Short-wave transverse instabilities of line solitons of the 2-D hyperbolic nonlinear Schrödinger equation

# Short-wave transverse instabilities of line solitons of the 2-D hyperbolic nonlinear Schrödinger equation

## Abstract

We prove that line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation are unstable with respect to transverse perturbations of arbitrarily small periods, i.e., short waves. The analysis is based on the construction of Jost functions for the continuous spectrum of Schrödinger operators, the Sommerfeld radiation conditions, and the Lyapunov–Schmidt decomposition. Precise asymptotic expressions for the instability growth rate are derived in the limit of short periods.

## 1 Introduction

Transverse instabilities of line solitons have been studied in many nonlinear evolution equations (see the pioneering work [14] and the review article [10]). In particular, this problem has been studied in the context of the hyperbolic nonlinear Schrödinger (NLS) equation

 iψt+ψxx−ψyy+2|ψ|2ψ=0, (1)

which models oceanic wave packets in deep water. Solitary waves of the one-dimensional (-independent) NLS equation exist in closed form. If all parameters of a solitary wave have been removed by using the translational and scaling invariance, we can consider the one-dimensional trivial-phase solitary wave in the simple form . Adding a small perturbation to the one-dimensional solitary wave and linearizing the underlying equations, we obtain the coupled spectral stability problem

 (L+−ρ2)U=−λV,(L−−ρ2)V=λU, (2)

where is the spectral parameter, is the transverse wave number of the small perturbation, and are given by the Schrödinger operators

 L+=−∂2x+1−6sech2(x),L−=−∂2x+1−2sech2(x).

Note that small corresponds to long-wave perturbations in the transverse directions, while large corresponds to short-wave transverse perturbations.

Numerical approximations of unstable eigenvalues (positive real part) of the spectral stability problem (2) were computed in our previous work [5] and reproduced recently by independent numerical computations in [13, Fig. 5.27] and [3, Fig. 2]. Fig. 2 from [5] is reprinted here as Figure 1. The figure illustrates various bifurcations at , , , and , as well as the behavior of eigenvalues and the continuous spectrum in the spectral stability problem (2) as a function of the transverse wave number .

An asymptotic argument for the presence of a real unstable eigenvalue bifurcating at for small values of was given in the pioneering paper [14]. The Hamiltonian Hopf bifurcation of a complex quartet at for was explained in [5] based on the negative index theory. That paper also proved the bifurcation of a new unstable real eigenvalue at for , using Evans function methods. What is left in this puzzle is an argument for the existence of unstable eigenvalues for arbitrarily large values of . This is the problem addressed in the present paper.

The motivation to develop a proof of the existence of unstable eigenvalues for large values of originates from different physical experiments (both old and new). First, Ablowitz and Segur [1] predicted there are no instabilities in the limit of large and referred to water wave experiments done in narrow wave tanks by J. Hammack at the University of Florida in 1979, which showed good agreement with the dynamics of the one-dimensional NLS equation. Observation of one-dimensional NLS solitons in this limit seems to exclude transverse instabilities of line solitons.

Second, experimental observations of transverse instabilities are quite robust in the context of nonlinear laser optics via a four-wave mixing interaction. Gorza et al. [6] observed the primary snake-type instability of line solitons at for small values of as well as the persistence of the instabilities for large values of . Recently, Gorza et al. [7] demonstrated experimentally the presence of the secondary neck-type instability that bifurcates at near .

In a different physical context of solitary waves in -symmetric waveguides, results on the transverse instability of line solitons were re-discovered by Alexeeva et al. [3]. (The authors of [3] did not notice that their mathematical problem is identical to the one for transverse instability of line solitons in the hyperbolic NLS equation.) Appendix B in [3] contains asymptotic results suggesting that if there are unstable eigenvalues of the spectral problem (2) in the limit of large , the instability growth rate is exponentially small in terms of the large parameter . No evidence to the fact that these eigenvalues have nonzero instability growth rate was reported in [3].

