ˜ n , k , namely, γ solves [13]

J γ γ t = ∇ γ x γ x ,

where E = E ( x , t ) ∈ GL ( n , C ) , J γ is the J 2 = ± 1 -Kähler structure of G ˜ n , k at the point γ , ∇ x the covariant derivative ∇ ∂ ∂ x on the pull-back bundle γ − 1 T G ˜ n , k induced from the Levi-Civita connection on G ˜ n , k , and by γ x the ∇ x γ .

Let γ = E − 1 ( t , x ) σ 3 E ( t , x ) be a map from the line R to G ˜ n , k , where σ 3 is given by (8). Without loss of generality, we may assume that E satisfies: E x = P E for some P ∈ m , where m fits with G l ( n , C ) = k ⊕ m . In fact, if E does not meet the requirement, we may make a transform:

E → A 0 0 B E ,

(this is because the form E − 1 σ 3 E is invariant up to the transform) such that by suitably choosing A and B (through solving a linear differential system of A and B ), P can be modified so that the new P satisfies P ∈ m . Based on this fact, it is shown that

∇ γ x γ = γ x = E − 1 [ σ 3 , P ] E (taking the tangent part) , ∇ γ x γ x = γ x x + E − 1 [ P , [ σ 3 , P ] ] E (taking the tangent part of) ( ∇ x γ ) x .

Hence, we have

□ γ t = ε 2 J γ ∇ γ x γ x = ε 2 [ γ , ∇ γ x γ x ] = ε 2 [ γ , γ x x ] + [ γ , E − 1 [ P , [ σ 3 , P ] ] E ] = ε 2 [ γ , γ x x ] + [ E − 1 σ 3 E , E − 1 [ P , [ σ 3 , P ] ] E ] = ε 2 [ γ , γ x x ] , ( since P ∈ m ) .

Hence, the Lax pair of equation (13) is

(14) δ x = − γ λ δ , δ t = ( γ λ 2 − ε 2 [ γ , γ x ] λ ) δ .

Let C S 2 , ε ( 1 ) = { ( z 1 , z 2 , z 3 ) ∈ C 3 : z 1 2 − ε 2 z 2 2 + z 3 2 = 1 } ↪ C 3 with the metric (i.e., d z 2 = d z 1 2 − ε 2 d z 2 2 + d z 3 2 ). Now, for vectors u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ) in C 3 , we define the cross-product of u and v by:

u × ε v = ( u 2 v 3 − u 3 v 2 , − ε 2 ( u 3 v 1 − u 1 v 3 ) , u 1 v 2 − u 2 v 1 ) .

Corollary 1

The equation of Schrödinger flow from R 1 to complex Grassmannian manifold G ˜ 2 , 1 is

(15) s t = ε 2 s × ε s x x .

Proof

∀ E = a b c d ∈ GL ( 2 , C ) , then

τ : G ˜ 2 , 1 → C S 2 , ε ( 1 ) , γ = E − 1 σ 3 E = 1 a d − b c d − b − c a ε 2 0 0 − ε 2 a b c d = ε 2 ( a d − b c ) a d + b c 2 b d − 2 a c − a d − b c ∈ G ˜ 2 , 1 ↦ s 3 s 1 − ε s 2 s 1 + ε s 2 − s 3 ≔ S ↔ s = ( s 1 , s 2 , s 3 ) ,

where ( s 1 , s 2 , s 3 ) = b d − a c a d − b c , ε ( b d + a c ) a d − b c , a d + b c a d − b c ∈ C 3 satisfying s 1 2 − ε 2 s 2 2 + s 3 2 = 1 . It is a direct computation that γ × ε γ x x is just [ τ − 1 γ , τ ∗ − 1 γ x x ] , where τ ∗ is the tangent map induced from τ . Hence, the equation of Schrödinger flow from R 1 to G ˜ 2 , 1 returns to CLL equations ( ε -CLL)

(16) S t = ε 2 [ S , S x x ] ,

where S = S ( x , t ) is a 2 × 2 matrix of the form

(17) S ( x , t ) = s 3 ( x , t ) s 1 ( x , t ) − ε s 2 ( x , t ) s 1 ( x , t ) + ε s 2 ( x , t ) − s 3 ( x , t ) ,

where s 1 2 − ε 2 s 2 2 + s 3 2 = 1 by the requirement S 2 = I , S is identified with a vector s = ( s 1 , s 2 , s 3 ) ∈ C S 2 , ε ( 1 ) ↪ C 3 , and the ε -CLL equation (16) becomes equation (15).□

