# Discrete Spectrum of 2 + 1-Dimensional Nonlinear Schrödinger Equation and Dynamics of Lumps

Authors: Javier Villarroel; Julia Prada; Pilar G. Estévez
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

We consider a natural integrable generalization of nonlinear Schrödinger equation to2+1 dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dynamical evolution of the associated pulses. For all cases under study we find that the relevant dynamics corresponds to acentral configuration of a certain N-body problem.


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## Body

## 1. The Physical System

In this paper we consider the system of equations:(1)iut+uxx+2u∂xn=0,ny+u2=0,where ux,y,t is a complex function, depending on three real variables x, y, t. We derive via the inverse scattering method several large classes of solutions that satisfy the boundary conditions (r2≡x2+y2),(2)limr→∞⁡ux,y,t=1,limr→∞ux=0,nx,y=-∞,t=0.Note that in this case the system reads(3)iut+uxx+2u∂x∫-∞y1-u2dy′=0,nx,y,t=-y+∫-∞y1-u2x,y′dy′.Due to the overall simplicity, elegance, and potential physical interest, the above system is a natural interesting nonlinear 2+1-dimensional integrable equation to study, even though it has not been related to any physical situation so far. To further motivate its study we note the following:(1)
The more general problem corresponding to the boundary conditions(4)limr→∞ux,y,t=C,limr→∞ux=nx,y=-∞,t=0,where C>0 is a givennonnull constant is also solvable and reducible to problem (2) as follows. Let u^x,y,t be the unique solution to (1) under the boundary conditions (2). Then one proves easily that ux,y,t≡Cu^Cx,Cy,C2t also solves (1) and satisfies (4).(2)
System (3) arises as the compatibility of a Lax pair (see Section 2 below) and hence it is integrable. We show here that the operator has also a discrete spectrum, corresponding to potentials that solve (1) and (2): that is, it can be written as u=1+u~, where u~ is a regular, weakly decaying function.(3)
Ifu and n only satisfy u2∈L1R2 and limy→-∞⁡nx,y=0, then (1) reads(5)iut+uxx-2u∂x∫-∞ydy′u2=0,nx,y,t=∫-∞yu2x,y′,tdy′.This equation was first noticed to be integrable by Fokas [1] and also in Shul’man [2] who also embedded it into a general multiparameter family of equations. Note that in potentials satisfying u2∈L1R2 there is not discrete spectrum (see [3, 4]).(4)
The further reduction to the manifoldx=y yields the nonlinear Schrödinger (NLS) equation. Hence (1) generalizes NLS to 2+1 dimensions.(5)
The relationship between the previous problem and that corresponding to the boundary conditions (2) and (6)λnx,y=-∞,t+1-λnx,y=∞,t=0,where λ∈R and r2≡x2+y2, has been investigated in [5]. We note that the conserved quantities depend on the election of λ.(6)
There is an infinite number of conservation laws for this equation (see [5]), which includes, in particular, the functional (7)Mt≡∫u2x,y-1dxdy.We note that the “amplitude” u2x,y-1 of interesting solutions to (3) is naturally expected to be the physical observable but itis not definite positive; hence, the renormalized mass of the field M≡M+-M- can take both signs.(7)
An interesting parabolic real version of (1) (i.e., letting iut→ut) is obtained as a particular reduction of the Self-Dual Yang Mills Equations; see [6].Lump configurations (i.e., regular and rationally decaying solutions) are paradigmatic solutions of integrable equations in2+1 dimensions and as such they have been extensively studied in the last years. Their spectral interpretation was first unlocked in relation to KPI [7] (see also [8, 9]). A description of the KP equation, its physical origins, integrability, and soliton solutions can be found in [10]. Subsequently, lumps have been found in other integrable equations like Davey-Stewartdon II (DSII) (see [11]) and the 2+1-Toda lattice (see [12–14]).The dynamics ofstandard lumps on the plane is trivial: the motion is uniform; further, upon interaction, only a parallel shift on the asymptotic trajectory is to be found. Such uniform motion has been considered to rule the dynamics of localized pulses of integrable equations up until the late nineties. Remarkably it was found that KPI possess a new class of localized, real valued solutions that have a nontrivial asymptotic dynamics. Even though the simplest of these solutions has been known for a long time, the interesting scattering properties that they exhibit went unnoticed up until 1995 [15, 16]. The spectral interpretation along with a derivation of the general class of those solutions was unlocked in [17, 18]. Nonstandard lumps were found to be associated with a new discrete spectrum of the time dependent Schrödinger operator corresponding to meromorphic eigenfunctions with poles of higher multiplicity and to what we term nonstandard pole divisors. (A comprehensive account of the spectrum of both KPI and KPII is given in [19].) The extension of these ideas and solutions to DSII equation via spectral analysis of the Dirac operator on the plane has been considered in [20]. We also note that KPI possesses, in addition, other localized,nondecaying solutions like line solitons [21, 22]. The solution of the Cauchy problem in such a background is considered in [14, 23].In this paper we show that the spectral operator that linearizes (3) possesses a rich discrete spectrum corresponding to smooth, rationally “decaying” lump configurationswhose dynamical evolution and scattering are nontrivial. The physical behavior of these solutions is reminiscent to the aforementioned discrete spectrum and nonstandard lumps for KPI. We find that they are associated with higher-order pole meromorphic eigenfunctions of a similar discrete spectrum. Characterization of the discrete spectrum of this operator involves giving the pole multiplicity and an adequate pole divisor, which could be, in the spirit of the ideas of [18, 20], associated with integer winding numbers.Direct methods to study this class of solutions have also been developed; see [24, 25] for KPI and [26] for DSII. See also [27] for related ideas. These direct algebraic methods, although they do not shed any light on the associated spectral problem, are quite powerful to undertake a general classification of the relevant class of solutions. In this regard note that in [28–31] Painleve’s test for (3) was first considered and some special solutions, like line solitons, lumps, and dromion solutions, were found.The organization of the paper is as follows. A convenient form of the linearizing Lax pair associated with (3) with boundary conditions (2) is introduced in Section 2, following the ideas of [32]. In Section 3 we study the discrete spectrum of meromorphic eigenfunctions. Different relations between the Laurent coefficients (LC) compatible with the Lax pair are established. In Section 4 we study several classes of lump solutions and their dynamics and physical properties. Particular interest is addressed to the problem of determining the number of solitons that ensue from given analytic structure and to determine the motion of these entities. We find that typically lump’s dynamics is a superposition of the “center of mass” motion, which proceeds in a uniform way and individual, lump depending, motion that behaves as tr with r<1. Interestingly for all cases analyzed the dynamics corresponds to acentral configuration of a certain N-body problem (a central configuration of the Euler-Lagrange N-body problem corresponds to interacting point masses wherein for each body the acceleration vector is directed towards the center of mass and proportional to the distance to the center of mass) where they fall to the mass center. Particularly interesting is the case, similar to the “homothetic Lagrange” solution in the tree body problem, when pulses are located at the vertexes of an equilateral triangle which collapses to the center along a straight line as t1/3.