Finally and even more recently, similar instabilities of line solitons in the hyperbolic NLS equation (1) were observed numerically in the context of the discrete nonlinear Schrödinger equation away from the anti-continuum limit [12].

The rest of this article is organized as follows. Section 2 presents our main results. Section 3 gives the analytical proof of the main theorem. Section 4 is devoted to computations of the precise asymptotic formula for the unstable eigenvalues of the spectral stability problem (2) in the limit of large values of . Section 5 summarizes our findings and discusses further problems.

## 2 Main results

To study the transverse instability of line solitons in the limit of large , we cast the spectral stability problem (2) in the semi-classical form by using the transformation

 ρ2=1+1ϵ2,λ=iωϵ2,

where is a small parameter. The spectral problem (2) is rewritten in the form

 (−ϵ2∂2x−1−6ϵ2sech2(x))U=−iωV,(−ϵ2∂2x−1−2ϵ2sech2(x))V=iωU. (3)

Note that we are especially interested in the spectrum of this problem for , which corresponds to in the original problem. Also, the real part of , which determines the instability growth rate for (2) corresponds, up to a factor of , to the imaginary part of .

Next, we introduce new dependent variables which are more suitable for working with continuous spectrum for real values of :

 φ:=U+iV,ψ:=U−iV.

Note that and are not generally complex conjugates of each other because and may be complex valued since the spectral problem (3) is not self-adjoint. The spectral problem (3) is rewritten in the form

 (−ϵ2∂2x+ω−1−4ϵ2sech2(x))φ−2ϵ2sech2(x)ψ=0,(−ϵ2∂2x−ω−1−4ϵ2sech2(x))ψ−2ϵ2sech2(x)φ=0. (4)

We note that the Schrödinger operator

 L0=−∂2x−4sech2(x) (5)

admits exactly two eigenvalues of the discrete spectrum located at and [11], where

 E0=(√17−12)2,E1=(√17−32)2. (6)

The associated eigenfunctions are

 φ0=sech√E0(x),φ1=tanh(x)sech√E1(x). (7)

In the neighborhood of each of these eigenvalues, one can construct a perturbation expansion for exponentially decaying eigenfunction pairs and a quartet of complex eigenvalues of the original spectral problem (4). This idea appears already in Appendix B of [3], where formal perturbation expansions are developed in powers of .

Note that the perturbation expansion for the spectral stability problem (4) is not a standard application of the Lyapunov–Schmidt reduction method [4] because the eigenvalues of the limiting problem given by the operator are embedded into a branch of the continuous spectrum. Therefore, to justify the perturbation expansions and to derive the main result, we need a perturbation theory that involves Fermi’s Golden Rule [9]. An alternative version of this perturbation theory can use the analytic continuation of the Evans function across the continuous spectrum, similar to the one in [5]. Additionally, one can think of semi-classical methods like WKB theory to be suitable for applications to this problem [2].

The main results of this paper are as follows. To formulate the statements, we are using the notation to indicate that for sufficiently small positive values of , there is an -independent positive constant such that . Also, denotes the standard Sobolov space of distributions whose derivatives up to order two are square integrable.

###### Theorem 1.

For sufficiently small , there exist two quartets of complex eigenvalues in the spectral problem (4) associated with the eigenvectors in .

Let be one of the two eigenvalue–eigenvector pairs of the operator in (5). There exists an such that for all , the complex eigenvalue in the first quadrant and its associated eigenfunction satisfy

 |ω−1−ϵ2E0|≲ϵ3,∥φ−φ0∥L2≲ϵ,∥ψ∥L∞≲ϵ, (8)

while the positive value of is exponentially small in .

###### Proposition 1.

Besides the two quartets of complex eigenvalues in Theorem 1, no other eigenvalues of the spectral problem (4) exist for sufficiently small .

