Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls-Nabarro Barrier
We study a class of discrete focusing nonlinear Schrödinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the case of nearest neighbor interactions to a large class of nonlocal short-range and long-range interactions. The appearance of distinct onsite and offsite states is a consequence of the breaking of continuous spatial translation invariance. The PNB plays a role in the dynamics of energy transport in such nonlinear Hamiltonian lattice systems.
Our class of nonlocal interactions is defined in terms of coupling coefficients, , where is the lattice site index, with and (Kac-Baker). For , the bifurcation is seeded by solutions of the (effective / homogenized) cubic focusing nonlinear Schrödinger equation (NLS). However, for , the bifurcation is controlled by the fractional nonlinear Schrödinger equation, FNLS, with replacing . The proof is based on a Lyapunov-Schmidt reduction strategy applied to a momentum space formulation. The PN barrier bounds require appropriate uniform decay estimates for the discrete Fourier transform of DNLS discrete solitary waves. A key role is also played by non-degeneracy of the ground state of FNLS, recently proved by Frank, Lenzmann & Silvestre.
Key words. Discrete nonlocal nonlinear Schrödinger equation, onsite and offsite solitary waves, bifurcation from continuous spectrum, Peierls-Nabarro energy barrier, intrinsic localized modes
In this paper we study the discrete, focusing and cubically nonlinear Schrödinger equations with nonlocal interactions. We consider the both short-range and long-range interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive plane waves. This is the limit of long waves of small amplitude. The profile of such states is, at leading order, expressible in terms of the ground state of the continuum, in general fractional, nonlinear Schrödinger equation. Further, we prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states.
The appearance of both onsite and offsite states is a discreteness effect, in particular a consequence of the breaking of continuous translation invariance, and is therefore not captured by continuum approximations (homogenization, effective media). The PNB has played a role in a physical understanding of the dynamics of energy transport in nonlinear Hamiltonian lattice systems since the pioneering work of Peyrard and Kruskal . In particular, the PNB is interpreted as the energy a localized discrete wave must expend (shed to dispersive radiation) in order to transit from one lattice site to the next.
Such radiation damping is the mechanism through which a traveling discrete localized structure slows and eventually gets trapped and pinned to a lattice site; see the discussion in the introduction to  and, for example, see [25, 38, 27]. On the general subject of the role of radiation damping in extended Hamiltonian systems and applications see, for example, [54, 44, 46, 45].
1.1 DNLS and FNLS
We consider the discrete and nonlocal nonlinear Schrödinger equation (DNLS)
Here, is a linear and nonlocal interaction operator:
The range of the nonlocal interaction is characterized by the rate of decay of the coupling sequence, . Nonlocal interactions arising in applications typically have coupling sequences, with polynomial decay , or exponential (Kac-Baker) decay: , . We refer to the case as the short-range interaction case, and the case as long-range interaction case. The case is the critical or marginal range interaction. The case and , the operator corresponds to the nearest-neighbor discrete Laplacian.
DNLS is a Hamiltonian system, expressible in the form
We note the following basic result concerning acting on .
Assume is non-negative (), symmetric (), and . Then,
is a bounded linear operator on .
The spectrum of is continuous and equal to , where .
Proof: Young’s inequality implies boundedness. The other details of the proof are presented in Appendix LABEL:appendix:operatorL.
The initial value problem is globally well-posed, in the sense that for each there exists a unique global solution . Time translation invariance implies that is time-invariant on solutions and the invariance implies the time-invariance of
In  it was proved by Kirkpatrick, Lenzmann and Staffilani that on any fixed time interval, , the family of solutions which is parametrized by a discretization parameter tending to zero, converges in some norm to the solution of the initial value problem for the, in general, fractional nonlinear Schrödinger equation (FNLS) with focusing cubic nonlinearity:
where depends on the interaction range parameter, . (In terms of the Fourier transform, and its inverse , we define .) Thus, if the interaction is sufficiently short-range (), the limiting equation is the standard cubic nonlinear Schrödinger equation, while if interactions are longer range () the limiting equation is the fractional nonlinear Schrödinger equation (FNLS) with fractional Laplacian, .