## 2. Linear Problem

As we have already pointed out system (3) arises as the compatibility of a pair of operators; see [1, 10, 33, 34] for genial ideas. Here we restrict our study to the class ux,y of potentials that are nonsingular and satisfy (2). Under these boundary conditions a convenient form of the Lax pair, depending on a complex spectral parameter k, is given by the pair of operators:(8)Lμ≡μxy+1kμx+μyk-uxu+μ1-1kuxu-u2=0,(9)Mμ≡μxx+iμt+2kμx+2mxμ=0.We assume the existence of an eigenfunction normalized to1 as k→∞. Then, (8) implies that μ(k) has the asymptotic expansion (with asymptotic coefficients m1,m2,…):(10)μ=1+m1k+m2k2+⋯,k⟶∞whereu2=1+∂ym1.This expression permits recovering thephysical amplitude|u|2. To determine the phase of the potential u consideration of the next order in the expansion is required. We find(11)uxu=∂xym1+∂ym2-1/2∂ym121+∂ym1.Thus, the potential is determined from(12)u2=1+∂ym1,i∂xarg⁡u=12∂y∂xm1-m12+2m21+∂ym1.


## 3. Discrete Spectrum and Relationships among Laurent Coefficients

Basic Assumption. The basic assumption of this work is the existence of a solution to (8) and (9), which has a finite or denumerable number of singularities kn,n=1,…,∞. We suppose that all singularities are higher-order poles and that μk is analytic away from the singularities. Thus, around any pole k=k1, μk must have a local Laurent expansion:(13)μk=μsing.k1k+μreg.k1k,where(14)μreg.k1k≡∑r=0∞νrk-k1r,μsing.k1k≡∑r=1mϕrk-k1r,are, respectively, the regular and singular parts of the eigenfunction at the pole and ϕr=ϕrx,y,t, νr=νrx,y,t are the Laurent coefficients (LC). Notice that μsing.k1k is also termed thesingularity principal part or the pole divisor.Unlike what happens in the regular case, where the eigenfunction is fixed by the corresponding∂¯-problem, when singularities exist the inverse problem does not fix uniquely the singular part. Additional information relating different coefficients of the poles divisor is required. In this section we consider examples of meromorphic eigenfunctions related to nonsingular potentials and determine different relationships between coefficients of the poles divisor.We posit the existence of a linear relationship between coefficients of the Laurent expansion in the form(15)νr=∑j=-mr-1fjx,y,tνjfor  a  certainr≥0,where m is the pole multiplicity and fj are certain functions to be determined. We say that the integer 1+r isthe index.


### 3.1. Simple Poles

We shall first suppose thatμ≡μx,y,t,· is an eigenfunction of (8), (9) with a meromorphic dependence on k and assume that k1 is a simple pole of μ. Let ϕ1,k1≡ϕ be the residue of μ(k) at k1. Equation (14) reads(16)μx,y,t,k=ϕx,y,tk-k1+μreg.k1x,y,t,k.Indeed, lettingk→k1 in (8), (9) we obtain, at dominant orders, that ϕ and ν≡ν0 must satisfy the system of equations:(17)ϕxy+1k1ϕx+ϕyk1-uxu+ϕ1-1k1uxu-u2=0,(18)νxy+1k1νx+νyk1-uxu+ν1-1k1uxu-u2+ϕy-1k12ϕx+1k12uxuϕ=0,(19)ϕxx+iϕt+2k1ϕx+2mxϕ=0,νxx+iνt+2k1νx+2mxν+2ϕx=0.


#### 3.1.1. Simple Poles of Index One

We first consider the simplest linear relationship when(20)r=0,ν=fϕfor  certainfx,y,t.Then, from the previous equations fx,y,t must satisfy(21)fxy+fxk1+k1fy-fy-1k12uxuϕ+ϕyfx+1+ϕxfy-1k12=0,ϕfxx+ift+2k1fx+2ϕxfx+1=0.We can satisfy these equations by requiring f to solve the following system ofconstant differential equations:(22)fy-1k12=fx+1=fxx+ift+2k1fx=0.It follows that(23)f=-x-2ik1t+yk12+γ,whereγ≡γR+iγI∈C.Therefore,eigenfunctions of (8)with the structure (16)can be constructed by requiring thatϕsolves (17)and that the relationship(24)ν0=-x-2ik1t+yk12+γϕholds for some complex constant γ. A similar relation was first established for KPI in [7]. Whenever such a situation holds we say that the pole divisor isstandard. We next study the possibility of having nonstandard pole divisors.


#### 3.1.2. Simple Poles of Index 2

We now show that the analytic structure (16) for eigenfunctions of (8)does not necessarily imply that (24) holds and different linear relationship between the LCs may hold; concretely we consider here the case when ν1,ν0,ν-1≡ϕ are related as(25)r=1,ν1=fν0+gϕforcertainfx,y,t,gx,y,t.In this case we also need to supplement (17), (18) with higher-order equations:(26)∂xx+i∂t+2k∂x+2mxν1+2∂xν0=0,(27)ν1xy+1k1ν1x+k1-uxuν1y+1-1k1uxu-u2ν1-ν0xk12+ν0y+uxk12uν0=0.By insertion of (25) into (26) we obtain(28)fxxν0+gxxϕ+2fxν0x+2gxϕx+iftν0+gtϕ+2k1fxν0+gxϕ+2ν0x-2fϕx=0.This equation will be satisfied if f is again given by (23) and g solves (29)gxx+igt+2k1gx=0,gx=f.By integration(30)ν1=fν0+gϕ≡-x-2ik1t+yk12+γν0-f22+it+yk13-δϕ,where γ and δ are complex constants. Equation (30) gives yet another relationship between the three first LCs ϕ, ν0, and ν1 of simple pole eigenfunctions compatible with the Lax pair.


