On the Riemann–Hilbert problem of a generalized derivative nonlinear Schr?dinger equation

Bei-Bei Hu,Ling Zhang,* and Tie-Cheng Xia

1 School of Mathematics and Finance,Chuzhou University,Anhui 239000,China

2 Department of Mathematics,Shanghai University,Shanghai 200444,China

Abstract In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schr?dinger(DNLS)equation.By establishing a matrix Riemann–Hilbert problem and reconstructing potential function q(x,t)from eigenfunctionsin the inverse problem,the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed.Moreover,we also obtain that the spectral functions f(η),s(η),F(η),S(η)are not independent of each other,but meet an important global relation.As applications,the generalized DNLS equation can be reduced to the Kaup–Newell equation and Chen–Lee–Liu equation on the half-line.

Keywords:Riemann–Hilbert problem,generalized derivative nonlinear Schr?dinger equation,initial-boundary value problems,unified transformation method

1.Introduction

In 1967,Gardner et al[1]proposed the famous inverse scattering method(ISM)when studying the fast decay initial value problem of the Korteweg–de Vries equation,which is a powerful tool for solving the initial value problem of nonlinear integrable systems.However,because the ISM was only used to discuss the initial value problem of nonlinear integrable equations and the limitation of the initial value conditions is suitable for infinity,how to extend ISM to the initial-boundary value problems(IBVPs)of nonlinear integrable systems is a major challenge for soliton theory research.In 1997,Fokas[2]extended the ISM and proposed a unified transformation method(UTM)to analyze the IBVPs of partial differential equations[3].In 2008,Lenells[4]used UTM to analyze the IBVPs of the following derivative nonlinear Schr?dinger(DNLS)equation[5–7]

Equation(1.1)has an important application in plasma physics,which is a model for Alfvén waves propagating parallel to the ambient magnetic field[8,9].Since then,more and more mathematical physicists have paid attention to the UTM to study the IBVPs of integrable equations[10–18].In 2012,Lenells extended UTM to integrable systems related to high-matrix spectral[19],and used UTM to analyze the IBVPs of the Degasperis–Procesi equation[20,21].In 2013,Xu and Fan discussed the IBVPs of the Sasa–Satsuma equation through UTM[22],and gave the proof of the existence and uniqueness of the solution of the IBVPs of the integrable equation with higherorder matrix spectrum through analyzing a three-wave equation[23].Subsequently,more and more scholars have studied the IBVPs of integrable equations with higher-order matrix spectral[24–27].Particularly,the soliton solutions and the long-time asymptotic behavior for the integrable models can be solved by constructing a Riemann–Hilbert(RH)problem.Such as,Wang and Wang investigated the long-time asymptotic behavior of the Kundu–Eckhaus equation[28].Yang and Chen obtained the high-order soliton matrix form solution of the Sasa–Satsuma equation[29].Ma analyzed multicomponent AKNS integrable hierarchies[30],etc.

In 1987,Clarkson and Cosgrove[31]proposed a generalized derivative NLS(GDNLS)equation in the form of

where q is the amplitude of the complex field envelope.The equation(1.2)has several applications in optical fibers,nonlinear optics,weakly nonlinear dispersion water waves,quantum field theory,and plasma physics[32,33],etc.As an example,equation(1.2)can be used to simulate single-mode propagation in the optical fibers,which enjoys traveling and stationary kink envelope solutions of monotonic and oscillatory type.However,it is well know that equation(1.2)has Painlevé property only ifholds.At this time,equation(1.2)is reduced to an integrable GDNLS model as follows

Given α=2β≠0,the equation(1.3)becomes to the DNLSI(Kaup–Newell)equation(1.1),and if α≠0,β=0,the equation(1.3)becomes to the DNLS-II(Chen–Lee–Liu)equation

whose IBVPs on the half-line has been solved[34].Recently,the conservation laws of equation(1.3)have been discussed[35].However,as far as we know,the IBVPs of equation(1.3)have not been analyzed.So we will utilize UTM to study the IBVPs of equation(1.3)on the half-line domain Γ={(x,t):0

The design structure of this paper is as follows.In section 2,we give spectral analysis of the Lax pair of equation(1.3).In section 3,some key functions f(η),s(η),F(η),S(η)are further analyzed.In section 4,the RH problem is proposed.Finally,some conclusions and discussions are given in section 5.

