## Abstract

The dynamics of a one-dimensional slowly modulated, nearly monochromatic localized wave train in deep water is described by a one-dimensional soliton solution of a two-dimensional nonlinear Schrödinger (NLS) equation. In this paper, the instability of such a wave train with respect to transverse perturbations is examined numerically in the context of the NLS equation, using Hill's method. A variety of instabilities are obtained and discussed. Among these, we show that the solitary wave is susceptible to an oscillatory instability (complex growth rate) due to perturbations with arbitrarily short wavelength. Further, there is a cut-off on the instability with real growth rates. We show analytically that the nature of this cut-off is different from what is claimed in previous works.

## 1. Introduction

The hyperbolic nonlinear Schrödinger (NLS) equation in non-dimensional variables is(1.1)for a complex wave function . It describes the evolution of gravity–capillary waves in deep water that may be two-dimensional, nearly monochromatic, and are slowly modulated. Subscripts in (1.1) are used to denote partial derivatives. A derivation of this equation may be found in Ablowitz & Segur (1981). Up to scaling, the elevation of the water wave surface is given by(1.2)where ‘Re’ denotes the real part, is the wavenumber of the carrier wave train, and is the carrier frequency given by the deep-water dispersion relation.

A trivial-phase one-soliton solution of the one-dimensional NLS equation (equation (1.1), ignoring dependence on *y*) is given by(1.3)This solution corresponds to a one-dimensional localized wave train:(1.4)A contour plot of is shown in figure 1.

*Remarks*

By using the Lie point symmetries of (1.1) (Sulem & Sulem 1999), this solution may be transformed into the most general form of a one-dimensional soliton solution with trivial spatial phase profile of (1.1). Thus, only considering (1.3) is not as restrictive as it appears.

Using a linear transformation on the spatial variables, the solution (1.3) may be oriented so as to propagate at a non-zero angle with the

*x*-axis (Saffman & Yuen 1978). The analysis of these solutions is similar to that considered here.

In order to examine the stability of (1.3) with respect to two-dimensional perturbations, consider(1.5)where *ϵ* is a small, real constant and , are real-valued functions. The resulting equation is polynomial in *ϵ*. Its constant term vanishes, since (1.3) is a solution of (1.1). The order *ϵ* term is dominant, and its coefficient is a linear partial differential equation for and . Separating its real and imaginary parts gives the following coupled set of partial differential equations:

(1.6)Since the coefficients of these equations do not explicitly depend on *y* or *t*, we may separate variables, and let(1.7)where , as only perturbations that are bounded in space are considered. Further, denotes that the complex conjugate of the previous term is added. Lastly, is a complex constant that determines the stability of the solitary wave solution with respect to perturbations with *x*-dependence given by and and *y*-dependence characterized by the wavenumber *ρ*. Then, , satisfy the coupled system of ordinary differential equations(1.8)where(1.9)Thus, if eigenfunctions , of the system are found for which has a strictly positive real part, the solitary wave solution (1.3) is unstable with respect to perturbations with wavenumber *ρ*. The growth rate of the perturbation is . For real , are real-valued, and the form of the perturbed solution is(1.10)This shows that a real positive eigenvalue gives rise to uniform exponential growth of the perturbation, i.e. the entire profile of the perturbation is multiplied by the same factor . On the other hand, for complex ,(1.11)In this case, the effect of the instability is not a mere amplification of the perturbation profile, as there is an additional oscillatory effect. We refer to instabilities due to non-real eigenvalues as oscillatory instabilities.

In this paper, the spectral problem (1.8) is investigated numerically and analytically. A review of previous works on this problem is given in §2. The numerical results are discussed in §3. Sections 4 and 5 describe analytical results on the spectrum of (1.8). Section 6 concludes the paper. Appendix A contains the outline of the numerical method.

## 2. Literature review

An overview of previous work was already presented in Pelinovsky (2001). For the sake of contrasting our results with those in the literature, this overview is partially repeated here, although different points are emphasized. Different reviews may be found in Yuen & Lake (1982), Kuznetsov *et al*. (1986) and Kivshar & Pelinovsky (2000). Similar considerations for a different but related problem are discussed in Skryabin (1999).