### 3.2. Poles of Order Two

We now show how the above ideas can be extended to cover the case when the eigenfunction has a pole of multiplicity2 at some point k=k1: that is, m=2 in (14) and μsing.k1)k≡ψ/k-k12+ϕ/k-k1, where, to avoid an awkward notation, we write ϕ2,k1≡ψ, ϕ1,k1≡ϕ and also ν0,k1≡ν. By letting k→k1 in (8) we find that the main coefficients must satisfy ψxx+iψt+2k1ψx+2mxψ=0, and also(31)ψxy+ψxk1+k1-uxuψy+1-1k1uxu-u2ψ=0,ϕxy+1k1ϕx+k1-uxuϕy+1-1k1uxu-u2ϕ+ψy-1k12ψx+1k12uxuψ=0,νxy+1k1νx+k1-uxuνy+1-1k1uxu-u2ν-1k12ϕx+ϕy+1k12uxuϕ+1k13ψx-1k13uxuψ=0,ϕxx+iϕt+2k1ϕx+2mxϕ+2ψx=0,νxx+iνt+2k1νx+2mxν+2ϕx=0.Again, the assumed analytic structure does not fix uniquely the way LCs are related. We next consider several possibilities.


#### 3.2.1. Poles of Order Two and Index Two

We first assume that the spanψ,ϕ,ν0 of all linear combinations of ψ, ϕ and ν0≡ν with coefficients dependent on x, y, t is generated by just one of them; concretely we require(32)Dim.ψ,ϕ,ν=1;alternativelyϕ=fψ,ν=hψ,where f, h depend on x, y, t. As previously mentioned, from the ϕ-equations we find f to be given by (23), while the remaining equations yield that h must satisfy(33)hxx+iht+2k1hx+2fx=0,hx+f=0or, alternatively, hf-f=0, iht=1. It follows that (32) is compatible with the given analytic structure provided f is given by (23) and for some complex constants γ, δ(34)h=f22-it-yk13+δ.


#### 3.2.2. Poles of Order Two and Index Three

A different possibility obtains by requiring now(35)Dim.ψ,ϕ=1,Dim.ψ,ϕ,ν0,ν1=2,(36)orϕ=fψ,ν1=Ax,y,tν0+Bx,y,tψ.In this case, letting k→k1 in (8) we obtain that(37)ψxx+iψt+2k1ψx+2mxψ=0,ψxy+ψxk1+k1-uxuψy+1-1k1uxu-u2ψ=0,ν1xy+ν1xk1+k1-uxuν1y+1-1k1uxu-u2ν1-1k12ν0x+ν0y+1k12uxuν0+1k13ϕx-1k13uxuϕ-1k14ψx+1k14uxuψ=0,∂xx+i∂t+2k1∂x+2mxν1+2∂xν0=0.Hence, in view of (36) (38)Axxν0+Bxxψ+2Axν0x+2Bxψx+iAtν0+Btψ+2k1Axν0+Bxψ+2ν0x-2Afψx-ψ=0.Requiring independence of ν0, ψ we obtain that A, B must solve(39)Axx+iAt+2k1Ax=Ax+1=0,Bxx+iBt+2k1Bx+2A=Bx-Af=0.Hence, we obtain that A=f+η and B=-1/3(f3+3/2ηf2)+iηt+r(y) where η is a complex constant and r(y) an arbitrary function of y. The y-equation gives (40)-By-1k13f+1k14-Ak12f+Ak13=0(41)orry=ηk13+1k14y+r0.Thus, (36) is satisfied provided f is given by (23) and(42)A=f+η,B=-13f3+32ηf2+iηt+ηk13+1k14y+r0.


## 3.1. Simple Poles

We shall first suppose thatμ≡μx,y,t,· is an eigenfunction of (8), (9) with a meromorphic dependence on k and assume that k1 is a simple pole of μ. Let ϕ1,k1≡ϕ be the residue of μ(k) at k1. Equation (14) reads(16)μx,y,t,k=ϕx,y,tk-k1+μreg.k1x,y,t,k.Indeed, lettingk→k1 in (8), (9) we obtain, at dominant orders, that ϕ and ν≡ν0 must satisfy the system of equations:(17)ϕxy+1k1ϕx+ϕyk1-uxu+ϕ1-1k1uxu-u2=0,(18)νxy+1k1νx+νyk1-uxu+ν1-1k1uxu-u2+ϕy-1k12ϕx+1k12uxuϕ=0,(19)ϕxx+iϕt+2k1ϕx+2mxϕ=0,νxx+iνt+2k1νx+2mxν+2ϕx=0.


### 3.1.1. Simple Poles of Index One

We first consider the simplest linear relationship when(20)r=0,ν=fϕfor  certainfx,y,t.Then, from the previous equations fx,y,t must satisfy(21)fxy+fxk1+k1fy-fy-1k12uxuϕ+ϕyfx+1+ϕxfy-1k12=0,ϕfxx+ift+2k1fx+2ϕxfx+1=0.We can satisfy these equations by requiring f to solve the following system ofconstant differential equations:(22)fy-1k12=fx+1=fxx+ift+2k1fx=0.It follows that(23)f=-x-2ik1t+yk12+γ,whereγ≡γR+iγI∈C.Therefore,eigenfunctions of (8)with the structure (16)can be constructed by requiring thatϕsolves (17)and that the relationship(24)ν0=-x-2ik1t+yk12+γϕholds for some complex constant γ. A similar relation was first established for KPI in [7]. Whenever such a situation holds we say that the pole divisor isstandard. We next study the possibility of having nonstandard pole divisors.


### 3.1.2. Simple Poles of Index 2

We now show that the analytic structure (16) for eigenfunctions of (8)does not necessarily imply that (24) holds and different linear relationship between the LCs may hold; concretely we consider here the case when ν1,ν0,ν-1≡ϕ are related as(25)r=1,ν1=fν0+gϕforcertainfx,y,t,gx,y,t.In this case we also need to supplement (17), (18) with higher-order equations:(26)∂xx+i∂t+2k∂x+2mxν1+2∂xν0=0,(27)ν1xy+1k1ν1x+k1-uxuν1y+1-1k1uxu-u2ν1-ν0xk12+ν0y+uxk12uν0=0.By insertion of (25) into (26) we obtain(28)fxxν0+gxxϕ+2fxν0x+2gxϕx+iftν0+gtϕ+2k1fxν0+gxϕ+2ν0x-2fϕx=0.This equation will be satisfied if f is again given by (23) and g solves (29)gxx+igt+2k1gx=0,gx=f.By integration(30)ν1=fν0+gϕ≡-x-2ik1t+yk12+γν0-f22+it+yk13-δϕ,where γ and δ are complex constants. Equation (30) gives yet another relationship between the three first LCs ϕ, ν0, and ν1 of simple pole eigenfunctions compatible with the Lax pair.


