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Based upon different methods such as a newly revised version of inverse scattering transform, Marchenko formalism, and Hirota’s bilinear derivative transform, this chapter aims to study and solve the derivative nonlinear Schrödinger (DNLS for brevity) equation under vanishing boundary condition (VBC for brevity). The explicit one-soliton and multi-soliton solutions had been derived by some algebra techniques for the VBC case. Meanwhile, the asymptotic behaviors of those multi-soliton solutions had been analyzed and discussed in detail.
Keywords
soliton
nonlinear equation
derivative nonlinear Schrödinger equation
inverse scattering transform
Zakharov-Shabat equation
Marchenko formalism
Hirota’s bilinear derivative transform
rogue wave
chapter and author info
Author
Zhou Guo-Quan*
Department of Physics, Wuhan University, Wuhan, P.R. China
Derivative nonlinear Schrödinger (DNLS for brevity) equation is one of the several rare kinds of integrable nonlinear models. Research of DNLS equation has not only mathematic interest and significance, but also important physical application background. It was first found that the Alfven waves in space plasma [1, 2, 3] can be modeled with DNLS equation. The modified nonlinear Schrödinger (MNLS for brevity) equation, which is used to describe the sub-picosecond pulses in single mode optical fibers [4, 5, 6], is actually a transformed version of DNLS equation. The weak nonlinear electromagnetic waves in ferromagnetic, anti-ferromagnetic, or dielectric systems [5, 6, 7, 8, 9] under external magnetic fields can also be modeled by DNLS equation.
Although DNLS equation is similar to NLS equation in form, it does not belong to the famous AKNS hierarchy at all. As is well known, a nonlinear integrable equation can be transformed to a pair of Lax equation satisfied by its Jost functions, the original nonlinear equation is only the compatibility condition of the Lax pair, that is, the so-called zero-curvature condition. Another fact had been found by some scholars that those nonlinear integrable equations which have the same first operator of the Lax pair belong to the same hierarchy and can deal with the same inverse scattering transform (IST for brevity). As a matter of fact, the DNLS equation has a squared spectral parameter of λ2in the first operator of its Lax pair, while the famous NLS equation, one typical example in AKNS hierarchy, has a spectral parameter of λ. Thus, the IST of the DNLS equation is greatly different from that of the NLS equation which is familiar to us. In a word, it deserves us to demonstrate several different approaches of solving it as a typical integrable nonlinear equation.
In this chapter, we will solve the DNLS equation under two kinds of boundary condition, that is, the vanishing boundary condition (VBC for brevity) and the non-vanishing boundary condition (NVBC for brevity), by means of three different methods – the revised IST method, the Marchenko formalism, and the Hirota’s bilinear derivative method. Meanwhile, we will search for different types of special soliton solution to the DNLS equation, such as the light/dark solitons, the pure solitons, the breather-type solitons, and the rogue wave solution, in one- or multi-soliton form.
2. An N-soliton solution to the DNLS equation based on a revised inverse scattering transform
For the VBC case of DNLS equation, which is just the concerned theme of the section, some attempts and progress have been made to solve the DNLS equation. Since Kaup and Newell proposed an IST with a revision in their pioneer works [10, 11], one-soliton solution was firstly attained and several versions of raw or explicit multi-soliton solutions were also obtained by means of different approaches [12, 13, 14, 15, 16, 17, 18, 19, 20]. Huang and Chen have got a N−soliton solution by means of Darboux transformation [15]. Steudel has derived a formula for N−soliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation [13]; but just as Chen points out in Ref. [16], Steudel’s multi-soliton solution is difficult to demonstrate collisions among solitons and still has a too complicate form to be used in the soliton perturbation theory of DNLS equation, although it can easily generate compute pictures. Since the integral kernel in Zakharov-Shabat (Z-S for brevity) equation does not tend to zero in the limit of spectral parameter λwith λ→∞, the contribution of the path integral along the big circle (the out contour) is also nonvanishing, the usual procedure to perform inverse scattering transform encounters difficulty and is invalid. Kaup thus proposed a revised IST by multiplying an additional weighing factor before the Jost solution Exλ, so that it tends to zero as ∣λ∣→∞, thus the modified Z-S kernel should lead to vanishing contribution of the integral along the big circle of Cauchy contour. Though the one-soliton solution has been found by the obtained Z-S equation of their IST, it is very difficult to derive directly its multi-soliton solution by their IST due to the existence of a complicated phase factor which is related to the solution itself [11]. We thus consider proposing a new revised IST to avoid the excessive complexity. Our N-soliton solution obviously has a standard multi-soliton form. It can be easily used to discuss its asymptotic behaviors and then develop its direct perturbation theory. On the other hand, in solving Z-S equation for DNLS with VBC, unavoidably we will encounter a problem of calculating determinant detI+Q1Q2, for two N × N matrices Q1and Q2, where Iis a N × N identity matrix. Our work also shows Binet-Cauchy formula and some other linear algebra techniques, (Appendices A.1–4 in Part 2), play important roles in the whole process, and actually also effective for some other nonlinear integrable models [21].