Remark 1.1 (Discrete nonlinear systems as physical models)
Discrete nonlinear dispersive systems arise in nonlinear optics, e.g. [2, 1, 10, 11, 35, 55, 53], the dynamics of biological molecules, e.g. [7, 9], and condensed matter physics. For example, they arise in the study of intrinsic localized modes in anharmonic crystal lattices, e.g. [3, 49]. See also [43, 20]. In these fields, discrete systems arise either as phenomenological models or as tight-binding approximations; see also [33, 39]. There is also a natural interest in such systems as discrete numerical approximations of continuum equations.
Nonlocal discrete models such as nonlocal DNLS addressed in this paper are of particular interest in several of the fields above where atomic interactions occur at length scales considerably larger than those accounted for by models with nearest-neighbor coupling [18, 34, 6, 26, 24, 16].
A case of particular interest with respect to the coupling potential (LABEL:eqn:Jdeftrue) is , which corresponds to dipole-dipole interactions in biopolymers ; this is the threshold value of which marks the transition between the standard and fractional continuum Laplacian limits; see Theorem LABEL:th:mainnonlocal. The case of Coulomb interactions corresponding to is also of interest in this context, though we do not address it here. The exponential (Kac-Baker) coupling sequence is of considerable interest in both models of statistical physics and again in the dynamics of lattice models of biopolymers [16, 24, 18].
Continuum models with nonlocal effects such as FNLS also arise in several contexts including path-integral formulations of quantum mechanics , deep water internal and small-amplitude surface wave fluid dynamics [4, 37, 52], semi-relativistic quantum mechanics (astrophysics) [32, 15], and as the continuum limits of the aforementioned long-range discrete models [28, 50, 51, 17].
Cubic FNLS has, for and for any frequency , a standing wave solution:
where is the unique positive nonlinear ground state solution to the nonlinear eigenvalue problem
Although the dynamics (LABEL:eqn:IVPdnlsnonlocal) are defined for any choice of in , we shall restrict our attention to the case in (LABEL:eqn:Jdeftrue) for . The constraint on the range of the nonlocal interaction is the range for which the limiting fractional NLS equation has nonlinear bound state solutions. See Proposition LABEL:prop:psinonlocal, Theorem LABEL:th:mainnonlocal, and Remark LABEL:remark:main2nonlocal.
A key recent advance in the analysis of fractional / nonlocal equations, which plays as central role in the current work, are the results of Frank, Lenzmann and Silvestrie  on the uniqueness and non-degeneracy of the ground state solitary waves of FNLS. Our bifurcation results are perturbative about the FNLS limit and are based on an application of an implicit function theorem to a nonlinear mapping acting on a space of functions defined on momentum space. That the differential of this mapping, evaluated at the FNLS ground state, is an isomorphism between appropriate spaces is a consequence of this non-degeneracy.
The continuum nonlinear Schroedinger equation (NLS), with local dispersion corresponding to is Galilean invariant. Hence, any solitary standing wave of NLS can be boosted to give a solitary traveling wave; see . In [29, 21], non-deforming traveling solitary waves of FNLS were been shown to exist. This is in direct contrast with the discrete setting, where the PNB phenomenon occurs due to the breaking of continuous translational symmetries. In , traveling waves of the form are shown to occur for . In , waves of the form for , are constructed.
1.2 Discrete nonlocal solitary standing waves near the continuum limit
In analogy with the case of nearest-neighbor DNLS , we seek discrete solitary standing waves of DNLS with nonlocal interactions:
Thus, satisfies the nonlinear algebraic eigenvalue problem:
where is defined by (LABEL:eqn:Ldef) and for and for . The precise expressions for is displayed below in (LABEL:eqn:Jdeftrue).
The goal of this work is to obtain a precise understanding of the onsite symmetric and offsite symmetric discrete solitary standing waves of DNLS with short- and long-range nonlocal interactions.
Definition 1.2 (Onsite symmetric and offsite symmetric states in one spatial dimension)
A solution to equation (LABEL:eqn:tiDNLSnonlocal) is referred to as onsite symmetric if for all , it satisfies In this case, is symmetric about .