The transverse instability of the one-dimensional NLS equation soliton (1.3) was first discussed by Zakharov & Rubenchik (1974). The one-dimensional (i.e. *ρ*=0) problem (1.8) has a quadruple zero eigenvalue, i.e. for *ρ*=0. Zakharov & Rubenchik used regular perturbation theory near to find asymptotic expressions for four branches . Two of these branches result in imaginary (neutrally stable) values of , whereas the other two give a real positive (unstable) and a real negative (stable) value of . In effect, this settles the global stability question of the one-dimensional soliton solution: it is unstable because it is unstable with respect to two-dimensional perturbations with long-wavelength (small *ρ*) transverse perturbations. The method of Zakharov & Rubenchik leads to(2.1)for the two real branches (unstable) and (stable), and(2.2)for the two imaginary branches and , both neutrally stable (Yajima 1974; Pereira *et al*. 1978; Anderson *et al*. 1979; Kivshar & Pelinovsky 2000).

Successively, attention focused on the transverse stability of (1.3) with respect to short waves (large *ρ*). Plotting the positive root of (2.1) may lead one to believe that there is a cut-off on the set of *ρ* values for which (1.3) is unstable with respect to transverse perturbations with wavenumber *ρ*. However, this argument is based on a perturbation expansion around , and is not valid for large *ρ*. Different authors examined this behaviour, see for instance Cohen *et al*. (1976), Saffman & Yuen (1978) and Anderson *et al*. (1979).

Cohen *et al*. (1976) appear to be the first to consider (1.8) from a numerical point of view, for a large range of *ρ* values. Their results agree well with the perturbation result (2.1) for small values of *ρ*. For larger values of *ρ*, they found that the instability growth rate due to real reaches a maximum, around , after which it decreases. Their numerical results for values of *ρ*>1 were sufficiently problematic that they were not displayed in their paper. Some unstable growth was observed for , but not for *ρ*=2. At *ρ*=1, they found that is at around 90% of its maximal value.

Saffman & Yuen (1978), apparently unaware of this work, solved the eigenvalue problem (1.8) numerically as well. They concluded that the growth due to real eigenvalues reaches a maximum around , after which it decreases to zero, at . Some of these results were extended to periodic solutions with increasingly longer periods by Martin *et al*. (1980). These authors also discussed instabilities due to complex eigenvalues corresponding to short-wave (large *ρ*) perturbations, but no concrete claims about the soliton limit were made.

Anderson *et al*. (1979) revisit earlier numerical results by Pereira *et al*. (1978). No references to Cohen *et al*. (1976) or Saffman & Yuen (1978) are made. Anderson *et al*. solve (1.8) numerically, using a leap-frog finite-difference method. They report excellent agreement with (2.1), almost up to the maximal value of , obtained at . As *ρ* is advanced beyond *ρ*=1, a bifurcation of the growth rate to complex values is observed, at . Their results are corroborated by use of the Rayleigh–Ritz variational quotient, using a variety of test functions.

Ablowitz & Segur (1979) considered this same problem, using mostly perturbation theory. Specifically, considering real , they found that in the short-wave limit (), . This establishes neutral stability at the leading order. These authors referred to experiments done in one-dimensional (i.e. narrow) wave tanks by J. Hammack (1979, unpublished work) at the University of Florida in 1979, which showed good agreement with the dynamics of (1.3) in a one-dimensional NLS equation setting. As the presence of transverse short-wave instabilities would appear to contradict these experimental observations, Ablowitz & Segur concluded that ‘short waves are not unstable. Indeed if they were, it would be difficult to observe envelope solitons even in narrow wave tanks.’

*Remarks*

It should be pointed out that many of the works discussed above claim to numerically solve the problem (1.8). In all cases, these authors presumably reduced (1.8) to a periodic problem, possibly with large period. The effect of this truncation is not discussed. The numerical method used in the present paper does this as well, but the effects due to a finite period are carefully controlled and can be made arbitrarily small by tuning the input parameters (see appendix A).

The majority of the works cited are interested in the dominant unstable mode at any

*ρ*. Starting from random perturbations, it is this mode which might be observed in experiments, or in numerical solutions of (1.1).