2.1 The revised inverse scattering transform and the Zakharov-Shabat equation for DNLS equation with VBC
2.1.1 The fundamental concepts for the IST theory of DNLS equation
DNLS equation for the one-dimension wave function uxtis usually expressed as
iut+uxx+iu2ux=0E1
with VBC, where the subscripts stand for partial derivative. Eq. (1) is also called Kaup-Newell (KN for brevity) equation. Its Lax pair is given by
L=−iλ2σ3+λU,U=0u−u¯0E2
and
M=−i2λ4σ3+2λ3U−iλ2U2σ3−λ−U3+iUxσ3E3
where λis a spectral parameter, and σ3is the third one of Pauli matrices σ1, σ2, σ3, and a bar over a letter, (e.g., u¯in (2)), represents complex conjugate. The first Lax equation is
∂xfxλ=LxλfxλE4
In the limit of ∣x∣→∞, u→0, and
L→L0=−iλ2σ3;M→M0=−i2λ4σ3E5
The free Jost solution is a 2×2matrix.
Exλ=e−iλ2xσ3;E•1xλ=10e−iλ2x,E•2xλ=01eiλ2xE6
The Jost solutions of (4) are defined by their asymptotic behaviors as x→±∞.
Ψxλ=ψ˜xλψxλ→Exλ,asx→∞E7
Φxλ=ϕxλϕ˜xλ→Exλ,asx→−∞E8
where ψxλ=ψ1xλψ2xλT, ψ˜xλ=ψ˜1xλψ˜2xλT, etc., and superscript “T” represents transposing of a matrix here and afterwards.
Since the first Lax equation of DNLS is similar to that of NLS, there are some similar properties of the Jost solutions. The monodromy matrix Tλis defined as
Then we can get the following reduction relation and symmetry properties
iσ2ψxλ¯¯=ψ˜xλE15
−iσ2φ¯xλ¯=φ˜xλE16
a˜¯λ¯=aλ;b˜¯λ¯=bλE17
and
ψx−λ=−σ3ψxλE18
ψ˜x−λ=σ3ψ˜xλE19
a−λ=aλ;b−λ=−bλa˜−λ=a˜λ;b˜−λ=−b˜λE20
2.1.2 Relation between Jost functions and the solutions to the DNLS equation
The asymptotic behaviors of the Jost solutions in the limit of ∣λ∣→∞can be obtained by simple derivation. Let υ=υ1υ2T≡ψ˜xλ; Eq. (4) can be rewritten as
υ1x+iλ2υ1=λuυ2,υ2x−iλ2υ2=−λu¯υ1E21
Then we have
υ1xx−uxυ1x+iλ2υ1/u+λ4υ1+λ2u2υ1=0E22
In the limit ∣λ∣→∞, we assume ψ˜1xλ=e−iλ2x+g, substituting it into Eq. (22), then we have
−iλ2+gx2+gxx−uxgx/u+λ4+λ2u2=0E23
In the limit ∣λ∣→∞, gxcan be expanded as series of λ−2j, j=1,2,⋯.
igx≡μ=μ0+μ22λ2−1+⋯E24
and
μ0=u2/2,μ2=−iu¯xu/2−u4/4,⋯E25
Eq. (21) leads to gxυ1=λuυ2. Considering (25), in the limit of λ→∞, we find a useful formula
u¯=i2lim∣λ∣→∞λψ˜2xλ/ψ˜1xλE26
which expresses the conjugate of solution uin terms of the Jost solutions as λ→∞.
On the other hand, the zeros of aλappear in pairs and can be designed by λn, n=1,2,⋯,Nin the I quadrant, and λn+N=−λnin the III quadrant. The discrete part of aλis [21, 22, 23].
aλ=∏n=1Nλ2−λn2λ2−λn2¯•λn2¯λn2E27
where a0=1. It comes from our consideration of the fact that, from the sum of two Cauchy integrals
in order to maintain that lnaλ→0,asλ→0,andlnaλis finiteasλ→∞, we then have to introduce a factor λ¯n2/λn2in (27). At the zeros of aλ, we have
ϕxλn=bnψxλn,ȧ−λn=−ȧλn,bn+N=−bnE28
Due to μ0≠0in (24) and (25), the Jost solutions do not tend to free Jost solutions Exλin the limit of ∣λ∣→∞. This is their most typical property which means that the usual procedure of constructing the equation of IST by a Cauchy contour integral must be invalid and abortive, thus a newly revised procedure to derive a suitable IST and the corresponding Z-S equation is proposed in our group.