## 3.1.1. Simple Poles of Index One

We first consider the simplest linear relationship when(20)r=0,ν=fϕfor  certainfx,y,t.Then, from the previous equations fx,y,t must satisfy(21)fxy+fxk1+k1fy-fy-1k12uxuϕ+ϕyfx+1+ϕxfy-1k12=0,ϕfxx+ift+2k1fx+2ϕxfx+1=0.We can satisfy these equations by requiring f to solve the following system ofconstant differential equations:(22)fy-1k12=fx+1=fxx+ift+2k1fx=0.It follows that(23)f=-x-2ik1t+yk12+γ,whereγ≡γR+iγI∈C.Therefore,eigenfunctions of (8)with the structure (16)can be constructed by requiring thatϕsolves (17)and that the relationship(24)ν0=-x-2ik1t+yk12+γϕholds for some complex constant γ. A similar relation was first established for KPI in [7]. Whenever such a situation holds we say that the pole divisor isstandard. We next study the possibility of having nonstandard pole divisors.


## 3.1.2. Simple Poles of Index 2

We now show that the analytic structure (16) for eigenfunctions of (8)does not necessarily imply that (24) holds and different linear relationship between the LCs may hold; concretely we consider here the case when ν1,ν0,ν-1≡ϕ are related as(25)r=1,ν1=fν0+gϕforcertainfx,y,t,gx,y,t.In this case we also need to supplement (17), (18) with higher-order equations:(26)∂xx+i∂t+2k∂x+2mxν1+2∂xν0=0,(27)ν1xy+1k1ν1x+k1-uxuν1y+1-1k1uxu-u2ν1-ν0xk12+ν0y+uxk12uν0=0.By insertion of (25) into (26) we obtain(28)fxxν0+gxxϕ+2fxν0x+2gxϕx+iftν0+gtϕ+2k1fxν0+gxϕ+2ν0x-2fϕx=0.This equation will be satisfied if f is again given by (23) and g solves (29)gxx+igt+2k1gx=0,gx=f.By integration(30)ν1=fν0+gϕ≡-x-2ik1t+yk12+γν0-f22+it+yk13-δϕ,where γ and δ are complex constants. Equation (30) gives yet another relationship between the three first LCs ϕ, ν0, and ν1 of simple pole eigenfunctions compatible with the Lax pair.


## 3.2. Poles of Order Two

We now show how the above ideas can be extended to cover the case when the eigenfunction has a pole of multiplicity2 at some point k=k1: that is, m=2 in (14) and μsing.k1)k≡ψ/k-k12+ϕ/k-k1, where, to avoid an awkward notation, we write ϕ2,k1≡ψ, ϕ1,k1≡ϕ and also ν0,k1≡ν. By letting k→k1 in (8) we find that the main coefficients must satisfy ψxx+iψt+2k1ψx+2mxψ=0, and also(31)ψxy+ψxk1+k1-uxuψy+1-1k1uxu-u2ψ=0,ϕxy+1k1ϕx+k1-uxuϕy+1-1k1uxu-u2ϕ+ψy-1k12ψx+1k12uxuψ=0,νxy+1k1νx+k1-uxuνy+1-1k1uxu-u2ν-1k12ϕx+ϕy+1k12uxuϕ+1k13ψx-1k13uxuψ=0,ϕxx+iϕt+2k1ϕx+2mxϕ+2ψx=0,νxx+iνt+2k1νx+2mxν+2ϕx=0.Again, the assumed analytic structure does not fix uniquely the way LCs are related. We next consider several possibilities.


### 3.2.1. Poles of Order Two and Index Two

We first assume that the spanψ,ϕ,ν0 of all linear combinations of ψ, ϕ and ν0≡ν with coefficients dependent on x, y, t is generated by just one of them; concretely we require(32)Dim.ψ,ϕ,ν=1;alternativelyϕ=fψ,ν=hψ,where f, h depend on x, y, t. As previously mentioned, from the ϕ-equations we find f to be given by (23), while the remaining equations yield that h must satisfy(33)hxx+iht+2k1hx+2fx=0,hx+f=0or, alternatively, hf-f=0, iht=1. It follows that (32) is compatible with the given analytic structure provided f is given by (23) and for some complex constants γ, δ(34)h=f22-it-yk13+δ.


### 3.2.2. Poles of Order Two and Index Three

A different possibility obtains by requiring now(35)Dim.ψ,ϕ=1,Dim.ψ,ϕ,ν0,ν1=2,(36)orϕ=fψ,ν1=Ax,y,tν0+Bx,y,tψ.In this case, letting k→k1 in (8) we obtain that(37)ψxx+iψt+2k1ψx+2mxψ=0,ψxy+ψxk1+k1-uxuψy+1-1k1uxu-u2ψ=0,ν1xy+ν1xk1+k1-uxuν1y+1-1k1uxu-u2ν1-1k12ν0x+ν0y+1k12uxuν0+1k13ϕx-1k13uxuϕ-1k14ψx+1k14uxuψ=0,∂xx+i∂t+2k1∂x+2mxν1+2∂xν0=0.Hence, in view of (36) (38)Axxν0+Bxxψ+2Axν0x+2Bxψx+iAtν0+Btψ+2k1Axν0+Bxψ+2ν0x-2Afψx-ψ=0.Requiring independence of ν0, ψ we obtain that A, B must solve(39)Axx+iAt+2k1Ax=Ax+1=0,Bxx+iBt+2k1Bx+2A=Bx-Af=0.Hence, we obtain that A=f+η and B=-1/3(f3+3/2ηf2)+iηt+r(y) where η is a complex constant and r(y) an arbitrary function of y. The y-equation gives (40)-By-1k13f+1k14-Ak12f+Ak13=0(41)orry=ηk13+1k14y+r0.Thus, (36) is satisfied provided f is given by (23) and(42)A=f+η,B=-13f3+32ηf2+iηt+ηk13+1k14y+r0.


## 3.2.1. Poles of Order Two and Index Two

We first assume that the spanψ,ϕ,ν0 of all linear combinations of ψ, ϕ and ν0≡ν with coefficients dependent on x, y, t is generated by just one of them; concretely we require(32)Dim.ψ,ϕ,ν=1;alternativelyϕ=fψ,ν=hψ,where f, h depend on x, y, t. As previously mentioned, from the ϕ-equations we find f to be given by (23), while the remaining equations yield that h must satisfy(33)hxx+iht+2k1hx+2fx=0,hx+f=0or, alternatively, hf-f=0, iht=1. It follows that (32) is compatible with the given analytic structure provided f is given by (23) and for some complex constants γ, δ(34)h=f22-it-yk13+δ.