Note that when ε = − 1 , C S 2 , ε ( 1 ) becomes the complex 2-sphere C S 2 ( 1 ) = { ( z 1 , z 2 , z 3 ) ∈ C 3 : z 1 2 + z 2 2 + z 3 2 = 1 } . From Corollary 1, the equation of Schrödinger flow the complex 2-sphere C S 2 ( 1 ) with the standard holomorphic metric (i.e., d z 2 = d z 1 2 + d z 2 2 + d z 3 2 ) is

s t = − s × s x x ,

where × denotes the cross-product of C 3 , which also refers to [23]. By Corollary 1, we can interpret C S 2 , ε ( 1 ) as a symmetric space G ˜ 2 , 1 = GL ( 2 , C ) ∕ GL ( 1 , C ) × GL ( 1 , C ) .

Theorem 2

Equations (11) and (13) are gauge equivalent to each other.

Proof

Let u : R 2 → m be a smooth solution of equation (11), we let G : R 2 → GL ( n , C ) satisfy the following linear system:

G x = − G u , G t = − ε 2 G P − 1 ( u ) , ( since ε 2 u t = P − 1 ( u ) x + [ P − 1 ( u ) , u ] , G exist ) .

By using the gauge transformation:

δ = G φ , γ = G σ 3 G − 1 ,

where φ satisfies equation (10), we see that

δ x = ( G φ ) x = G x φ + G φ x = − G u φ + G ( − σ 3 λ + u ) φ = − G σ 3 λ φ = − γ λ δ

and

δ t = ( G φ ) t = G t φ + G φ t = − ε 2 G P − 1 ( u ) φ + G ( σ 3 λ 2 − u λ + ε 2 P − 1 ( u ) ) φ = ( G σ 3 λ 2 − G u λ ) φ = ( γ λ 2 − G u G − 1 λ ) δ = ( γ λ 2 − ε 2 [ γ , γ x ] λ ) δ , since [ γ , γ x ] = ε 2 G u G − 1 .

Namely, γ = γ ( x , t ) = G ( x , t ) σ 3 G ( x , t ) − 1 is a solution of the Schrödinger flow (13) on G ˜ n , k . Hence, we have a gauge transform from equation (11) to equation (13).

Next, if γ = E − 1 ( t , x ) σ 3 E ( t , x ) ∈ G ˜ n , k is a solution of the Schrödinger flow equation (13). Without loss of generality, we may assume that E satisfies (see [39]):

E x E − 1 = 0 R 1 R 2 0 ≔ P 1 ∈ m .

By γ t = ε 2 [ γ , γ x x ] , it is a direct verification that

[ σ 3 , E t E − 1 ] = P 1 x , since J γ ( γ x x ) = ε 2 E − 1 P 1 x E , γ t = E − 1 [ σ 3 , E t E − 1 ] E .

Hence, we have

E t E − 1 = ξ 11 R 1 x ε − R 2 x ε ξ 22 ≔ P 2 .

Moreover, from the integrability condition: E x t = E t x , i.e., P 1 t = P 2 x + [ P 2 , P 1 ] , we can obtain

ξ 11 = − 1 ε R 1 R 2 + C 1 ( t ) , ξ 22 = 1 ε R 2 R 1 + C 2 ( t ) ,

where C ( t ) = C 1 ( t ) 0 0 C 2 ( t ) depend only on t , and

E t E − 1 = 1 ε − R 1 R 2 R 1 x − R 2 x R 2 R 1 + C 1 ( t ) 0 0 C 2 ( t ) = 2 ε 2 σ 3 ( P 1 x − P 1 2 ) + C ( t ) .

In order to vanish C 1 and C 2 , we take

ρ = ρ 1 0 0 ρ 2 ,

where ρ 1 = ρ 1 ( t ) , ρ 2 = ρ 2 ( t ) depend only on t , satisfying

ρ 1 t = − ρ 1 C 1 , ρ 2 t = − ρ 2 C 2 .