2.1.3 The revised IST and Zakharov-Shabat equation for DNLS equation with VBC
The 2 × 1 column function Θxλcan be introduced as usual
Because in the limit of λ→∞, lim∣λ∣→∞eiλ2x=0,asx>0,Imλ2>0,λin theIIIIquadrants,x<0,Imλ2<0,λin theIIIVquadrants,then the integral path Γshould be chosen as shown in Figure 1, where the radius of big circle tends to infinite, while the radius of small circle tends to zero. And the factor λ−2is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile, our modification produces no new poles since Lax operator L→0, as λ→0. In the reflectionless case, the revised IST equation gives
where ȧλn=daλ/dλλ=λn. Similarly, an alternative form of IST equation is proposed as follows:
1λΘ2xλeiλ2x=12πi∫Γdλ′1λ′−λ1λ′Θ2xλ′eiλ′2xE32
where a factor λ−1is introduced for the same reason as λ−2in Eq. (30). Then in the reflectionless case, we can attain
ψ˜2xλ=∑n=12N1λnλλ−λnbnȧλnψ2xλneiλn2xe−iλ2xE33
Taking the symmetry and reduction relation (18) and (28) into consideration, from (31) and (33), we can obtain the revised Zakharov-Shabat equation for DNLS equation with VBC, that is,
It is obvious that formula (52) has the usual standard form of soliton solution. Here in formula (52), some algebra techniques have been used and can be found in Appendix A.1 in Part 2.
2.3 Explicit expression of N-soliton solution
2.3.1 Verification of standard form for the N-soliton solution
We only need to prove that Eq. (55) holds. Firstly, we define N × N matrices P1, P2, Q1, Q2, respectively, as
where Q1n1n2⋯nrm1m2⋯mrdenotes a minor, which is the determinant of a submatrix of Q1 consisting of elements belonging to not only rows (n1, n2,…, nr) but also columns (m1,m2,…, mr). Here use is made of Binet-Cauchy formula in the Appendices A.2–4 in Part 2. Then
Comparing (58) with (62) and making use of (63), we thus complete verification of Eq. (55). The soliton solution is surely of a typical form as that in NLS equation and can be expressed as formula (52).
2.3.2 Introduction of time evolution function
The time evolution factor of the scattering data can be introduced by standard procedure [21]. Due to the fact that the second Lax operator M→−i2λ4σ3in the limit of ∣x∣→∞, it is easy to derive the time dependence of scattering date.
with n,n′∈n1n2⋯nrand m,m′∈m1m2⋯mr. Where use is made of Binet-Cauchy formula which is numerated in Appendix A. 3–4 in Part 2. Substituting expression (67) into formula (58) thus complete the calculation of determinant D¯.
About the calculation of the most complicate determinant Ain (52), we introduce a N×(N + 1) matrix Ω1and a (N + 1) × N matrix Ω2defined as
The above summation obviously can be decomposed into two parts: one is extended to m1 = 0, the other is extended to m1≥1. Subtracted from (69), the part that is extended to m1≥1, the remaining parts of (69) is just Ain (52) (with m1=0and m2≥1). Due to (68), we thus have
Here n,n′∈n1n2⋯nrand especially m,m′∈m2⋯mr, which completes the calculation of determinant Ain formula (52). Substituting the explicit expressions of D, D¯, and A into (52), we finally attain the explicit expression of N-soliton solution to the DNLS equation under VBC and reflectionless case, based upon a newly revised IST technique.
An interesting conclusion is found that, besides a permitted well-known constant global phase factor, there is also an undetermined constant complex parameter bn0 before each of the typical soliton factor e−θneiφn, (n = 1,2,…, N). It can be absorbed into e−θneiφnby redefinition of soliton center and its initial phase factor. This kind of arbitrariness is in correspondence with the unfixed initial conditions of the DNLS equation.
2.4 The typical examples for one- and two-soliton solutions
We give two concrete examples – the one- and two-soliton solutions as illustrations of the general explicit soliton solution.