## 3. Numerical results

In this paper, the spectral problem (1.8) is solved numerically, for a discrete set of *ρ*-values, ranging from *ρ*=0 to 1.5. The numerical method is described in appendix A. The results of these numerical runs are presented in figures 2 and 3. Also indicated are solid curves, depicting known and new analytical results. Note that figure 2*a* could be extended to stable modes, by including negative . The resulting figure is reflection symmetric with respect to the horizontal axis. For values of *ρ*>1, a spurious cloud of eigenvalues surrounds the eigenvalue . This is a numerical effect. The size of this cloud can be made arbitrarily small by using larger values of *L* and *N* (see appendix A). For the runs here, for any of the numerical eigenvalues in the spurious cloud. These points were removed before producing the plots shown here.

Figure 4 displays the different bifurcations as *ρ* changes from *ρ*=0 to 1.5. Note the fourfold symmetry due to the Hamiltonian character of (1.1). Finally, a variety of computed eigenfunctions and the water wave surface they give rise to, using (1.10) or (1.11), is shown in figures 5–7, corresponding to real and complex values of .

From these figures, the following conclusions may be drawn.

*Small-ρ instability*. As predicted by Zakharov & Rubenchik (1974) (and verified by all cited in the previous section), as*ρ*moves away from 0, the quadruple eigenvalue at the origin splits into four, resulting in two real eigenvalues and two imaginary eigenvalues . This is illustrated in figure 4*a*. The numerical results show excellent agreement with the perturbation results (2.1) and (2.2). A representative eigenfunction for the unstable eigenvalue is shown in figure 5i. As noted by Zakharov & Rubenchik, this eigenfunction is odd, and it gives rise to a so-called ‘snaking’ instability, whereby the solitary wave ridge oscillates forward and backward.Also at

*ρ*=0, a discrete eigenvalue emerges from the edge point of each component of the continuous spectrum located at , with . Note that Saffman & Yuen (1978) already reported the emergence from the continuous spectrum at*ρ*=0 of the isolated eigenvalue . Such a bifurcation is known as an edge bifurcation. It may be analysed using a modified Evans function (Kapitula & Sandstede 1998, 2002; Pelinovsky*et al*. 1998).*An oscillatory instability*. At , the isolated imaginary eigenvalues collide with the imaginary eigenvalues . This gives rise to two complex conjugate pairs of eigenvalues and , past , due to a Hamiltonian–Hopf bifurcation. This bifurcation may be explained from the collision of eigenvalues of opposite (negative and positive) Krein signatures, where the Krein signature is the sign of the Hamiltonian of (1.6) computed for the eigenvector corresponding to the colliding imaginary eigenvalues (Kapitula*et al*. 2004).The real part of and on these curves reaches a maximum at , after which it monotonically decreases. The corresponding eigenfunction and the resulting water wave surface are shown in figure 7vii. These eigenfunctions appear to be even, resulting in ‘neck’ instability, where the ridge of the soliton alternates between being high and narrow, and low and broad. For all