Let g = ρ E ; hence,

(18) g x g − 1 = ρ E x E − 1 ρ − 1 = ρ P 1 ρ − 1 = ρ 0 R 1 R 2 0 ρ − 1 ≔ 0 Ψ 1 Ψ 2 0 ≔ u ,

and moreover,

(19) g t g − 1 = ( ρ t E + ρ E t ) E − 1 ρ − 1 = ρ t E + ρ 2 ε 2 σ 3 ( P 1 x − P 1 2 ) E + C ( t ) E E − 1 ρ − 1 = 2 ε 2 ρ σ 3 ( P 1 x − P 1 2 ) ρ − 1 = 2 ε 2 σ 3 ( ρ P 1 x ρ − 1 − ρ P 1 2 ρ − 1 ) = 2 ε 2 σ 3 ( u x − u 2 ) ≔ 1 ε 2 P − 1 ( u ) ,

where P − 1 ( u ) = 2 σ 3 ( u x − u 2 ) . Let G = g − 1 , φ = G − 1 δ , where δ satisfies equation (14). From equations (18) and (19), we obtain

G x = − G u , G t = − 1 ε 2 G P − 1 ( u ) = − ε 2 G P − 1 ( u ) .

Moreover, we have

φ x = − G − 1 G x G − 1 δ + G − 1 δ x = ( − σ 3 λ + u ) φ , φ t = − G − 1 G t G − 1 δ + G − 1 δ t = ( σ 3 λ 2 − u λ + ε 2 P − 1 ( u ) ) φ ,

i.e., φ satisfies equation (10). Hence, we have proved that equation (13) is gauge equivalent to equation (10).□

From equations (10), (11), (13), (14), and Theorem 2, we have the following:

Theorem 3

The coupled matrix NLS equations (1) on G l ( n , C ) = k ⊕ m are gauge equivalent to the equation (13) of Schrödinger flow from R 1 to complex Grassmannian manifold G ˜ n , k .

Corollary 2

The matrix NLS equation (6) on the first subclass u ( n ) = k 1 ⊕ m 1 or the second subclass u ( k , n − k ) = k 2 ⊕ m 2 is gauge equivalent to the equation (13) of Schrödinger flow from R 1 to complex Grassmannian manifold U ( n ) ∕ U ( k ) × U ( n − k ) or U ( k , n − k ) ∕ U ( k ) × U ( n − k ) (see [13,18]).

Corollary 3

The matrix version of the nonlinear heat equations

Ψ 1 t − Ψ 1 x x + 2 Ψ 1 Ψ 2 Ψ 1 = 0 , Ψ 2 t + Ψ 2 x x − 2 Ψ 2 Ψ 1 Ψ 2 = 0 ,

which are a reduction of equation (1) by ε = 1 , and Ψ 1 = Ψ 1 ( x , t ) , Ψ 2 = Ψ 2 ( x , t ) are k × ( n − k ) and ( n − k ) × k real matrix-valued function, on the third subclass G l ( n , R ) = k 3 ⊕ m 3 is gauge equivalent to equation (13) of Schrödinger flow from R 1 to manifold GL ( n , R ) ∕ GL ( k , R ) × GL ( n − k , R ) (see [18]).

4 Darboux transformations and explicit soliton solutions

In this section, we use the loop factorization method given in [25] to construct Darboux transformation for G ˜ n , k -NLS, and we apply Darboux transformation (Theorems 4 and 5) to:

(1) the trivial solution u = 0 of the coupled NLS equations

(20) ε Ψ 1 t − Ψ 1 x x + 2 Ψ 1 Ψ 2 Ψ 1 = 0 , ε Ψ 2 t + Ψ 2 x x − 2 Ψ 2 Ψ 1 Ψ 2 = 0 ,

where Ψ 1 = Ψ 1 ( x , t ) , Ψ 2 = Ψ 2 ( x , t ) are the complex valued function and ε 2 = ± 1 , to obtain soliton solutions;

(2) the constant map solution γ ( x , t ) = a and to the solutions of the Schrödinger flow (13) on G ˜ 2 , 1 .

Let α 1 , α 2 ∈ C , { v 1 , v 2 } a basis of C 2 , π the projection of C 2 onto C v 1 along C v 2 and

f α 1 , α 2 , π ( λ ) = I 2 × 2 + α 1 − α 2 λ − α 1 ( I 2 × 2 − π ) .