In the case of one-soliton solution, N = 1, λ2=−λ1, λ1=ρ1eiβ1=μ1+iν1, and
The complex conjugate of one-soliton solution u¯1xtin (75) is u1xt, which is just in conformity with that gotten from pure Marchenko formalism [24] (see the next section), up to a permitted global constant phase factor. In the case of two-soliton solution, N = 2, λ3=−λ1, λ4=−λ2and
Substituting (81) and (84) into formula (52), we thus get the two-soliton solution to the DNLS equation with VBC
u¯2=−i2A2D2/D¯22E86
Once again we find that, up to a permitted global constant phase factor, the above two-soliton solution is equivalent to that gotten in Ref. [23, 24], verifying the validity of our formula of N-soliton solution and the reliability of those linear algebra techniques. As a matter of fact, a general and strict demonstration of our revised IST for DNLS equation with VBC has been given in one paper by use of Liouville theorem [25].
2.5 The asymptotic behaviors of N-soliton solution
The complex conjugate of expression (52) gives the explicit expression of N-soliton solution as
uN=i2A¯ND¯N/DN2E87
Without the loss of generality, for λn=μn+ivn,Vn=−4μn2−vn2,n=1,2,⋯,N,we assume V1<V2<⋯<Vn<⋯VNand define the n’th vicinity area as Γn:x−xno−Vnt∼0,n=1.2⋯N.
As t→−∞, Nvicinity areas Γn,n=1,2…,N, queue up in a descending series
ΓN,ΓN−1,⋯,Γ1E88
and in the vicinity of Γn, we have (note that κj>0)
θj=4κjx−xj0−Vjt→−∞,forj<n+∞,forj>nE89
Here the complex constant 2cn0in expression (65) has been absorbed into e−θneiφnby redefinition of the soliton center xn0and the initial phase αn0.
Introducing a typical factor Fn=−e−2θn/λn2−λn2¯2>0,n=1,2,⋯,N; then
Each u1θnφn, (1,2,⋯,n) is a one-soliton solution characterized by one parameter λn, moving along the positive direction of the x-axis, queuing up in a series with descending order number nas in series (88). As t→∞, in the vicinity of Γn, we have (note that κj>0)
That is to say, the N-soliton solution can be viewed as Nwell-separated exact one- solitons, queuing up in a series with ascending order number n: Γ1,Γ2,⋯,ΓN.In the course going from t→−∞to t→∞, the n’th one-soliton overtakes the solitons from the first to n−1’th and is overtaken by the solitons from n+1’th to N’th. In the meantime, due to collisions, the n’th soliton got a total forward shift Δθn−/κnfrom exceeding those slower soliton from the first to n−1’th, and got a total backward shift Δθn+/κnfrom being exceeded by those faster solitons from n+1’th to N’th, and just equals to the summation of shifts due to each collision between two solitons, together with a total phase shift Δφn, that is,
Δxn=Δθn+−Δθn−/κnE103
Δφn=Δφn+−Δφn−E104
2.6 N-soliton solution to MNLS equation
Finally, we indicate that the exact N-soliton solution to the DNLS equation can be converted to that of MNLS equation by a gauge-like transformation. A nonlinear Schrödinger equation including the nonlinear dispersion term expressed as
i∂tυ+∂xxυ+iα∂xυ2υ+2βυ2υ=0E105
is also integrable [23] and called modified nonlinear Schrödinger (MNLS for brevity) equation. It is well known that MNLS equation well describes transmission of femtosecond pulses in optical fibers [4, 5, 6] and is related to DNLS equation by a gauge-like transformation [23] formulated as
υxt=uXTei2ρX+i4ρ2TE106
with x=α−1X+4ρT, t=α−2T; X=αx−4βt, T=α2t; ρ=βα−2. Using a method that is analogous to reference [16], and applying above gauge-like transformation to Eq. (105), the MNLS equation with VBC can be transformed into DNLS equation with VBC.
i∂Tu+∂XXu+i∂Xu2u=0E107
with u=uXT. So according to (106), the N-soliton solution to MNLS equation can also be attained by a gauge-like transformation from that of DNLS equation.
The N-soliton solution to the DNLS equation with VBC has been derived by means of a IST considered anew and some special linear algebra techniques. The one- and two-soliton solutions have been given as two typical examples in illustration of the general formula of the N-soliton solution. It is found to be perfectly in agreement with that gotten in the following section based on a pure Marchenko formalism or Hirota’s Bilinear derivative transformation [24, 26, 27]. The demonstration of the revised IST considered anew for DNLS equation with VBC has also been given by use of Liouville theorem [25].
The newly revised IST technique for DNLS equation with VBC supplies substantial foundation for its direct perturbation theory.
3. A simple method to derive and solve Marchenko equation for DNLS equation
Gel’fand-Levitan-Marchenko (GLM for brevity) equations can be viewed as an integral-transformed version of IST for those integrable nonlinear equations [21, 24, 28].