A solution to equation (LABEL:eqn:tiDNLSnonlocal) is referred to as offsite symmetric or bond-centered symmetric if for all , it satisfies In this case, is symmetric about the point halfway between and .
We seek spatially localized onsite and offsite solutions of (LABEL:eqn:tiDNLSnonlocal) in the long wave limit as . Examples of such states are displayed in figure LABEL:fig:onoff. Here, there is large scale separation between the width of the discrete standing wave and the unit lattice spacing. Solutions are expected to approach a (homogenized or averaged) continuum FNLS equation with fractional power to be determined.
We study the limit through the introduction of a single small parameter, . Introduce the continuous function where , where as is appropriately chosen. Then, satisfies the nonlinear eigenvalue problem
The precise form of is displayed below in (LABEL:eqn:kappa-def) and depends on whether the coupling interaction operator is long-range (), short-range (), or critical (). Thus, the study of nonlocal DNLS in the limit is equivalent to the study of . Onsite and offsite nonlinear bound states of (LABEL:eqn:tiDNLSrescalednonlocal) arise as bifurcations of non-trivial localized states from the zero state at frequency , the bottom of the continuous spectrum of (Proposition LABEL:prop:Lprops).
In , Kirkpatrick et. al. address the so-called continuum limit, of the following nonlocal DNLS equation:
Here, is the lattice spacing. This is a limit of interest in numerical computations. Solutions are expected to approach, as , those of continuum FNLS, , with fractional power to be determined. Equation (LABEL:eqn:KLSDNLS) may be mapped to (LABEL:eqn:IVPdnlsnonlocal) via the substitution .
Time-periodic bound states of (LABEL:eqn:KLSDNLS) satisfy
Substituting and taking , we obtain (LABEL:eqn:tiDNLSrescalednonlocal).
1.3 Main results
Our main results concern the existence and energetic properties of localized solutions of (LABEL:eqn:tiDNLSrescalednonlocal) for , and therefore , small:
Theorem LABEL:th:mainnonlocal; Bifurcation of onsite and offsite states of nonlocal DNLS: Let the sequence of coupling coefficients, , of nonlocal DNLS be (1) the polynomially decaying sequence (LABEL:eqn:Jdeftrue) with coupling decay rate , or (2) the exponentially decaying sequence (LABEL:eqn:Jdeftrue) for which we set . Then there exist families of onsite (vertex-centered) symmetric and offsite (bond-centered) symmetric solitary standing waves of (LABEL:eqn:tiDNLSrescalednonlocal) (in the sense of Definition LABEL:defn:onoff), which bifurcate for from the continuum limit ground state solitary wave of fractional NLS (LABEL:eqn:introtiNLS) with fractional power .
Theorem LABEL:th:PNnonlocal; Exponential smallness of the Peierls-Nabarro barrier: Let . Then, there exist positive constants and such that for all , we have:
Theorem LABEL:th:PNnonlocal is subtle - to any polynomial order in the small parameter, , and are discrete translates of one another by a half integer, but differ at exponentially small order in .
1.4 Strategy of proofs of Theorem LABEL:th:mainnonlocal and Theorem LABEL:th:PNnonlocal
The limit for in (LABEL:eqn:tiDNLS2nonlocal) is related to the continuum FNLS limit. In order to compare the spatially discrete and spatially continuous problems, it is natural to work with the discrete and continuous Fourier transforms respectively; both are functions of a continuous (momentum) variable.
Let denote the discrete Fourier transform on of the sequence and let denote the continuous Fourier transform on of ; see Section LABEL:section:introDFT for definitions and Appendix LABEL:appendix:DFT for a discussion of key properties. The following proposition characterizes onsite symmetric and offsite symmetric states on the integer lattice, , in terms of the discrete Fourier transform. The proof follows from a direct calculation using the definition of the discrete Fourier transform and its inverse .
Let be real and onsite symmetric in the sense of Definition LABEL:defn:onoff. Then, , the discrete Fourier transform of , is real-valued and symmetric. Conversely, if is real and symmetric, then , its inverse discrete Fourier transform, is real and onsite symmetric.