*ρ*, the growth rate along this curve is less than that of some other curve of unstable eigenvalues (i.e. for any*ρ*in the domain of , at least one other eigenvalue exists such that , see below). Thus, this curve of complex eigenvalues and its eigenfunctions lead to oscillatory instabilities, but these are always dominated by other instabilities. The birth of these complex eigenvalues is illustrated in figure 4*b*. To our knowledge, the existence of the branch of oscillatory instabilities (complex eigenvalues) for the solitary wave solution (1.3) is a new result. Similar oscillatory instabilities for periodic waves were reported in Martin*et al*. (1980).*Maximal growth rate*. At , reaches its maximum , resulting in the largest growth rate for perturbations of the solution (1.3) of (1.1). For , decreases monotonically. This agrees qualitatively with the results in Cohen*et al*. (1976) and Saffman & Yuen (1978) and quantitatively with Anderson*et al*. (1979). A numerical computation of the resulting eigenfunction is shown in figure 5ii. The resulting instability is still of snaking type.*Behaviour at ρ*=1. The behaviour at*ρ*=1 has been the subject of much discussion. The continuous spectrum for (1.8) is always confined to the imaginary -axis. It may be found exactly by considering the behaviour as in (1.8), leading to(3.1)as an explicit characterization of the continuous spectrum. Thus, for a given the continuous spectrum is given by(3.2a)(3.2b)Note that the edge of the continuous spectrum is given by . Our numerical results are in excellent agreement with this analytical result, as can be seen in figure 2*b*. At*ρ*=1, the two previously disconnected parts of the continuous spectrum merge, after which they overlap for larger values of*ρ*. As depicted in figure 4*c*, the consequence of this merger is the ejection of a pair of real eigenvalues for*ρ*>1, given by(3.3)which is proven in §5. To our knowledge this is a new result. Two eigenfunctions on this branch are shown in figure 6v,vi.*Existence of a cut-off for real**instabilities*. There is a largest*ρ*value such that for no instabilities due to real exist. Approximately, . At , the two eigenvalues and collide. Their corresponding eigenfunctions near the collision appear qualitatively similar. They are shown in figure 5iii,iv.*Large-ρ instability*. The solitary wave is unstable with respect to perturbations with arbitrarily large transverse wavenumber*ρ*. This instability is due to a pair of complex conjugate eigenvalues (emerging from the collision of and at ), as well as the pair of complex conjugate eigenvalues . For both branches, , but , and thus the instability is oscillatory in nature. The behaviour near is illustrated in figure 4*d*. This appears to be in agreement with the results of Anderson*et al*. (1979), and the results hinted at by Cohen*et al*. (1976). The eigenfunction corresponding to and the resulting water wave surface are shown in figure 7viii. These eigenfunctions are odd, resulting in ‘snake’ instability.

Note that the existence of a large-*ρ* instability does not contradict the conclusions reached by Ablowitz & Segur (1979) because those authors only considered real . It follows from their results that no *real* discrete eigenvalues exist for large *ρ*.

## 4. Bifurcations in the spectrum for

The spectra of the operators are known analytically. The continuous spectrum of is and the discrete spectrum has an isolated eigenvalue at *λ*=0. Similarly, the continuous spectrum of is as well. In this case the discrete spectrum has two isolated eigenvalues at *λ*=0 and −3.

When , the stability problem (1.8) satisfies the assumptions of the recent work (Pelinovsky 2005). Two negative eigenvalues of and one negative eigenvalue of must match the unstable eigenvalues and the eigenvalues of negative Krein signature of the problem (1.8) (see proposition 8.3 in Pelinovsky 2005). Indeed, there is one real (unstable) eigenvalue and a pair of eigenvalues of negative Krein signatures for , where . (The fact that the pair has negative Krein signature follows from Lyapunov–Schmidt reduction, see Skryabin (1999) for a similar analysis). The other pair of eigenvalues has positive Krein signature, which follows from general properties of edge bifurcations (Kapitula *et al*. 2004). The number of unstable eigenvalues (total three: one real and two complex) is preserved after the Hamiltonian–Hopf bifurcation for .

## 5. Analysis of the behaviour of the spectrum near

It was previously hypothesized in the literature that the curve of the real unstable eigenvalue touches the horizontal axis at some cut-off value (see the discussion in Pelinovsky (2001)), and various values were mentioned (for example in Saffman & Yuen (1978)), with *ρ*=1 always being a contender. Our numerical results show that the curve does not touch the horizontal axis. Instead, a new curve of eigenvalues emanates from the origin for *ρ*>1 as a consequence of the collision of the two parts of the continuous spectrum, as mentioned in §3. Therefore, in this section, we prove the asymptotic formula (3.3).

When , the assumptions of Pelinovsky (2005) are violated since the negative index of the operators and becomes infinite. It was suggested in Pelinovsky (2001) that at a modified edge bifurcation takes place, which would be responsible for the disappearing of the unstable eigenvalues as is increased past 1. An analytical method was developed based on eigenfunction decompositions and asymptotic analysis of singular integral equations (adopted from Pelinovsky *et al*. (1998)). Unfortunately, a miscalculation was made. The integral kernel defined below eqn (12) in Pelinovsky (2001) is singular in the limit , while it was assumed in eqn (13) that . With this correction in mind, the results of §4 of Pelinovsky (2001) on the modified edge bifurcations cannot be justified. Note that the results of §5 of Pelinovsky (2001) remain valid, as they are based on regular perturbation theory.