###### Proposition 2.

The instability growth rates for the two complex quartets of eigenvalues in Theorem 1 are given explicitly as by

 Re(λ)=Im(ω)ϵ2∼2p+32π2[Γ(p)]2ϵ3−2pe−√2πϵ,Re(λ)=Im(ω)ϵ2∼2p+52π2q2[Γ(q)]2ϵ1−2qe−√2πϵ, (9)

where and .

Note that the result of Theorem 1 guarantees that the two quartets of complex eigenvalues that we can see on Figure 1 remain unstable for all large values of the transverse wave number in the spectral stability problem (2).

## 3 Proof of Theorem 1

By the symmetry of the problem, we need to prove Theorem 1 only for one eigenvalue of each complex quartet, e.g., for in the first quadrant of the complex plane. Let and rewrite the spectral problem (4) in the equivalent form

 (−∂2x−4sech2(x))φ−2sech2(x)ψ=−Eφ,−2ψ−ϵ2(∂2x+E+4sech2(x))ψ=2ϵ2sech2(x)φ. (10)

At the leading order, the first equation of system (10) has exponentially decaying eigenfunctions (7) for and in (6). However, the second equation of system (10) does not admit exponentially decaying eigenfunctions for these values of because the operator

 Lϵ(E):=−2−ϵ2(∂2x+E+4sech2(x))

is not invertible for these values of . The scattering problem for Jost functions associated with the continuous spectrum of the operator admits solutions that behave at infinity as

 ψ(x)∼eikx,\rm wherek2=E+2ϵ2.

If , then . The Sommerfeld radiation conditions as correspond to solutions that are exponentially decaying in when is extended from real positive values for to complex values with for . Thus we impose Sommerfeld boundary conditions for the component satisfying the spectral problem (10):

 ψ(x)→a{eikx,x→∞,σe−ikx,x→−∞,k=1ϵ√2+ϵ2E, (11)

where is the radiation tail amplitude to be determined and depends on whether is even or odd in . To compute , we note the following elementary result.

###### Lemma 1.

Consider bounded (in ) solutions of the second-order differential equation

 ψ′′+k2ψ=f, (12)

where with and , whereas is a given function, either even or odd. Then

 ψ(x)=12ik∫x−∞eik(x−y)f(y)dy+12ik∫+∞xe−ik(x−y)f(y)dy (13)

is the unique solution of the differential equation (12) with the same parity as that satisfies the Sommerfeld radiation conditions (11) with

 a=12ik∫+∞−∞f(y)e−ikydy. (14)
###### Proof.

Solving (12) using variation of parameters, we obtain

 ψ(x)=eikx[u(0)+12ik∫x0f(y)e−ikydy]+e−ikx[v(0)−12ik∫x0f(y)eikydy],

where and are arbitrary constants. We fix these constants using the Sommerfeld radiation conditions (11), which yields

 u(0)=12ik∫0−∞f(y)e−ikydy,v(0)=12ik∫+∞0f(y)eikydy.

Using these expressions and the definition , we obtain (13) and (14). It is easily checked that has the same parity as . ∎

To prove Theorem 1, we select one of the two eigenvalue–eigenvector pairs of the operator in (5) and proceed with the Lyapunov–Schmidt decomposition

 E=E0+E,φ=φ0+ϕ,ϕ⊥φ0.

To simplify calculations, we assume that is normalized to unity in the norm. The orthogonality condition is used with respect to the inner product in and is assumed in the decomposition.