If is real and offsite symmetric in the sense of Definition LABEL:defn:onoff, then
where is real and symmetric. Conversely, if , where is real and symmetric, then is real and offsite symmetric.
Motivated by Proposition LABEL:prop:off-on, we first rewrite the equation for a DNLS standing wave profile (onsite or offsite) , (LABEL:eqn:tiDNLSrescalednonlocal), in discrete Fourier space for , where satisfies . Here, corresponds to the onsite case and to the offsite case. Note that is determined by its restriction, , to the fundamental cell (Brillouin zone); we often suppress the dependence on , , and for notational convenience.
We obtain the following equation for , defined for :
where is the characteristic function on . Here, is the discrete Fourier symbol of the operator : see (LABEL:LC-def) and Lemma LABEL:lemma:nonlocaltransform. In particular, we take
The positive constant, , is chosen so that the expansion of , for small , leads to an effective limiting () continuum equation in which has a coefficient equal to one (unit effective mass).
Toward a formal determination of a limiting equation, we introduce rescalings of the momentum: and , where is to be determined. Substitution into (LABEL:eqn:phieqnnonlocal0) and dividing by , we obtain the following equation for :
Here, is characteristic function of , and
Note that the dependence of the nonlinear operator, on designates the case of onsite or offsite states. Note that for :
Therefore, choosing to depend on the nonlocality parameter, :
we obtain the formal scaled convergence as ,
Returning to equation (LABEL:eqn:Phieqn3-intro), we obtain a formal balance of terms in the equation by choosing so that or equivalently The formal limit of (LABEL:eqn:Phieqn3-intro) is then
the equation for , the Fourier transform on of the continuum FNLS solitary wave, . Here, we have used: (a) , for bounded , as , (b) as , and (c) as for localized; see Lemma LABEL:lemma:expconvononlocal. We therefore expect, for small , that .
Therefore, the onsite and offsite discrete standing solitary waves, and can be constructed via rescaling : where , respectively and inverting the discrete Fourier transform.
The proof of bounds on the Peierles-Nabbaro (PN) barrier (Theorem LABEL:th:PNnonlocal) follows the outline  for the case of nearest-neighbor interaction. The PN barrier bounds depend on exponential decay bounds, uniformly in small, for . The extension of these bounds to the nonlocal case is presented in Appendix LABEL:appendix:expogeneral.
Reduction to FNLS employs a Lyapunov-Schmidt reduction strategy. We first solve for the high frequency components of for , ( and as ) in terms of those for (low frequency components of ). This yields a closed system for the low-frequency components, which we study perturbatively about the continuum FNLS limit using the implicit function theorem.
1.5.1 General coupling sequences
We note that our results for non-local coupling sequences can be extended to general non-local coupling sequences , with asymptotic decay rate . If , one may only extend under appropriate assumptions on the rate of decay of as .
The results may also be extended to coupling sequences with asymptotic decay faster than any algebraic power in . Note that this case includes any coupling sequence with compact support, including that of the nearest-neighbor (centered-difference) Laplacian.
1.5.2 Higher dimensional cubic lattices
In , our results apply to cubic DNLS with nearest-neighbor coupling in dimensions . This range of spatial dimensions corresponds to those for which the continuum NLS equation has a non-trivial ground state, as seen using classical virial identities. For the current context of nonlocal lattice equations, we expect our results on the one-dimensional cubic nonlocal DNLS to extend to dimensions, , satisfying: ; see Remark LABEL:remark:pohoone.
1.6 Outline of the paper
In Section LABEL:section:preliminaries, we provide a summary of basic facts about the discrete and continuous Fourier transforms and also provide a list of notations and conventions used throughout the paper.
In Section LABEL:section:psinonlocal, we summarize the properties of the continuum FNLS solitary standing wave.
In Section LABEL:section:mainresults, we state our result on the bifurcation of onsite and offsite solitary waves, Theorem LABEL:th:mainnonlocal. The exponentially small bound on the PN-barrier, the energy difference between onsite and offsite solutions, is stated in Theorem LABEL:th:PNnonlocal.