In this section, a simple method is used to derive and solve Marchenko equation (or GLM equation) for DNLS E with VBC [28]. Firstly, starting from the first Lax equation, we derive two conditions to be satisfied by the kernel matrix Nxyof GLM by applying the Lax operator ∂x−Lupon the integral representation of Jost function for DNLSE. Secondly, based on Lax equation, a strict demonstration has been given for the validness of Marchenko formalism. At last, the Marchenko formalism is determined by choosing a suitable Fx+yand Gx+y, and their relation (135) has been constructed. The one and multi-soliton solution in the reflectionless case is attained based upon a pure Marchenko formalism by avoiding direct use of inverse scattering data and verified by using direct substitution method with Mathematica.
3.1 The lax pair and its Jost functions of DNLS equation
DNLS equation is usual expressed as
iut+uxx+iu2ux=0E108
with vanishing boundary, ∣x∣→∞, u→0. Here the subscript denotes partial derivative. Its Lax pair is given by
L=−iλ2σ3+λU,U=0u−u¯0E109
M=−i2λ4σ3+2λ3U−iλ2U2σ3−λ−U3+iUxσ3E110
The first Lax equation is
∂xfxλ=LxλfxλE111
In the case of ∣x∣→∞, u→0, L→L0=−iλ2σ3, the free Jost solution is
Exλ=e−iλ2xσ3,E•1xλ=10e−iλ2x;E•2xλ=01eiλ2xE112
where λ2is a real squared parameter, Exλexpresses two independent solutions with two components. The Jost solutions of (4) are defined by their asymptotic properties at x→±∞,
Ψxλ=ψ˜xλψxλ→Exλ,asx→∞E113
Φxλ=φxλφ˜xλ→Exλ,asx→−∞E114
3.2 The integral representation of Jost function
As usual, we introduce the integral representation,
Ψxλ=Exλ+∫x∞dyλ2Ndxy+λNo(xy)EyλE115
where the superscripts d and o mean the diagonal and off-diagonal elements, respectively. According to the conventional operation in IST, the time variable is suppressed temporarily. Here
Ndxy=Nxy1100Nxy22, Noxy=0Nxy12Nxy210
Due to the symmetry of the first Lax operator λ2−iσ311=λ2−iσ3¯22and λU21=−λU¯12, the kernel matrix Nxyof the integral representation of Jost function should have the same symmetry as follows:
λ2Ndxy11=λ2Ndxy22¯;λNoxy21=−λNoxy12¯E116
Substitute Eq. (115) into the first Lax Eq. (111). By simply partial integration, we have the following terms:
According to equation ∂x−LΨxλ=0, adding up the l.h.s. and r.h.s., respectively, of Eq. (117)–(120), (121). We obtain two equations involving with terms λ2and λoutside of the integral ∫dy⋯as follows:
λ2:−Ndxx+σ3Ndxxσ3=0E122
λ1:−Ux−Noxx+σ3Noxxσ3+iUxNdxxσ3=0E123
Or
U12=ux=−2N12xx1+iN¯11xx,E124
and the equations in the integral ∫dyS=0, where Sis equal to
Therefore, Eq. (125) gives two conditions to be satisfied by the kernel matrix Nxyin the integral representation of Jost solution
λ2terms:Axy≡Nxdxy+σ3Nydxyσ3−UxNoxy=0E126
λ1terms:Bxy≡Nxoxy+σ3Nyoxyσ3+iUxNydxyσ3=0E127
Since (122) is an identity, Eq. (123) or (124) gives the solution Uxor uxin terms of N(x,x), thus the first Lax equation gives two conditions (126) and (127) which should be satisfied by the integral kernel N(x,y). Note that the time variable of uxin (124) is suppressed temporarily.
3.3 Marchenko equation for DNLSE and its demonstration
In Eq. (115), the Ndxyand Noxyappear in different manner, we assume the form of Marchenko equation for DNLSE with VBC is
Ndxy+∫x∞dzNoxzFz+y=0E128
Noxy+Fx+y+∫x∞dzNdxzGz+y=0E129
where Fx+yis only with off-diagonal terms. Gx+yis considered as another function with only off-diagonal terms. We notice that the Marchenko equation needn’t involve obviously the function of spectral parameter λ.
We now show the kernel Nxydetermined by (128) and (129) indeed satisfy the conditions (126) and (127) as long as we choose a suitable form of expression for Gx+y.