The spectral problem (10) is rewritten in the form

 (L0+E0)ϕ−2sech2(x)ψ=−E(φ0+ϕ),Lϵ(E0+E)ψ=2ϵ2sech2(x)(φ0+ϕ). (15)

Because , the correction term is uniquely determined by projecting the first equation of the system (15) onto :

 E=2∫∞−∞sech2(x)φ0(x)ψ(x)dx. (16)

If , then . Let be the orthogonal projection from to the range of . Then, is uniquely determined from the linear inhomogeneous equation

 P(L0+E0+E)Pϕ=2sech2(x)ψ−2φ0∫∞−∞sech2(x)φ0(x)ψ(x)dx, (17)

where is invertible with a bounded inverse and is assumed. On the other hand, is uniquely found using the linear inhomogeneous equation

 ψ′′+k2ψ=f,wheref=−2sech2(x)(φ0+ϕ+2ψ), (18)

subject to the Sommerfeld radiation condition (11), where is assumed. Note that is not real because of the Sommerfeld radiation condition (11) and depends on because of the -dependence of in

 k=1ϵ√2+ϵ2E0+ϵ2E. (19)

We are now ready to prove Theorem 1.

Proof of Theorem 1. The function on the right-hand-side of (18) is exponentially decaying as if . From the solution (13), we rewrite the equation into the integral form

 ψ(x) = iϵ√2+ϵ2E0+ϵ2E∫x−∞eik(x−y)sech2(y)(φ0+ϕ+2ψ)(y)dy (20) +iϵ√2+ϵ2E0+ϵ2E∫+∞xe−ik(x−y)sech2(y)(φ0+ϕ+2ψ)(y)dy.

The right-hand-side operator acting on is a contraction for small values of if and are bounded as , and for (yielding ). By the Fixed Point Theorem [4], we have a unique solution of the integral equation (20) for small values of such that as . This solution can be substituted into the inhomogeneous equation (17).

Since as and the operator is invertible with a bounded inverse, we apply the Implicit Function Theorem and obtain a unique solution of the inhomogeneous equation (17) for small values of such that as . Note that by Sobolev embedding of to , the earlier assumption for finding in (18) is consistent with the solution .

This proves bounds (8). It remains to show that for small nonzero values of . If so, then the real eigenvalue bifurcates to the first complex quadrant and yields the eigenvalue of the spectral problem (4) with . Persistence of such an isolated eigenvalue with respect to small values of follows from regular perturbation theory. Also, the eigenfunction in (20) is exponentially decaying in at infinity if . As a result, the eigenvector is defined in for small nonzero values of , although diverges as .

To prove that for small but nonzero values of , we use (11) and (18), integrate by parts, and obtain the exact relation

 2∫∞−∞sech2(x)(φ0+ϕ)ψ(x)dx = ∫∞−∞¯ψ(x)(−∂2x−k2−4sech2(x))ψ(x)dx = (−¯ψψx+¯ψxψ)∣∣∣x→+∞x→−∞ text+∫∞−∞ψ(x)(−∂2x−k2−4sech2(x))¯ψ(x)dx = 4ik|a(ϵ)|2+2∫∞−∞sech2(x)(φ0+ϕ)¯ψ(x)dx.

By using bounds (8), definition (14), and projection (16), we obtain

 Im(E) = 2Im∫∞−∞sech2(x)φ0(x)ψ(x)dx=2k|a(ϵ)|2(1+O(ϵ)) (21) = 2k∣∣∣∫+∞−∞sech2(x)φ0(x)e−ikxdx∣∣∣2(1+O(ϵ)),

which is strictly positive. Note that this expression is referred to as Fermi’s Golden Rule in quantum mechanics [9]. Since as , the Fourier transform of at this is exponentially small in . Therefore, is exponentially small in . The statement of the theorem is proved.

## 4 Proofs of Propositions 1 and 2

To prove Proposition 1, let us fix to be -independent and different from and in (6). We write for some small -dependent values of . The spectral problem (10) is rewritten as

 (22)

Proof of Proposition 1. If is real and negative, the system (22) has only oscillatory solutions, hence exponentially decaying eigenfunctions do not exist for values of near . Furthermore, note that the Schrödinger operator in (5) has no end-point resonances. Therefore no bifurcation of isolated eigenvalues may occur if . Thus, we consider positive values of if is real and values with if is complex.