In this section, we develop an alternative analysis based on the modified Evans function (following Kapitula & Sandstede (1998)). We confirm that the point does indeed give rise to a modified edge bifurcation, which leads to a new pair of real eigenvalues for , but to no additional eigenvalues for .

### (a) The Evans function

In order to define the Evans function for the stability problem (1.8), we define a map from the parameters to the new parameters ,(5.1)Near the point of interest , the new parameters provide a multiple cover of the original parameter plane . Using the new parameters, the Evans function can be locally extended in an analytic way. To find the eigenvalues of the stability problem (1.8), some care is required to ensure that the zeros of the extended Evans function are found on the correct Riemann sheet. Using the new parameters , the stability problem (1.8) is rewritten as(5.2a)

(5.2b)Let . The eigenfunctions corresponding to the discrete spectrum of (5.2a) and (5.2b) decay exponentially as as in the no-potential case. In the domain , we define four fundamental solutions of (5.2a) and (5.2b) specified by their exponential behaviour as :

(5.3)and

(5.4)The Evans function is defined as the Wronskian determinant of these fundamental solutions:

(5.5)By Abel's theorem, this Wronskian determinant is independent of *x*. In the no-potential case, an easy calculation gives . Following standard techniques (e.g. Kapitula & Sandstede 1998), the fundamental solutions (5.3) and (5.4) and the Evans function (5.5) are well-defined when the potential functions decay exponentially. Zeros (including multiplicities) of in the domain and coincide with eigenvalues of the systems (5.2a) and (5.2b) with an exponentially decaying eigenfunction. The analyticity properties of are described in proposition 5.1.

*The fundamental solutions* *and* *characterized by* *(5.3) and (5.4)* *and the Evans function* *defined by* *(5.5)* *are analytic functions near* .

In the domain , we express the fundamental solutions in the form(5.6)where the component functions solve the coupled system(5.7a)

(5.7b)subject to the boundary conditions(5.8)Since the systems (5.7a) and (5.7b) depend analytically on the parameters and the boundary conditions (5.8) are independent of , the fundamental solutions depend analytically on these parameters near the point , see Gardner & Zumbrun (1998). The same statement extends to the fundamental solutions and hence to the Evans function computed at any . ▪

### (b) Eigenvalues at

The point corresponds to the case , so that . We compute , the value of at , to determine the possible values of at which bifurcations may occur. The value depends on whether or .

For and , the system (1.8) decouples, and its solutions may be written in terms of the exact eigenfunctions of the operators and :(5.9)where are given explicitly in Pelinovsky (2001) as

with . It is easy to verify that (5.9) satisfies (5.3) and (5.4) with the correspondence . Substituting (5.9) in the Evans function (5.5), we obtain the exact value of for :(5.10)For and , the fundamental solutions of (1.8) have another representation:(5.11)whereandwith . As before, it is easy to verify that (5.11) satisfies (5.3) and (5.4) with the correspondence . Substituting (5.11) in the Evans function (5.5), we obtain the exact value of for :(5.12)We see from (5.10) and (5.12) that at , 0 and 1, which are the only bifurcation points from the origin (see also Pelinovsky (2001)). The first case is not of interest here, because it results in an imaginary transverse wavenumber. Note that for the elliptic NLS equation, this case corresponds to the cut-off on the range of transversely unstable wavenumbers of the soliton solution (1.3), as shown Janssen & Rasmussen (1983). The second case has been analysed in the past, originally by Zakharov & Rubenchik (1974), as reviewed in §2. We focus on the case in what follows.

### (c) Bifurcation analysis near

The bifurcation point and corresponds to the origin in the systems (5.2a) and (5.2b). Using proposition 5.1, we study the Taylor series expansion of near . Let the fundamental solutions be represented by (5.6). The Taylor series approximation for the scalar functions up to quadric terms in can be written as(5.13)and

(5.14)where is defined by (5.1), and the terms represent the remainder of the Taylor series.