Making partial derivation in (128) with respect to x and y, respectively, we obtain
Since Fxis off-diagonal, Fxσ3=−σ3Fx. Thus the terms involving with Fx+youtside of integral are equal to −iUxNdxxσ3Fx+yby use of Eq. (123). Then (133) can be rewritten as
We find that, as long as we choose a suitable form for Gx+yas well as Fx+yaccording to Eq. (135), Eq. (128) and (129) will just satisfy the two conditions (126) and (127) derived from the first Lax Eq. (111). On the other hand, owing to the symmetry properties of Noxyand Ndxy, the function fx+yin (128) and (129) can only has off-diagonal elements, we write
Fx+y=0−fx+y¯fx+y0;Gz+y=0−hz+y¯hz+y0=iσ3F′z+yE144
Considering the dependence of the Jost solutions on the squared spectral parameter λ2, in the reflectionless case, we choose
fx+y=∑n=1NCnteiλn2x+yE145
where Cntcontains a time-dependent factor ei4λn4t, which can be introduced by a standard procedure [29], due to a fact of the Lax operator M→−i2λ4σ3as x→±∞.
As is well known, Lax equations are linear equation so that a constant factor can be introduced in its solution, that is, Cn=eβn+iαnei4λn4t. It means that βnis related to the center of soliton and αnexpresses the initial phase up to a constant factor. Thus, the time-independent part of Cn is inessential and can be absorbed or normalized only by redefinition of the soliton center and initial phase. On the other hand, notice the terms generated by partial integral in (133)–(142), in order to ensure the convergence of the partial integral, we must let limx→∞eiλn2x=0, so we only consider the Nzero points of aλin the first quadrant of complex plane of λ(also in the upper half part of the complex plane of λ2), that is, the discrete spectrum for λ1,λ2,⋯⋯λN, although −λn,n=12⋯Nin the third quadrant of the complex plane of λare also the zero points of aλdue to symmetry of Lax operator and transition matrix. Then Eq. (145) corresponds to the N-soliton solution in the reflectionless case, and we have completed the derivation and manifestation of Marchenko equation (128) and (129), (144), and (145) for DNLSE with VBC.
3.4 A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism
When there are N simple poles λ1,λ2,⋯,λNin the first quadrant of the complex plane of λ, the Marchenko equation will give a N-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that
fx+y=F21x+y=∑n=1Ngnxthny≡GxtHyTE146
where gnxt≡Cnteiλn2x,hny≡eiλn2y,n=1,2,⋯,N, and
Gxt≡g1xtg2xt⋯gNxt;HyT≡h1yh2y⋯hNyTE147
Here and hereafter the superscript T represents transposing of a matrix. On the other hand, we assume that
By means of some linear algebraic techniques, especially the Binet-Cauchy formula for some special matrices (see the Appendices 2–3 in Part2), the determinant D and C can be expanded explicitly as a summation of all possible principal minors. Firstly, we can prove identity (164) by means of Binet-Cauchy formula.
where M1n1n2⋯nrm1m2⋯mrdenotes a minor, which is the determinant of a submatrix of M1, consisting of elements belonging to not only (n1, n2,…, nr) rows but also columns (m1, m2,…, mr).
where n,n′∈n1n2⋯nr,m,m′∈m1m2⋯mr. Comparing (172) and (174), we find the following permutation symmetry between them
M1n1n2⋯nrm1m2⋯mrM2m1m2⋯mrn1n2⋯nr
=Q1m1m2⋯mrn1n2⋯nrQ2n1n2⋯nrm1m2⋯mr
Using above identity, comparing (169), (172), (173), and (174), we find that identity (164) holds and complete the computation of D.
Secondly, we compute the most complicate determinant Cin (163). In order to calculate detI+iΔ1Δ2+H¯TG¯, we introduce an N×N+1matrix Ω1and an N+1×Nmatrix Ω2
The above summation obviously can be decomposed into two parts: one is extended to m1 = 0 and the other extended to m1 ≥ 1. Subtracted from (176), the part that is extended to m1 ≥ 1, the remaining parts of (176) is just C in Eq. (163) (with m1=0, m2≥1). Due to (175), we have
here n,n′∈n1n2⋯nr, m,m′∈m2⋯mrin (178)–(180). Finally, substituting (174) into (173), (180) into (177), and (173 and 177) into (162), we thus attain the explicit N-soliton solution to the DNLS equation with VBC under the reflectionless case, based on a pure Marchenko formalism and in no need of the concrete spectrum expression of aλ. Obviously, the N-soliton solution permits uncertain complex constants cn0n=12⋯Nas well as an arbitrary global constant phase factor.
3.5 The special examples for one- and two-soliton solutions
In the case of one simple pole and one-soliton solution as N=1, according to (173), (177), (174), and (180), we have
By further redefinition of its soliton center and initial phase, the single soliton solution can be further rewritten as usual standard form. It is easy to find, up to a permitted well-known constant global phase factor, the one-soliton solution to DNLS equation gotten in the pure Marchenko formalism is in perfectly agreement with that gotten from other approaches [23, 24, 26, 27].