2.The spectral analysis

The GDNLS equation(1.3)enjoys a Lax pair as follows[35]

where Φ=(Φ1,Φ2)Tis the vector eigenfunction,the 2×2 matrices U(x,t,η),V(x,t,η)are given by the following form

2.1.The exact one-form

The equations(2.1a),(2.1b)is equivalent to

where α≠β,complex number η is a spectral parameter and

One can introduce Ψ(x,t,η)by

hence,equations(2.4a),(2.4b)become to

where[σ3,Ψ]=σ3Ψ?Ψσ3,it is easy to see that the above equations give the following full differential

One supposes that the following asymptotic expansion

is a solution of equations(2.6a),(2.6b).Substituting equation(2.8)into equation(2.6a)and comparing the coefficients for ηj,one can get

From O(η2),one finds that D0enjoys a diagonal matrix form denoted as

From O(η1),one obtains

Through tedious calculation,one gets

since equations(2.1a),(2.1b)admit the following conservation law

the equations(2.10)and(2.12)for D0are consistent,then,one defines

where Ω is the closed one-form and given by

Since the integration of equation(2.13)is independent of the integration path and Ω is independent of η,one can introduce a key function G(x,t,η)by

then,equation(2.7)is equal to

where

It follows from M(x,t,η),N(x,t,η)and Ω that

with

Figure 1.The three contours γ1,γ2,γ3 in the(x,t)-domain.

then equation(2.16)becomes to

2.2.The three important functions

For(x,t)∈Γ,we suppose thatq(x,t)∈S,one defines three eigenfunctionsof equations(2.19a),(2.19b)given by

where I=diag{1,1}is a 2×2 unit matrix,Aj(ξ,τ,η)is given by equation(2.17),just replacing G(ξ,τ,η)with Gj(ξ,τ,η),the integral path(xj,tj)→(x,t)is a directed smooth curve and(x1,t1)=(0,0),(x2,t2)=(0,T),(x3,t3)=(∞,t).Since the integral of equation(2.20)has nothing to do with the integral path,we select a special integral path parallel to the coordinate axis as shown in figure 1,then we have

The first column of equation(2.20)enjoysand the following inequalities

On the other hand,the second column of equation(2.20)contains opposite index terms

Consequently,if we remember that1,2 represent k-column ofone can get

Figure 2.The areas Li,i=1,…,4 division on the complex η-plane.

and

To construct the RH problem of GDNLS equation(1.3),we must define another two important special functions ψ(η)and φ(η)by

upon evaluation at(x,t)=(0,0)and(x,t)=(0,T),respectively,from equations(2.27a)and(2.27b)we can get

It follows from(2.27a),(2.27b)and equation(2.28)that

Particularly,one also obtains G1(x,t,η),G2(x,t,η)at x=0

and G1(x,t,η),G3(x,t,η)at t=0

Assume that u0(x)=q(x,t=0),v0(t)=q(x=0,t),v1(t)=qx(x=0,t)are initial condition and boundary conditions of q(x,t)and qx(x,t),then,one get

with

2.3.The other properties of the eigenfunctions

Proposition 2.1.The functions

Proof.Indeed,according to the definition of function Gj(x,t,η)in equation(2.20)and combining with equations(2.25),(2.26),we can easily get this proposition.

To better analyze special functions ψ(η)and φ(η),one can get the following proposition according to the ISM theory.

Proposition 2.2.It follows from equation(2.28)that functions ψ(η),φ(η)can be expressed by

Assume that ψ(η),φ(η)possess the following 2×2 matrix from,respectively

It follows from equations(2.28)and(2.33a),(2.33b)that the following key properties are ture

2.4.The basic RH problem

To facilitate subsequent calculations,we remember that the following symbolic assumptions

then,one obtains

and the W(x,t,η)is defined by

These definitions imply that

In the following,one only gives the case of α>β for jump condition and residue relation,and we can discuss the case of α<β similarly.

Theorem 2.3.For α>β,setq(x,t)∈,and the function W(x,t,η)is given by equation(2.36),then equation(2.36)meets the following jump relation on the curve.

where

and

Proof.From equations(2.27a),(2.27b)and(2.34),one finds that

and

then,the equations(2.41a),(2.42b)and(2.35)give rise to

It follows from the equations(2.36)and(2.39)that

Therefore,the equations(2.44a)–(2.44d)lead to the jump matricesdefined by equation(2.40).

Assumption 2.4.One makes assumptions about the simple zeros of functions f(η)and h(η)as follows

Proposition 2.5(The residue conditions).Letone enjoys the following residue conditions:therefore,the equation(2.48)can lead to the equation(2.45a),and the other three equations(2.45b)–(2.45d)can be similarly proved.

2.5.The inverse problem

The inverse problem includes the reconstruction of potential function q(x,t)from spectral functionsIt follows from equation(2.10)thatSince asymptotic expansion in equation(2.8)is a solution of equation(2.7),which implies that

where G(x,t,η)is related to Ψ(x,t,η)as shown in equation(2.15)and given by211replaces of w(x,t).It follows from equation(2.49)and its complex conjugate that

Meanwhile, G(x, t, η) is the solution of equation (2.16) ifreplaces of w(x, t). It follows from equation (2.49)and its complex conjugate that

Then,the one-form Ω given by equation(2.13)can be expressed by w(x,t)

Proof.One only shows the equation(2.45a).As result ofone finds that the zerosof f(η)are the poles ofThen,one gets

taking η=?jinto the first and second equations of(2.36),we can get

together with equations(2.46)and(2.47),one obtains

Hence,one can solve the inverse problem according to the following steps successively:

(i)One utilizes any one of the functionsto calculate w(x,t)by

(ii)One gets Ω(x,t)from equation(2.50).