Theorem 4

(Darboux transformation for the G ˜ 2 , 1 -NLS equation (2))

Let u = 0 Ψ 1 Ψ 2 0 be a solution of the G ˜ 2 , 1 -NLS equation (2), and E ( x , t , λ ) a frame of u . Let v 1 = c 1 c 2 ∈ C 2 , v 2 = d 1 d 2 ∈ C 2 , c i , d i , α 1 , α 2 ∈ C , i = 1 , 2 , and α 1 ≠ α 2 , det ( v 1 , v 2 ) ≠ 0 , and π = 1 c 1 d 2 − c 2 d 1 c 1 d 2 − c 1 d 1 c 2 d 2 − c 2 d 1 . If

v 1 ˜ = ( c ˜ 1 , c ˜ 2 ) T ≔ E ( x , t , α 1 ) − 1 ( v 1 ) , v 2 ˜ = ( d ˜ 1 , d ˜ 2 ) T ≔ E ( x , t , α 2 ) − 1 ( v 2 ) ,

and suppose v 1 ˜ , v 2 ˜ are linearly independent. Then, we have the following.

u ˜ = u + ( α 1 − α 2 ) [ σ 3 , π ˜ ] is a solution of the G ˜ 2 , 1 -NLS equation (2), and
E ˜ ( x , t , λ ) = f α 1 , α 2 , π ˜ ( λ ) E ( x , t , λ ) f α 1 , α 2 , π ˜ ( x , t ) ( λ ) − 1
is a frame for u ˜ , where
π ˜ = 1 c ˜ 1 d ˜ 2 − c ˜ 2 d ˜ 1 c ˜ 1 d ˜ 2 − c ˜ 1 d ˜ 1 c ˜ 2 d ˜ 2 − c ˜ 2 d ˜ 1 .
u ˜ satisfies d u ˜ = θ ( ⋅ , ⋅ , α ) u ˜ , where
θ ( x , t , α ) = ( − σ 3 α + u ) d x ( σ 3 α 2 − u α + ε 2 P − 1 ( u ) ) d t ,
σ 3 = ε 2 1 0 0 − 1 , u = 0 Ψ 1 Ψ 2 0 , P − 1 ( u ) = ε − Ψ 1 Ψ 2 Ψ 1 x − Ψ 2 x Ψ 2 Ψ 1 , i.e.,
(21) V ˜ x = − α 2 ε Ψ 1 Ψ 2 α 2 ε V ˜ , V ˜ t = α 2 2 ε − ε 3 Ψ 1 Ψ 2 − Ψ 1 α + ε 3 Ψ 1 x − α Ψ 2 − ε 3 Ψ 2 x − α 2 2 ε + ε 3 Ψ 2 Ψ 1 V ˜ .
Moreover, if α 1 ≠ α 2 , and v 1 ˜ ( x , t ) = ( c ˜ 1 ( x , t ) , c ˜ 2 ( x , t ) ) T , v 2 ˜ ( x , t ) = ( d ˜ 1 ( x , t ) , d ˜ 2 ( x , t ) ) T are solutions of equation (21) with α = α 1 and α = α 2 , respectively. Suppose v 1 ˜ , v 2 ˜ are linearly independent, then the formula of u ˜ given in (i) is a solution of the G ˜ 2 , 1 -NLS equation (2).

Theorem 5

(Darboux transformation for the Schrödinger flow on G ˜ 2 , 1 ). Let γ be a solutions of Schrödinger flow (13) on G ˜ 2 , 1 , g ( x , t ) a Schrödinger frame of γ , and u ≔ g − 1 g x the solution of the G ˜ 2 , 1 -NLS equation (2). Let E ( x , t , λ ) = g ( 0 , 0 ) for all λ ∈ C . Let α 1 , α 2 ∈ C \ R be a constant, π a constant Hermitian projection onto V 1 ∈ C 2 along V 2 ∈ C 2 , and π ˜ ( x , t ) the Hermitian projection onto V ˜ 1 ( x , t ) ≔ E ( x , t , α 1 ) − 1 ( V 1 ) along V ˜ 2 ( x , t ) ≔ E ( x , t , α 1 ) − 1 ( V 2 ) . Then,

γ ( x , t ) = g ( x , t ) f α 1 , α 2 , π ˜ ( 0 ) ( x , t ) − 1 σ 3 f α 1 , α 2 , π ˜ ( 0 ) ( x , t ) g ( x , t ) − 1

is a new solution of (13).