As N=2in the case of two-soliton solution corresponding to double simple poles, we have
Up to a permitted constant global phase factor, the two-soliton solution gotten above is actually equivalent to that gotten from both IST and Hirota’s method [23, 24, 26, 27], verifying the validity of the algebraic techniques that is used and our formula of the generalized multi-soliton solution. Because Marchenko equations (128), (129), (144), and (145) had been strictly proved, the multi-soliton solution is certainly right as long as we correctly use the algebraic techniques, especially Binet-Cauchy formula for the principal minor expansion of some special matrices.
4. Soliton solution of the DNLS equation based on Hirota’s bilinear derivative transform
Bilinear derivative operator D had been found and defined in the early 1970s by Hirota R., a Japanese mathematical scientist [30, 31, 32, 33]. Hirota’s bilinear-derivative transform (HBDT for brevity) can be used to deal with some partial differential equation and to find some special solutions, such as soliton solutions and rogue wave solutions [26, 27, 32]. In this section, we use HBDT to solve DNLS equation with VBC and search for its soliton solution. The DNLS equation with VBC, that is,
iut+uxx+iu2ux=0,E189
is one of the typical integrable nonlinear models, which is of a different form from the following equation:
iut+uxx+i2u2ux=0,E190
which had been solved in Ref. [14] by using HBDT. We have paid special attention to the following solution form in it [14]:
u=g/f,E191
where f,gare usually complex functions. Solution (191) is suitable for Eq. (190) and NLS equation, and so on, but not suitable for the DNLS equation. Just due to this fact, their work cannot deal with Eq. (189) at the same time. As is well known, rightly selecting an appropriate solution form is an important and key step to apply Hirota’s bilinear derivative transform to an integrable equation like Eq. (189). Refs. [13, 16, 17, 23], etc., have proved the soliton solution of the DNLS equation must has following standard form
u=gf¯/f2E192
here and henceforth a bar over a letter represents complex conjugation.
In view of the existing experiences of dealing with the DNLS equation, in the present section, we attempt to use the solution form (192) and HBDT to solve the DNLS equation. We demonstrate our solving approach step by step, and naturally extend our conclusion to the n-soliton case in the end.
4.1 Fundamental concepts and general properties of bilinear derivative transform
For two differentiable functions Axt,Bxtof two variables xand t, Hirota’s bilinear derivative operator, D, is defined as
where Axt,Bxtare two functions derivable for an arbitrary order, and the dot ·between them represents a kind of ordered product. Hirota’s bilinear derivative has many interesting properties. Some important properties to be used afterwards are listed as follows:
➀DtnDxmA·B=−1n+mDtnDxmB·AE195
for example, DxA·B=−DxB·A; DxA·A=0; Dx2A·B=Dx2B·A; DxnA·1=∂xnA; Dxn1·A=−1n∂xnA
➁DxnA·B=Dxn−mDxmA·B,m<nE196
➂ Suppose ηi=Ωit+Λix+η0i, i=1,2, Ωi, Λi, η0iare complex constants, then
DtnDxmexpη1·expη2=Ω1−Ω2nΛ1−Λ2mexpη1+η2E197
Especially, we have DtnDxmexpη1·expη2=0as Ω1=Ω2or Λ1=Λ2. Some other important properties are listed in the Appendix.
4.2 Bilinear derivative transform of DNLS equation
After a suitable solution form, for example, (192) has been selected, under the Hirota’s bilinear derivative transform, a partial differential equation like (189) can be generally changed into [20, 26, 27].
F1DtDx⋯g1·f1+F2DtDx⋯g2·f2=0E198
where FiDtDx⋯, i=1,2are the polynomial functions of Dt, Dx⋯; and gi,fi, i=1,2, are the differentiable functions of two variables xand t. Using formulae in the Appendix and properties ①–③ of bilinear derivative transform numerated in the last chapter, with respect to (192), we have
From (226) and (228), we can get an expression of f2
f2=i2Λ1Λ1+Λ¯12eη1+η¯1E229
Due to (224) and (229), we can also easily verify that
iDt+Dx2g1·f2=0E230
which immediately leads to
i∂t+∂x2g3=0E231
in Eq. (215). Then from (215), we can select g3=0. For the same reason, from (216)–(223), we can select f3,g4, g5, …; f4, f5, …all to be zero. Thus the series (207) and (208) have been successfully cut off to have limited terms as follows:
g1=eη1E232
f1=1+i2Λ1Λ1+Λ¯12eη1+η¯1E233
where εihas been absorbed into the constant eη10by redefiniing η10. In the end, we attain the one-soliton solution to the DNLS equation with VBC
u1xt=g1f¯1/f12E234
which is characterized with two complex parameters Λ1and η10and shown in Figure 1. If we redefine the parameter Λ1as Λ1≡−i2λ¯12and λ1≡μ1+iv1, then
It is easy to find, up to a permitted constant global phase factor eiπ=−1, the one-soliton solution (234) or (237) gotten in this paper is in perfect agreement with that gotten from other approaches [16]. By further redefining its soliton center, initial phase and λ1=ρ1eiβ1, the one-soliton solution can be changed into the usual typical form [16, 23, 26, 27].