(iii)One computes potential function q(x,t)by equation(2.49).

2.6.The global relation

In this subsection,one gives the spectral functions f(η),s(η),F(η),S(η)which are not independent but admit a significant relationship.In fact,at the boundary of the region(ξ,τ):0<ξ<∞,0<τ

On the one hand,since ψ(η)=G3(0,0,η),together with equation(2.31b),one can find that the first term of the equation(2.51)is

Set x=0 in the equation(2.27a),we obtain

then

On the other hand,it follows from equations(2.53)and(2.30a)that the second term of the equation(2.51)is

Letq(x,t)∈for x→∞,then,equation(2.51)turns into

where the first column of equation(2.54)is valid for η2in the lower half-plane and the second column of equation(2.54)is valid for η2in the upper half-plane,and the expression of φ(t,η)is

Denoting φ(η)=φ(T,η)and letting t=T,one finds that the equation(2.54)turns into

Hence,the(21)-component of equation(2.55)is

where E(η)is expressed by

Indeed,equation(2.56)is the so-called global relation.

3.The functions f(η),s(η),F(η)and S(η)

Definition 3.1.(f(η)and s(η))Letone defines the mapping

in terms of

where G3(x,0,η)is given by

with M1(x,0,η)expressed by equation(2.32a).

Proposition 3.2.The f(η)and s(η)possess the properties as following

where W(x)(x,η)admits RH problem as follows.

Proof.(i)–(iv)follow from the investigation in section 2.3,and the deduction of(v)can be obtained following[4],where the derivation of u0(x)is given in the inverse problem(see section 2.5).

Definition 3.3.(F(η)and S(η))the mapping

in terms of

where G1(0,t,η)is given by

and N1(0,t,η)is expressed by equation(2.32b).

Proposition 3.4.The F(η)and S(η)possess the properties as follows

where

and the functions w(j)(t),j=1,2,3 are determined by

where W(t)(t,η)admits RH problem as follows

Proof.(i)–(iv)follow from the investigate in section 2.3,and the deduction of(v)can be obtained following[4],where the derivation of v0(t)and v1(t)are given in appendix.

4.The RH problem

Theorem 4.1.Letthe matrix functions ψ(η)and φ(η)in terms of f(η),s(η),F(η),S(η)are given by equation(2.34),respectively.Assume that the possible simple zerosof function f(η)andof function h(η)are given by assumption 2.4.Therefore,the matrix-value function W(x,t,η)conforms to the following RH problem:

Hence,the function W(x,t,η)is uniquely existing.Then,one can use W(x,t,η)to define q(x,t)as

thus,the function q(x,t)is a solution of the GDNLS equation(1.3).Furthermore,u(x,0)=u0(x),u(0,t)=v0(t),ux(0,t)=v1(t).

Proof.Indeed,one can manifest the above RH problem following[4].

5.Conclusions and discussions

In this paper,we use UTM to discuss the IBVPs of the generalized DNLS equation(1.3),one can also discuss the equation(1.3)on a finite interval,and analyze the asymptotic behavior of the solution for the equation(1.3)by the Deift–Zhou method[36].Since the RH problem is equivalent to Gel’fand–Levitan–Marchenko(GLM)theory,one can obtain the soliton solution of the equation(1.3)by solving the GLM equation following[37],which are our future investigation work.

Acknowledgments

This work is supported by the Natural Science Foundation of China(Nos.11 601 055,11 805 114 and 11 975 145),the Natural Science Research Projects of Anhui Province(No.KJ2019A0637),and University Excellent Talent Fund of Anhui Province(No.gxyq2019096).

Appendix.Recovering v0(t)and v1(t)

In this appendix,we will give a proof of equation(3.3),that is,derive v0(t)and v1(t)from W(t).Let G(x,t,η)is a solution of equation(2.16).According to equation(2.11),one gets

where Ψ(x,t,η)is the solution of equation(2.7)and enjoys the following form

Since Ψ(x,t,η)is defined by equation(2.15)and related to G(x,t,η)as follows

then,one gets

If seeking

then the(21)-entry of equation(A.1)gives

Taking the complex conjugate yields

At the same time,from equation(2.49),one finds

It follows from equations(A.2)–(A.4)that

which means that the coefficientof dt in the differential form Ω defined in equation(2.14)can be expressed as

with

where the functions w(j)(t),j=1,2,3 are determined by