By Lemma 1, we rewrite the second equation of the system (22) in the integral form

 ψ(x) = iϵ√2+ϵ2Ec+ϵ2E∫x−∞eik(x−y)sech2(y)(φ+2ψ)(y)dy (23) +iϵ√2+ϵ2Ec+ϵ2E∫+∞xe−ik(x−y)sech2(y)(φ+2ψ)(y)dy.

Again, the right-hand-side operator on is a contraction for small values of if and are bounded as , and for (yielding ). By the Fixed Point Theorem, under these conditions we have a unique solution of the integral equation (23) for small values of such that as . This solution can be substituted into the first equation of the system (22).

The operator is invertible with a bounded inverse if is complex or if is real and positive but different from and . By the Implicit Function Theorem, we obtain a unique solution of this homogeneous equation for small values of and for any value of as long as is small as (since is fixed independently of ). Next, with , the unique solution of the integral equation (23) is , hence is not an eigenvalue of the spectral problem (10).

To prove Proposition 2, we compute in Theorem 1 explicitly in the asymptotic limit . It follows from (19) and (21) that

 Im(ω)=√2ϵ3∣∣∣∫+∞−∞sech2(x)φ0(x)e−ikxdx∣∣∣2(1+O(ϵ)),

where .

Proof of Proposition 2. Let us consider the first eigenfunction in (7) for the lowest eigenvalue in (6). Using integral 3.985 in [8], we obtain

 I0 = ∫+∞−∞sech2(x)φ0(x)e−ikxdx=2∫∞0sechp(x)cos(kx)dx=2p−1Γ(p)∣∣∣Γ(p+ik2)∣∣∣2,

where . Since and , we have use the asymptotic limit 8.328 in [8]:

 lim|y|→∞|Γ(x+iy)|eπ2|y||y|12−x=√2π, (24)

from which we establish the asymptotic equivalence:

 I0 = 2p−1Γ(p)∣∣∣Γ(p+ik2)∣∣∣2∼2πΓ(p)k1−pe−π2k∼2p+12πΓ(p)ϵ1−pe−π√2ϵ.

Therefore, the leading asymptotic order for is given by

 Im(ω)∼2p+32π2[Γ(p)]2ϵ5−2pe−√2πϵ. (25)

Next, let us consider the second eigenfunction in (7) for the second eigenvalue in (6). Using integral 3.985 in [8] and integration by parts, we obtain

 I1 = ∫+∞−∞sech2(x)φ1(x)e−ikxdx=−2ikq∫∞0sechq(x)cos(kx)dx=−ik2q−1qΓ(q)∣∣∣Γ(q+ik2)∣∣∣2,

where . Using limit (24), we obtain

 I1 = −ik2q−1qΓ(q)∣∣∣Γ(q+ik2)∣∣∣2∼−2πikqΓ(q)k1−qe−π2k∼−i2p+22πqΓ(q)ϵ−pe−π√2ϵ.

Therefore, the leading asymptotic order for is given by

 Im(ω)∼2p+52π2q2[Γ(q)]2ϵ3−2qe−√2πϵ. (26)

In both cases (25) and (26), the expression for have the algebraically large prefactor in with the exponent and . Nevertheless, is exponentially small as .

## 5 Conclusion

We have proved that the spectral stability problem (2) has exactly two quartets of complex unstable eigenvalues in the asymptotic limit of large transverse wave numbers. We have obtained precise asymptotic expressions for the instability growth rate in the same limit.

It would be interesting to verify numerically the validity of our asymptotic results. The numerical approximation of eigenvalues in this asymptotic limit is a delicate problem of numerical analysis because of the high-frequency oscillations of the eigenfunctions for large values of , i.e., small values of , as discussed in [5]. As we can see in Figure 1, the existing numerical results do not allow us to compare with the asymptotic results of our work. This numerical problem is left for further studies.

Acknowledgments: The work of DEP, EAR, and OAK is supported by the Ministry of Education and Science of Russian Federation (Project 14.B37.21.0868). BD acknowledges support from the National Science Foundation of the USA through grant NSF-DMS-1008001.

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