On the other hand, just like in Ref. [13], we can rewrite g1and f1in a more appropriate or “standard” form
g1=eη1+φ1E239
f1=1+eη1+φ1+η¯1+φ1′+θ11′E240
Here
eφ1=1,eφ1′=i/Λ¯1,eθ11′=iΛ1−iΛ¯1/2Λ1+Λ¯12,E241
which makes us easily extend the solution form to the case of n-soliton solution.
4.3.2 The two-soliton solution
For the two-soliton case, again from (209), we can select g21as
g21=eη1+eη2,ηi=Ωit+Λix+ηi0,Ωi=iΛi2,i=1,2.E242
The similar procedures to that used in the one-soliton case can be used to deduce g2and f2. From (210) and (211), we can select f21=0, then from (212), we has to select g22=0. From (213) and (214), we can get the expressions of f22−f¯22and f22+f¯22, then attain f22to be
Substituting the expressions of g21,g22,g23,f21,f22into (216) and (217), we can select that f23=0. Then from the expressions of g21,g22,g23,f21,f22,f23and (218), we can select g24=0. From (219) and (220), we can get the expressions of f24−f¯24and f24+f¯24, then get f24to be
Due to (243) and (244), we can also easily verify that
iDt+Dx2g23·f22=0E246
Then from (244), (245), (246), and (221), we can select g25=0. From (222)–(223) and so on, we find that the series of (207) and (208) can be cut off by selecting g25,f25;g26,f26⋯, all to be zero. We thus attain the last result of g2,f2to be
therein ∑κj=0,1lrepresents a summation over κj=0,1under the condition ∑j=1Nκj=l+∑j=1Nκj+N.
Here, we have some discussion in order. Because what concerns us only is the soliton solutions, our soliton solution of DNLS equation with VBC is only a subset of the whole solution set. Actually in the whole process of deriving the bilinear-form equations and searching for the one and two-soliton solutions, some of the latter results are only the sufficient but not the necessary conditions of the former equations. Thereby some possible modes might have been missing. For example, the solutions of Eqs. (209)–(211) are not as unique as in (224) and (225), some other possibilities thus get lost here. This is also why we use a term “select” to determine a solution of an equation. In another word, we have selected a soliton solution. Meanwhile, we have demonstrated in Figures 2 and 3, the three-dimensional evolution of the one- and two-soliton amplitude with time and space, respectively. The elastic collision of two solitons in the two-soliton case has been demonstrated in Figure 4(a–d) too. It can be found that each soliton keeps the same form and characteristic after the collision as that before the collision. In this section, by means of introducing HBDT and employing an appropriate solution form (192), we successfully solve the derivative nonlinear Schrödinger equation with VBC. The one- and two-soliton solutions are derived and their equivalence to the existing results is manifested. The N-soliton solution has been given by an induction method. On the other hand, by using simple parameter transformations (e.g., (235) and (252)), the soliton solutions attained here can be changed into or equivalent to that gotten based on IST, up to a permitted global constant phase factor. This section impresses us so greatly for a fact that, ranked with the extensively used IST [23] and other methods, the HBDT is another effective and important tool to deal with a partial differential equation. It is especially suitable for some integrable nonlinear models.
Figure 2.
The evolution of one-soliton solution with time and space under parameter Λ1=−1+0.2i,η10=1 in (234).
Figure 3.
The evolution of two-soliton solution with time and space under parameter Λ1=1+0.3i,Λ2=1−0.3i,η10=η20=1 in (251).
Figure 4.
The elastic collision between two solitons at 4 typical moments: (a) t = −10(normalized time); (b) t = −1; (c) t = 1; (d) t = 10, from −10 before collision to 10 after collision.
Zhou Guo-Quan (September 5th 2020). Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1, Nonlinear Optics - From Solitons to Similaritons, İlkay Bakırtaş and Nalan Antar, IntechOpen, DOI: 10.5772/intechopen.93438. Available from:
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