Example 4.1

(1-Soliton solutions for the coupled NLS equations (20))

Note that

E ( x , t , λ ) = exp ( σ 3 ( − λ x + λ 2 t ) ) = e − B ( x , t , λ ) 0 0 e B ( x , t , λ ) ,

where B ( x , t , λ ) = ε 2 ( − λ x + λ 2 t ) , is the frame of the solution u = 0 of the G ˜ n , k -NLS equation (2) with E ( 0 , 0 , λ ) = I . Let π be a constant Hermitian projection onto V 1 ∈ C 2 along V 2 ∈ C 2 , where v 1 ( x , t ) = ( c 1 ( x , t ) , c 2 ( x , t ) ) T , v 2 ( x , t ) = ( d 1 ( x , t ) , d 2 ( x , t ) ) T , c i = c i ( x , t ) , d i = d i ( x , t ) ∈ C , i = 1 , 2 , we have

π = 1 c 1 d 2 − c 2 d 1 c 1 d 2 − c 1 d 1 c 2 d 2 − c 2 d 1 .

Then,

v ˜ 1 = E ( x , t , α ) − 1 ( v 1 ) = ( e B ( x , t , α 1 ) c 1 , e − B ( x , t , α 1 ) c 2 ) T ≔ ( c ˜ 1 , c ˜ 2 ) T , v ˜ 2 = E ( x , t , α ) − 1 ( v 2 ) = ( e B ( x , t , α 2 ) d 1 , e − B ( x , t , α 2 ) d 2 ) T ≔ ( d ˜ 1 , d ˜ 2 ) T ,

and

π ˜ = 1 c ˜ 1 d ˜ 2 − c ˜ 2 d ˜ 1 c ˜ 1 d ˜ 2 − c ˜ 1 d ˜ 1 c ˜ 2 d ˜ 2 − c ˜ 2 d ˜ 1 = 1 e η c 1 d 2 − e − η c 2 d 1 e η c 1 d 2 − e ξ c 1 d 1 e − ξ c 2 d 2 − e − η c 2 d 1 ,

namely,

u ˜ = u + ( α 1 − α 2 ) [ σ 3 , π ˜ ( x , t ) ] = ε ( α 1 − α 2 ) e − η c 2 d 1 − e η c 1 d 2 0 e ξ c 1 d 1 e − ξ c 2 d 2 0

is a solution of the G ˜ n , k -NLS equation (2), where ξ = ε 2 ( − ( α 1 + α 2 ) x + ( α 1 2 + α 2 2 ) t ) and η = ε 2 ( − ( α 1 − α 2 ) x + ( α 1 2 − α 2 2 ) t ) . Hence, Ψ ˜ 1 = ε ( α 1 − α 2 ) e − η c 2 d 1 − e η c 1 d 1 e ξ c 1 d 1 and Ψ ˜ 2 = ε ( α 1 − α 2 ) e − η c 2 d 1 − e η c 1 d 2 e − ξ c 2 d 2 are a solution of the coupled NLS equations (20).

If c 1 = c 2 = d 1 = d 2 = 1 , α 1 = r + i s , and α 2 = r − i s , then

π ˜ ( x , t ) = 1 e i ε ( − s x + 2 r s t ) − e − i ε ( − s x + 2 r s t ) e i ε ( − s x + 2 r s t ) − e ε ( − r x + ( r 2 − s 2 ) t ) e − ε ( − r x + ( r 2 − s 2 ) t ) − e − i ε ( − s x + 2 r s t ) ,

i.e.,

Ψ ˜ 1 ( x , t ) = 2 i s ε e ε ( − r x + ( r 2 − s 2 ) t ) e − i ε ( − s x + 2 r s t ) − e i ε ( − s x + 2 r s t ) , Ψ ˜ 2 ( x , t ) = 2 i s ε e ε ( − ( − r x + ( r 2 − s 2 ) t ) ) e − i ε ( − s x + 2 r s t ) − e i ε ( − s x + 2 r s t )

is a soliton solution of the coupled NLS equations (20).