## 3.2.2. Poles of Order Two and Index Three

A different possibility obtains by requiring now(35)Dim.ψ,ϕ=1,Dim.ψ,ϕ,ν0,ν1=2,(36)orϕ=fψ,ν1=Ax,y,tν0+Bx,y,tψ.In this case, letting k→k1 in (8) we obtain that(37)ψxx+iψt+2k1ψx+2mxψ=0,ψxy+ψxk1+k1-uxuψy+1-1k1uxu-u2ψ=0,ν1xy+ν1xk1+k1-uxuν1y+1-1k1uxu-u2ν1-1k12ν0x+ν0y+1k12uxuν0+1k13ϕx-1k13uxuϕ-1k14ψx+1k14uxuψ=0,∂xx+i∂t+2k1∂x+2mxν1+2∂xν0=0.Hence, in view of (36) (38)Axxν0+Bxxψ+2Axν0x+2Bxψx+iAtν0+Btψ+2k1Axν0+Bxψ+2ν0x-2Afψx-ψ=0.Requiring independence of ν0, ψ we obtain that A, B must solve(39)Axx+iAt+2k1Ax=Ax+1=0,Bxx+iBt+2k1Bx+2A=Bx-Af=0.Hence, we obtain that A=f+η and B=-1/3(f3+3/2ηf2)+iηt+r(y) where η is a complex constant and r(y) an arbitrary function of y. The y-equation gives (40)-By-1k13f+1k14-Ak12f+Ak13=0(41)orry=ηk13+1k14y+r0.Thus, (36) is satisfied provided f is given by (23) and(42)A=f+η,B=-13f3+32ηf2+iηt+ηk13+1k14y+r0.


## 4. Determination of Classes of Potentials

In this section we determine several classes solutions of (1) by considering proper election of meromorphic eigenfunctions and relationships between LCs of the form (15).


### 4.1. Potentials Corresponding to Simple Poles of Index One

Suppose first thatμk is a meromorphic eigenfunction that has the representation (14) with a finite even number 2N<∞ of simple poles. We make the following assumptions (denoted as condition C1):(1)
Poles come in pairskj, kj+N with corresponding residues ϕj, ϕj¯:(43)μk=1+∑j=1Nϕjk-kj+ϕj-k+k-j.(2)
At every polekα, α=1,…,2N the pole divisor is standard and hence satisfies (24):(3)
(44)kjR≠0,kj+N=-k-j,γj+N=γ-j,∀j=1,…,N.Suppose thatμ has just two poles k1, -k¯1. In this case from the previous assumptions and (24) we must have(45)ν0k1≡1+ϕ1¯k1+k¯1=f1ϕ1,ν0-k¯1≡1-ϕ1k¯1+k1=f¯1ϕ1¯.Solving this system of linear equations and using (10) we find the potential(46)u2=1+∂yϕ1+ϕ1¯=1-∂yxlog⁡Δwhilen=-y+∂xlog⁡Δ,where Δ is the determinant of the associated matrix:(47)Δ≡f1f-1+1k1+k-12.The fieldu given by (46) will be called the basic lump solution of (3). Obviously, if k1≡k1R+ik1I≡a+ib, where a≡k1R≠0, the solution is regular on the entire plane and decreases rationally.Consider a new Galilean framex′,y′ moving with velocities c,-bc2:(48)x′=x-ct-γ~,y′=y+bc2t+γ0Iρ,(49)c=a2+b2b,β=a2-b2a2+b22,ρ=-2aba2+b22,γ~≡γ0R-a2-b22abγ0I.One finds also convenient to go to a frame of skew coordinatesz, y′ where x′, and y′ are defined in (48) and z≡x′-βy′ is a shear transformation along the x-axis, where β represents a slanting coefficient. In this frame the potential reads(50)υ≡u2-1=-2βz2+ρ2y′ββy′-2x′-14a21Δ2,whereΔ=z2+ρ2y′2+14a2.Thus, in this inertial frame the solution is at rest for all time (objects moving uniformly with velocities (c,-bc2) in the unprimed frame remain at rest in the transformed primed frame) and strongly localized. However, the maxima structure is richer than what might have been expected. Inspection shows that critical points solve x′=σy′/a2+b2, σ=±1. With σ=1 we find that critical points are x′,y′=0,0 and(51)x′=±14ab3b2-a2,y′=±a2+b24ab3b2-a2while for σ=-1 one has(52)y′=±a2+b24a23a2-b2,x′=∓14a23a2-b2.These points are all candidates to maxima and minima; the number of them will vary depending on the values of the parameters. The parameter space is a two-dimensional plane deprived of the straight line a=0. To describe the situation in the general case we restrict, with no loss of generality, to the first quadrant on the parameter space. The situation varies according to which of the regions (53)C1≡a,b:a≤b3,C2≡a,b:b3≤a≤3b,C3≡a,b:0≤3b≤ado parameters belong. If a,b is in the interior of C2 then two of the four points described above are maxima and the other two are minima. Point x′,y′=0,0 is a saddle point between the formers at which u=3a2-b2/a2+b2.Particular cases areb=0 (previously studied) and a=b. In this case (54)υ≡u2-1=x′y′a4x′2+y′2/4a4+1/4a22.It has two symmetric maxima located at p=±2(1/4a,a/2) at which υ≡u2-1=1 while minima are to be found at the mirror images points at which υ=-1.When parameters satisfya=3b we obtain a configuration having a maximum x′,y′=0,0 at which υ=3; the minimum lump’s amplitude υ=-1 is to be found at the points x′,y′=±8/12b1,-4b2. At the other boundary point b=3a>0 the situation is opposite and is rather reminiscent of dark solitons; it corresponds to a configuration having minimum amplitude υ=-1 at the origin and maximum amplitude υ=3/25 at the points x′,y′=±8/43a1,4a2. Thus, generically the configuration is multipeaked, localized on an entire region containing several maxima.