At zeroth order in , we find decoupled ordinary differential equations for the functions satisfying the boundary conditions (5.8), so that(5.15)

The higher-order corrections solve a sequence of linear inhomogeneous problems with zero boundary conditions as . The correction functions solve the equations(5.16a)

(5.16b)for . Equations (5.16a) and (5.16b) can be solved analytically, since all solutions with are known in closed form, see (5.9) and (5.11). For instance, the first few solutions of the inhomogeneous problems (5.16*a*) are found explicitly as follows:

(5.17a)

(5.17b)

(5.17c)The correction functions solve the equations

(5.18a)

(5.18b)for and . Since the homogeneous problems (5.18a) and (5.18b) have non-zero solutions, solutions of the inhomogeneous problems with zero boundary conditions as grow algebraically as . This follows easily from variation of parameters. For instance grow as , grow as , and so on. Similarly, the correction functions satisfy inhomogeneous problems, from which it follows that these correction functions grow as in the limit .

Replacing the other set of fundamental solutions by its asymptotic values (5.4) as , we express the Evans function in its explicit form (after some simplification):(5.19)The function is expanded using the Taylor series expansions (5.13) and (5.14) for and the explicit analytical solutions (5.15) and (5.17a)–(5.17c). Explicit computations using symbolic manipulations in either Maple or Mathematica results in

(5.20)where is a numerical coefficient, given by

(5.21)Although the correction functions and grow algebraically as , one readily shows using the inhomogeneous problems (5.18a) and (5.18b) that the algebraically growing terms cancel and a finite value for exists. This fact is also implied from proposition 5.1, since the function is well-defined, *x*-independent and analytic for small . Let us define new variables

(5.22)This allows us to rewrite the Taylor series approximation (5.20) using these variables as

(5.23)The main consequence for our purposes of the Taylor series approximation is formulated below.

*There exists only one family of zeros of* *in a local neighbourhood of the point* . *This family of zeros is given asymptotically by*(5.24)

Since is analytic near , all zeros of in a neighbourhood of may be obtained using the Newton polygon technique (see §2.8 in Chow & Hale (1982)). Constructing the convex Newton polygon from the fourth-order terms (5.23), we obtain only one line segment, namely between the terms and , which proves the leading-order result (5.24) by invoking the Implicit Function Theorem (see §2.2 in Chow & Hale (1982)), modulo the actual coefficient of , which is easily found using a regular scaling argument.

In order to prove that the next-order term in (5.24) is in the order of , we need to prove that the term is absent in the remainder terms of the expansion (5.23). Since this term should also be present if we let *α*=0, this may be inferred from the exact solution (5.12) for . Indeed, for that case we have , with , , *α*=0 and . Since the expression (5.12) is even in *k*, the term is absent; therefore, the term with is absent from (5.23). ▪

### (d) Proof of the asymptotic formula (3.3)

The inverse mapping of (5.1) is(5.25)where we consider the right half-plane of the complex plane of . The two branches of the square root function are defined as usual:(5.26)with . When , , and , the numbers lie in the first (+) and fourth (−) quadrants of the complex plane. When , , and , the numbers lie in the second (+) and third (−) quadrants. For both of these cases, the + square root function (5.26) is used for the mapping and the − square root function is used for the mapping . Thus, maps to the first quadrant, while maps to the fourth quadrant. Since the new eigenvalue is real, we only need to consider the mappings in a narrow strip around the real axis, as above. Thus, the parameters *α* and *β* in proposition 5.2 are defined explicitly in terms of *ρ* and as(5.27a)

(5.27b)The balance in (5.24) implies that cannot be of order as . Therefore, we consider three separate cases, when is of different orders in and the definitions of *α* and *β* can be simplified.

Let and . The Taylor series expansions of the mappings (5.27a) and (5.27b) converge and result inWith this expansion, the asymptotic result (5.24) is equivalent to the pair of real eigenvaluesTherefore, we have established an asymptotic approximation for the real unstable eigenvalue , bifurcating for from the point :(5.28)

Let and . Then, the Taylor series expansions of the mappings (5.27a) and (5.27b) converge and giveUsing this expansion, the asymptotic result (5.24) is equivalent to four eigenvalues defined byHowever, the scaling violates the assumption that as . Therefore, this case does not produce non-zero eigenvalues of the discrete spectrum.