Example 4.2

(1-Soliton solutions for Schrödinger flow on G ˜ 2 , 1 )

The constant map γ ( x , t ) = σ 3 is a solution of the Schrödinger flow (3.1) on G ˜ 2 , 1 , g ( x , t ) = I 2 × 2 is a Schrödinger frame for γ , u = 0 is the corresponding solution of the G ˜ n , k -NLS equation (2), and

E ( x , t , λ ) = exp ( − σ 3 λ x + σ 3 λ 2 t )

is the frame of u = 0 , with E ( 0 , 0 , λ ) = I 2 × 2 .

By Theorem 5 to γ ( x , t ) = σ 3 , g ( x , t ) = I 2 × 2 , and use π ˜ ( x , t ) given in Example 4.1 to obtain soliton solution for Schrödinger flow on G ˜ 2 , 1 . We choose α 1 = i s , α 2 = − i s , and c i = d i = 1 , i = 1 , 2 ; then, we have

f i s , − i s , π ˜ ( 0 ) ( x , t ) = 2 e − ε s x i e − ε s x i − e ε s x i − 1 − 2 e − ε s 2 t e − ε s x i − e ε s x i 2 e ε s 2 t e − ε s x i − e ε s x i − 2 e ε s x i e − ε s x i − e ε s x i − 1 .

Hence,

γ ˜ ( x , t ) = g ( x , t ) f i s , − i s , π ˜ ( 0 ) ( x , t ) − 1 σ 3 f i s , − i s , π ˜ ( 0 ) ( x , t ) g ( x , t ) − 1 = ε ( 6 e 2 ε s x i + e 4 ε s x i + 1 ) 2 ( e 2 ε s x i − 1 ) 2 − 4 ε c o s ( ε s x ) e 2 ε s x i e ε s 2 t + e ε s ( 4 x i + s t ) − 2 e ε s ( 2 x i + s t ) 2 ε ( e ε s ( x i + s t ) + e ε s ( 3 x i + s t ) ) ( e 2 ε s x i − 1 ) 2 − ε ( 6 e 2 ε s x i + e 4 ε s x i + 1 ) 2 ( e 2 ε s x i − 1 ) 2

is a soliton solution for Schrödinger flow on G ˜ 2 , 1 (CLL equations (16)). From equation (17), we have

s = ( s 1 , s 2 , s 3 ) = − 2 ε cos ( ε s x ) e 2 ε s x i e ε s 2 t + e ε s ( 4 x i + s t ) − 2 e ε s ( 2 x i + s t ) + ε ( e ε s ( x i + s t ) + e ε s ( 3 x i + s t ) ) ( e 2 ε s x i − 1 ) 2 , 2 cos ( ε s x ) e 2 ε s x i e ε s 2 t + e ε s ( 4 x i + s t ) − 2 e ε s ( 2 x i + s t ) + e ε s ( x i + s t ) + e ε s ( 3 x i + s t ) ( e 2 ε s x i − 1 ) 2 , ε ( 6 e 2 ε s x i + e 4 ε s x i + 1 ) 2 ( e 2 ε s x i − 1 ) 2

is a soliton solution of equation (15).

Theorem 3 gives a geometric interpretation of the matrix generalization of the coupled NLS equation (i.e., CLL equation) via Schrödinger flow to the complex Grassmannian manifold GL ( n , C ) ∕ GL ( k , C ) × GL ( n − k , C ) . And Example 4.2 shows that 1-soliton solution for Schrödinger flow on C S 2 , ε ( 1 ) is given. There are many questions unclear in this aspect. For example, it is also of interest to consider N-soliton solutions for Schrödinger flow on C S 2 , ε ( 1 ) . As you know, there are many models for NLS equation such as the coupled NLS-type equations (see [40]), the extended coupled nonlinear Schrödinger equations (see [41]). Is it possible to give a geometric interpretation of this systems via Schrödinger flow? These questions deserve study in the future.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions.

Funding information: The authors are supported by the Natural Science Foundation of Jiangxi Province (No. 20212BAB211005), Science & Technology Project of Jiangxi Educational Committee (No. GJJ2201202), and Jiangxi Province Training Program of Innovation and Entrepreneurship for Undergraduates (No. S202210418008).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2022-07-27

Revised: 2023-05-30

Accepted: 2023-06-06

Published Online: 2023-07-03

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https://doi.org/10.1515/math-2022-0600

Keywords for this article

coupled matrix nonlinear Schrödinger-type equations; complex Grassmannian manifold; Schrödinger flow; gauge equivalence; soliton solutions

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