### 4.2. Potentials Corresponding to Index Two

Here we investigate potentials associated with meromorphic eigenfunctions satisfying the following conditions (C2):(1)
Poles come in pairskj, kj+N, j=1,…,N with kj+N=-k¯j:(55)μk=1+∑j=1Nϕjk-kj+ϕj¯k+k¯j+ψj¯k+k¯j2.(2)
Forj=1,…,N thepoleskjare simple with index 2(i.e., they satisfy (30)). Besides kj, j=N+1,…,2N,are double poles with index 2 in the sense of (32), (34) with constants satisfying kjR≠0, γj¯=γ¯j, δj¯=-δ¯j.We suppose thatμ has just two poles k1, -k¯1 with the above properties. In this case one can prove that(56)u2=1+∂yϕ1+ϕ1¯=1-∂yxlog⁡Δ,where Δ is the following definite-positive polynomial: (57)Δ≡h12+f1+12a214a2+116a4,h1=-f122+it+yk13-δ,f1=-x-2ik1t+yk12+γ.Note how it depends on3 complex parameters k1, γ1, δ.We next study the dynamical properties of the fieldu of (56), (57). Clearly u2 is a rational function, regular on the entire plane which decreases as 1/r2 as r2≡x2+y2→∞. Again we find it convenient to use coordinates defined by (48) (see also (49)) in terms of which the “τ”-function reads (we drop some irrelevant parameters)(58)Δ=hR2+hI2+14a2z+12a2+ρ2y′2+116a4.Note also that(59)hR=12z2-ρ2y′22+εby′a2+b22-t,hI=-ρzy′+b3a2-b2a2+b23y′-4a2a2+b2t,where z≡x′-βy′, ε=a2-3b2/a2+b2a/b, k1≡a+ib, h1¯=hR+ihI.Inspection of the solution shows that it corresponds to a coherent two pulsed structures. Each of the pulses has a rich internal structure with several maxima and minima in a similar disposition to that corresponding to the standard lump.The dynamical behavior of this configuration is different and far more subtle than previously found. It turns out thatlumps are not at rest in any inertial frame.As indicated we consider a system of skew coordinatesz, y′ defined in (48). We claim that the solutions to(60)Δz±∞,y′±∞,t=Ot,wherez±∞t,y′±∞t≡p±∞give the pulse’s positions as t→±∞. Indeed, (61)∂yΔ,∂zΔz±∞,y′±∞=Ot,uz±∞,y′±∞,t=O1.Note that -p±∞ solves (60) if p±∞ does; thus, the second pulse is the mirror image of the first respect to the origin, and it suffices to describe the motion of the first of them.We first consider the casea=b for which the dynamics and scattering process are easier to understand. In this case the moving frame (48) and solution are given by (62)x′=x-2at-γR,y′=y+4a3t+2a2γ0I,ρ=-12a2,Δ=x′22-y′28a4-y′4a3+t2+x′y′2a2+y′4a3-2t2+14a2x′+12a2+y′24a4+14a2.We then find from (60) that asymptotically the first of the pulses has coordinates p±∞(t)≡x′±∞t,y′±∞t, where ζ±=5∓11/2, ι±=±4a2/ζ±, and(63)x′±∞=ζ±t+x0±∞+Ot-1/2,y′±=ι±t+y0±+Ot-1/2.Thus, inthe moving reference frame the trajectories as t→±∞ arestraight lines with different slopes. Notice that initially (t→-∞) pulses are located in the second and fourth quadrants and will move to the first and third ones as t→∞. For moderate times they collide head-on henceforth undergoing a scattering process. In Figures 1 and 2 we show the collision path and form of the pulses in solution (57) before and after scattering. The scattering angle Ω is easily found to be given by(64)cos⁡Ω=1-4a41+4a42+4a4.Note that cos⁡Ω attains a maximum value 1 if (formally) a=0 and decreases towards cos⁡Ω=-1 as a→∞. Thus, transparent scattering is obtained when a=b=0 while perpendicular scattering corresponds to lumps with a2=1/2.Figure 1
The path of the two pulses for the cases incoming (t=-∞) and outgoing (t=∞) corresponding to a=0.3 with the scattering angle indicated.Figure 2
Location and form of the two pulses: incoming (t=-10) and outgoing (t=10) corresponding to the election of a=b=1. The scattering process is clear.
(a) (b)In the rest at frame the motion is the composition of a uniform motion2at,-4a3t and the slower one given by (63). Indeed when a=b both coordinate frames are related by a pure Galilean transformation and objects at rest in the primed frame pick a uniform motion with velocities 2a,-4a3 in the original frame. Hence, in the stationary frame thepath of the second pulse is no longer the mirror image of the first.


### 4.3. Potentials Corresponding to Double Poles with Index Three

We now study potentials that correspond to eigenfunctions with double poles with index three. Concretely, assume the following:(1)
Poles come in pairskj, kj+N, j=1,…,N with kj+N=-k¯j≡kj¯. Hence,(65)μk=1+∑j=1Nϕjk-kj+ψjk-kj2+ϕj¯k+k¯j+ψj¯k+k¯j2.(2)
The multiplicity of every pole is two and the index three; namely, all pole divisors satisfy (36).We consider the simplest such potential whenN=1.


#### 4.3.1. Pulses of Index Three

We consider in detail the physical properties and dynamics in the caseN=1. The potential is given by formula (46) where the tau function is(66)Δ=Bk12+42a6+fk142a2+4Imfk122a4≥0.Here k1≡a+ib and η, δ are complex constants, v, ρ are defined in (49), and (67)C≡-iη+bv2ω,ω=1k131k1+η.The position and dynamics of the associated pulses are determined mainly by the constant η (alternatively by C≡CR+iCI). Concretely, we have the following.Proposition 1.
(1) If the constant C=0 the solution is a multipeaked traveling wave of solitonic nature (i.e., stationary in the frame moving with the soliton).
(2) When C≠0 the solution is nonstationary. There are exactly three traveling pulses describing “homothetic motion.” Namely, the motion in the frame at rest is a superposition of a “center of mass” uniform motion and an individual and slower motion proportional to t1/3.
(3) In the “center of mass frame” pulses are located at the vertex of a triangle with center of gravity at the origin (see Figure 3 where this situation is shown). The time evolution contracts but does not deform the whole structure. As a result, pulses collapse onto the origin whereupon the configuration regains shape. No deflection angle appears.Figure 3
Homothetic solution att=-200. The equilateral triangle is easily appreciated.To prove the claims we note that the solution reads neater with a skew transformation and a dilatation of coordinates defined by (48) and z≡x′-βy′, ξ=ρy′. Then dropping some irrelevant constants we have (see (42)) (68)B≡BR+iBI=-f33-ηf22+ωy′-Ct,BR=-z33+ξ2z-ηR2z2-ξ2+ηIξz+ωRξ-tCR,BI=ξ33-ξz2-ηI2z2-ξ2-ηRξz+ωIξ-tCI.We simply consider here the case whenC=0, namely, when the parameters are chosen to satisfy(69)bv21+ηk1=iηk14.The t-dependence of Δ drops out and hence the entire configuration ux,y,t moves, with respect to the frame at rest, with constant velocity given by v,-bv2. A plot of u shows that u2-1 decreases to zero away from three regions. Thus, the configuration behaves like amultipeaked solution of solitonic nature.