Let . The Taylor series expansions of the mappings (5.27a) and (5.27b) converge and result inwhere . With this expansion the asymptotic result (5.24) produces only one solution , which is consistent with the assumptions only if *ρ*^{2}=1. Therefore, this case does not produce non-zero eigenvalues of the discrete spectrum either.

The above analysis proves theorem 5.3.

*The modified edge bifurcation for the stability problem* *(1.8)* *at* *results in a pair of discrete eigenvalues* *in the domain* *given by the asymptotic approximation* *(3.3)*. *No eigenvalues bifurcate from* *at* *into the domain* .

The result (5.28) is plotted in figures 2 and 8 as a solid curve. Figure 8 shows the real parts of eigenvalues zoomed near the point *ρ*=1. This figure illustrates the correctness of the asymptotic approximation (5.28), but it also demonstrates its small range of validity. This is somewhat expected. First, (5.28) is a low-order approximation. The inclusion of higher-order terms should improve the range of validity of the approximation. Second, at the bifurcation takes place, and one cannot expect the approximation (5.28) to be valid near or beyond where this other bifurcation occurs.

## 6. Concluding remarks

In this paper we have considered the linear stability problem (1.8) for the parameter . We have considered both real and complex eigenvalues, and the discrete and the continuous spectrum were computed. Wherever both are available, the agreement between numerical and analytical results is excellent, serving to validate both.

From the point of physical applications, the most important result presented in this paper is the existence of and , which implies the instability of the one-dimensional NLS soliton (1.3) with respect to perturbations of all wavelengths, including arbitrarily short ones. As outlined in §2, it was known previously that (1.3) is unstable with respect to long-wave (small-*ρ*) perturbations. Instabilities with respect to short-wave (large-*ρ*) perturbations were first hinted at by Cohen *et al*. (1976). The presence of such oscillatory instabilities was shown in more detail by Anderson *et al*. (1979).

How is this consistent with the remarks of Ablowitz & Segur (1979)? They state that the observations of J. Hammack (1979, unpublished work) imply the stability of (1.3) with respect to short-wave perturbations. This issue may be resolved by a closer look at figure 2*a*. It is clear from that figure that the growth rate associated with such instabilities is smaller by an approximate factor of at least 4: the maximal growth rate at is approximately 0.657. The largest growth rate associated with is approximately 0.2 at . The experiments performed by J. Hammack (1979, unpublished work) were done in a long, narrow wave tank. In such a setting, the smallest value of *ρ* resulting in a transverse perturbation of wavelength that will fit in the transverse direction of the wave tank is found significantly beyond the critical value . For those values of *ρ*, the associated growth rate is small, and observing this instability experimentally is difficult. One possibility is to employ a longer or wider tank, which is not a practical suggestion. The other possibility is to seed the instability by letting the wave pattern evolve according to (1.11), as opposed to according to (1.3). This might be accomplished using a suitably shaped paddle at the wave maker.

From a mathematical point of view, the main result of this paper is the bifurcation of the eigenvalues from the continuous spectrum, at the parameter value *ρ*=1 before which both parts of the continuous spectrum are disconnected. This result was established both numerically and analytically, resulting in the asymptotic approximation (5.28).

Mathematically, it would be desirable to also have asymptotic approximation results for the discrete but complex eigenvalue branches and . Since these bifurcations do not occur from the origin, the bifurcation analysis may be more complicated. Since for both and the imaginary part→∞ as *ρ*→∞, it seems reasonable that one may be able to establish a large *ρ* result, valid near the point at infinity on the complex sphere.

## Acknowledgements

The work presented here was sponsored by NSF-DMS-FRG-0351466 (BD), NSERC and PREA grants (DEP), and NSF-DMS-FRG-0139771 (JC). Further, we are grateful to Arnd Scheel for useful discussions regarding §5.

## Footnotes

- Received December 6, 2005.
- Accepted January 6, 2006.

- © 2006 The Royal Society