## 4.1. Potentials Corresponding to Simple Poles of Index One

Suppose first thatμk is a meromorphic eigenfunction that has the representation (14) with a finite even number 2N<∞ of simple poles. We make the following assumptions (denoted as condition C1):(1)
Poles come in pairskj, kj+N with corresponding residues ϕj, ϕj¯:(43)μk=1+∑j=1Nϕjk-kj+ϕj-k+k-j.(2)
At every polekα, α=1,…,2N the pole divisor is standard and hence satisfies (24):(3)
(44)kjR≠0,kj+N=-k-j,γj+N=γ-j,∀j=1,…,N.Suppose thatμ has just two poles k1, -k¯1. In this case from the previous assumptions and (24) we must have(45)ν0k1≡1+ϕ1¯k1+k¯1=f1ϕ1,ν0-k¯1≡1-ϕ1k¯1+k1=f¯1ϕ1¯.Solving this system of linear equations and using (10) we find the potential(46)u2=1+∂yϕ1+ϕ1¯=1-∂yxlog⁡Δwhilen=-y+∂xlog⁡Δ,where Δ is the determinant of the associated matrix:(47)Δ≡f1f-1+1k1+k-12.The fieldu given by (46) will be called the basic lump solution of (3). Obviously, if k1≡k1R+ik1I≡a+ib, where a≡k1R≠0, the solution is regular on the entire plane and decreases rationally.Consider a new Galilean framex′,y′ moving with velocities c,-bc2:(48)x′=x-ct-γ~,y′=y+bc2t+γ0Iρ,(49)c=a2+b2b,β=a2-b2a2+b22,ρ=-2aba2+b22,γ~≡γ0R-a2-b22abγ0I.One finds also convenient to go to a frame of skew coordinatesz, y′ where x′, and y′ are defined in (48) and z≡x′-βy′ is a shear transformation along the x-axis, where β represents a slanting coefficient. In this frame the potential reads(50)υ≡u2-1=-2βz2+ρ2y′ββy′-2x′-14a21Δ2,whereΔ=z2+ρ2y′2+14a2.Thus, in this inertial frame the solution is at rest for all time (objects moving uniformly with velocities (c,-bc2) in the unprimed frame remain at rest in the transformed primed frame) and strongly localized. However, the maxima structure is richer than what might have been expected. Inspection shows that critical points solve x′=σy′/a2+b2, σ=±1. With σ=1 we find that critical points are x′,y′=0,0 and(51)x′=±14ab3b2-a2,y′=±a2+b24ab3b2-a2while for σ=-1 one has(52)y′=±a2+b24a23a2-b2,x′=∓14a23a2-b2.These points are all candidates to maxima and minima; the number of them will vary depending on the values of the parameters. The parameter space is a two-dimensional plane deprived of the straight line a=0. To describe the situation in the general case we restrict, with no loss of generality, to the first quadrant on the parameter space. The situation varies according to which of the regions (53)C1≡a,b:a≤b3,C2≡a,b:b3≤a≤3b,C3≡a,b:0≤3b≤ado parameters belong. If a,b is in the interior of C2 then two of the four points described above are maxima and the other two are minima. Point x′,y′=0,0 is a saddle point between the formers at which u=3a2-b2/a2+b2.Particular cases areb=0 (previously studied) and a=b. In this case (54)υ≡u2-1=x′y′a4x′2+y′2/4a4+1/4a22.It has two symmetric maxima located at p=±2(1/4a,a/2) at which υ≡u2-1=1 while minima are to be found at the mirror images points at which υ=-1.When parameters satisfya=3b we obtain a configuration having a maximum x′,y′=0,0 at which υ=3; the minimum lump’s amplitude υ=-1 is to be found at the points x′,y′=±8/12b1,-4b2. At the other boundary point b=3a>0 the situation is opposite and is rather reminiscent of dark solitons; it corresponds to a configuration having minimum amplitude υ=-1 at the origin and maximum amplitude υ=3/25 at the points x′,y′=±8/43a1,4a2. Thus, generically the configuration is multipeaked, localized on an entire region containing several maxima.


## 4.2. Potentials Corresponding to Index Two

Here we investigate potentials associated with meromorphic eigenfunctions satisfying the following conditions (C2):(1)
Poles come in pairskj, kj+N, j=1,…,N with kj+N=-k¯j:(55)μk=1+∑j=1Nϕjk-kj+ϕj¯k+k¯j+ψj¯k+k¯j2.(2)
Forj=1,…,N thepoleskjare simple with index 2(i.e., they satisfy (30)). Besides kj, j=N+1,…,2N,are double poles with index 2 in the sense of (32), (34) with constants satisfying kjR≠0, γj¯=γ¯j, δj¯=-δ¯j.We suppose thatμ has just two poles k1, -k¯1 with the above properties. In this case one can prove that(56)u2=1+∂yϕ1+ϕ1¯=1-∂yxlog⁡Δ,where Δ is the following definite-positive polynomial: (57)Δ≡h12+f1+12a214a2+116a4,h1=-f122+it+yk13-δ,f1=-x-2ik1t+yk12+γ.Note how it depends on3 complex parameters k1, γ1, δ.We next study the dynamical properties of the fieldu of (56), (57). Clearly u2 is a rational function, regular on the entire plane which decreases as 1/r2 as r2≡x2+y2→∞. Again we find it convenient to use coordinates defined by (48) (see also (49)) in terms of which the “τ”-function reads (we drop some irrelevant parameters)(58)Δ=hR2+hI2+14a2z+12a2+ρ2y′2+116a4.Note also that(59)hR=12z2-ρ2y′22+εby′a2+b22-t,hI=-ρzy′+b3a2-b2a2+b23y′-4a2a2+b2t,where z≡x′-βy′, ε=a2-3b2/a2+b2a/b, k1≡a+ib, h1¯=hR+ihI.Inspection of the solution shows that it corresponds to a coherent two pulsed structures. Each of the pulses has a rich internal structure with several maxima and minima in a similar disposition to that corresponding to the standard lump.The dynamical behavior of this configuration is different and far more subtle than previously found. It turns out thatlumps are not at rest in any inertial frame.As indicated we consider a system of skew coordinatesz, y′ defined in (48). We claim that the solutions to(60)Δz±∞,y′±∞,t=Ot,wherez±∞t,y′±∞t≡p±∞give the pulse’s positions as t→±∞. Indeed, (61)∂yΔ,∂zΔz±∞,y′±∞=Ot,uz±∞,y′±∞,t=O1.Note that -p±∞ solves (60) if p±∞ does; thus, the second pulse is the mirror image of the first respect to the origin, and it suffices to describe the motion of the first of them.We first consider the casea=b for which the dynamics and scattering process are easier to understand. In this case the moving frame (48) and solution are given by (62)x′=x-2at-γR,y′=y+4a3t+2a2γ0I,ρ=-12a2,Δ=x′22-y′28a4-y′4a3+t2+x′y′2a2+y′4a3-2t2+14a2x′+12a2+y′24a4+14a2.We then find from (60) that asymptotically the first of the pulses has coordinates p±∞(t)≡x′±∞t,y′±∞t, where ζ±=5∓11/2, ι±=±4a2/ζ±, and(63)x′±∞=ζ±t+x0±∞+Ot-1/2,y′±=ι±t+y0±+Ot-1/2.Thus, inthe moving reference frame the trajectories as t→±∞ arestraight lines with different slopes. Notice that initially (t→-∞) pulses are located in the second and fourth quadrants and will move to the first and third ones as t→∞. For moderate times they collide head-on henceforth undergoing a scattering process. In Figures 1 and 2 we show the collision path and form of the pulses in solution (57) before and after scattering. The scattering angle Ω is easily found to be given by(64)cos⁡Ω=1-4a41+4a42+4a4.Note that cos⁡Ω attains a maximum value 1 if (formally) a=0 and decreases towards cos⁡Ω=-1 as a→∞. Thus, transparent scattering is obtained when a=b=0 while perpendicular scattering corresponds to lumps with a2=1/2.Figure 1
The path of the two pulses for the cases incoming (t=-∞) and outgoing (t=∞) corresponding to a=0.3 with the scattering angle indicated.Figure 2
Location and form of the two pulses: incoming (t=-10) and outgoing (t=10) corresponding to the election of a=b=1. The scattering process is clear.
(a) (b)In the rest at frame the motion is the composition of a uniform motion2at,-4a3t and the slower one given by (63). Indeed when a=b both coordinate frames are related by a pure Galilean transformation and objects at rest in the primed frame pick a uniform motion with velocities 2a,-4a3 in the original frame. Hence, in the stationary frame thepath of the second pulse is no longer the mirror image of the first.


## 4.3. Potentials Corresponding to Double Poles with Index Three

We now study potentials that correspond to eigenfunctions with double poles with index three. Concretely, assume the following:(1)
Poles come in pairskj, kj+N, j=1,…,N with kj+N=-k¯j≡kj¯. Hence,(65)μk=1+∑j=1Nϕjk-kj+ψjk-kj2+ϕj¯k+k¯j+ψj¯k+k¯j2.(2)
The multiplicity of every pole is two and the index three; namely, all pole divisors satisfy (36).We consider the simplest such potential whenN=1.


### 4.3.1. Pulses of Index Three

We consider in detail the physical properties and dynamics in the caseN=1. The potential is given by formula (46) where the tau function is(66)Δ=Bk12+42a6+fk142a2+4Imfk122a4≥0.Here k1≡a+ib and η, δ are complex constants, v, ρ are defined in (49), and (67)C≡-iη+bv2ω,ω=1k131k1+η.The position and dynamics of the associated pulses are determined mainly by the constant η (alternatively by C≡CR+iCI). Concretely, we have the following.Proposition 1.
(1) If the constant C=0 the solution is a multipeaked traveling wave of solitonic nature (i.e., stationary in the frame moving with the soliton).
(2) When C≠0 the solution is nonstationary. There are exactly three traveling pulses describing “homothetic motion.” Namely, the motion in the frame at rest is a superposition of a “center of mass” uniform motion and an individual and slower motion proportional to t1/3.
(3) In the “center of mass frame” pulses are located at the vertex of a triangle with center of gravity at the origin (see Figure 3 where this situation is shown). The time evolution contracts but does not deform the whole structure. As a result, pulses collapse onto the origin whereupon the configuration regains shape. No deflection angle appears.Figure 3
Homothetic solution att=-200. The equilateral triangle is easily appreciated.To prove the claims we note that the solution reads neater with a skew transformation and a dilatation of coordinates defined by (48) and z≡x′-βy′, ξ=ρy′. Then dropping some irrelevant constants we have (see (42)) (68)B≡BR+iBI=-f33-ηf22+ωy′-Ct,BR=-z33+ξ2z-ηR2z2-ξ2+ηIξz+ωRξ-tCR,BI=ξ33-ξz2-ηI2z2-ξ2-ηRξz+ωIξ-tCI.We simply consider here the case whenC=0, namely, when the parameters are chosen to satisfy(69)bv21+ηk1=iηk14.The t-dependence of Δ drops out and hence the entire configuration ux,y,t moves, with respect to the frame at rest, with constant velocity given by v,-bv2. A plot of u shows that u2-1 decreases to zero away from three regions. Thus, the configuration behaves like amultipeaked solution of solitonic nature.


## 4.3.1. Pulses of Index Three

We consider in detail the physical properties and dynamics in the caseN=1. The potential is given by formula (46) where the tau function is(66)Δ=Bk12+42a6+fk142a2+4Imfk122a4≥0.Here k1≡a+ib and η, δ are complex constants, v, ρ are defined in (49), and (67)C≡-iη+bv2ω,ω=1k131k1+η.The position and dynamics of the associated pulses are determined mainly by the constant η (alternatively by C≡CR+iCI). Concretely, we have the following.Proposition 1.
(1) If the constant C=0 the solution is a multipeaked traveling wave of solitonic nature (i.e., stationary in the frame moving with the soliton).
(2) When C≠0 the solution is nonstationary. There are exactly three traveling pulses describing “homothetic motion.” Namely, the motion in the frame at rest is a superposition of a “center of mass” uniform motion and an individual and slower motion proportional to t1/3.
(3) In the “center of mass frame” pulses are located at the vertex of a triangle with center of gravity at the origin (see Figure 3 where this situation is shown). The time evolution contracts but does not deform the whole structure. As a result, pulses collapse onto the origin whereupon the configuration regains shape. No deflection angle appears.Figure 3
Homothetic solution att=-200. The equilateral triangle is easily appreciated.To prove the claims we note that the solution reads neater with a skew transformation and a dilatation of coordinates defined by (48) and z≡x′-βy′, ξ=ρy′. Then dropping some irrelevant constants we have (see (42)) (68)B≡BR+iBI=-f33-ηf22+ωy′-Ct,BR=-z33+ξ2z-ηR2z2-ξ2+ηIξz+ωRξ-tCR,BI=ξ33-ξz2-ηI2z2-ξ2-ηRξz+ωIξ-tCI.We simply consider here the case whenC=0, namely, when the parameters are chosen to satisfy(69)bv21+ηk1=iηk14.The t-dependence of Δ drops out and hence the entire configuration ux,y,t moves, with respect to the frame at rest, with constant velocity given by v,-bv2. A plot of u shows that u2-1 decreases to zero away from three regions. Thus, the configuration behaves like amultipeaked solution of solitonic nature.



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*Source: 8620473-2016-10-27.xml*