Sections LABEL:section:1dproofnonlocal through LABEL:section:rescalinglow0nonlocal contain the proof of Theorem 3.1.
Sections LABEL:section:nonlocalleadingorderproof and LABEL:section:rescalinglow0nonlocal construct the corrector in the rigorous asymptotic expansion of the solution to DNLS.
In Section LABEL:section:nonlocalleadingorderproof, we derive equations for the corrector. We also construct the high frequency component of the corrector as a functional of the low frequency components and the small parameter .
In Section LABEL:section:rescalinglow0nonlocal, we construct the low frequency component of the corrector and map our asymptotic expansion back to the solution to DNLS, completing the proof of Theorem LABEL:th:mainnonlocal.
In Section LABEL:section:higherorder, we outline the steps required to prove Theorem LABEL:th:SJalpha-error, which generalizes the leading order expansion given in Theorem LABEL:th:mainnonlocal to a higher order expansion in the small parameter .
There are several appendices. Appendix LABEL:appendix:subadd contains an elementary and very useful subadditivity lemma. In Appendix LABEL:appendix:DFT, we discuss properties of the discrete Fourier transform. In Appendix LABEL:appendix:operatorL, we discuss spectral properties of the operator . In Appendix LABEL:appendix:IFT, we provide a formulation of the implicit function theorem, which we can apply directly to our setting. In Appendix LABEL:appendix:asymptotics, we discuss the asymptotic properties of several special functions, which are related to the dispersion relation for discrete and nonlocal NLS. Appendix LABEL:appendix:expogeneral addresses the important exponential decay properties of solitary waves in Fourier ( momentum) space.
This work was supported in part by National Science Foundation grants DMS-1412560 and DGE-1069420 (the Columbia Optics and Quantum Electronics IGERT), and a grant from the Simons Foundation (#376319, MIW). The authors thank O. Costin, R. Frank, and P. Kevrekides for very stimulating discussions.
1.8 Preliminaries, Notation and Conventions
1.8.1 Discrete Fourier Transform
For a sequence , define the discrete Fourier transform (DFT)
Since , we shall view as being defined on the torus , where is fundamental period cell (Brillouin zone), is completely determined by its values on . We shall also make use of the scaled Brillouin zone, , with The inverse discrete Fourier transform is given by
A summary of key properties of the discrete Fourier transform is included in Appendix LABEL:appendix:DFT. For and in , we define the convolution on by
Let ,, and denote pseudo-periodic functions. Then, and
1.8.2 Continuous Fourier Transform on
For , the continuous Fourier transform and its inverse are given by
The standard convolution of functions on is defined by
Since our differential equations are cubically nonlinear, we must often work with the expression for a triple convolution:
1.8.3 Notations, Conventions and a useful Lemma
A function is periodic if for all , . For , a function is pseudo-periodic if for all .
The inner product on is given by where is the complex conjugate of .
, the space of functions satisfying .
, the space of functions satisfying .
, the space of functions satisfying , where .
and , are subspaces of and consisting of functions, , which satisfy . We also refer to these as symmetric functions.
, the space of functions such that , with .
For , , define the forward difference operator
, the indicator function for a set , and .
if for all , .
For , we write if there exists a constant , independent of and , such that .
We shall frequently derive norm bounds of the type . Here, the implied constant is independent of and but may depend on the spaces and .
Generic constants are denoted by etc.
We shall often write, for example, rather than
We shall make frequent use of the following
if and only if .
Fix . Then, if then, the product . Moreover, we have
Fix . If then, their convolution and we have:
The implied constants in (LABEL:eqn:algebraA) and (LABEL:eqn:algebraB) are independent of and , but depend on .
Finally, we find it convenient to record certain - and -dependent functions, as . These arise throughout the our analysis:
2 Properties of the continuum solitary wave of FNLS, , on
A central role is played by the characterization of the continuum FNLS standing wave, and the linearized operator about a solution of (LABEL:eqn:introtiNLS) :
Let . Let denote a solution of (LABEL:eqn:introtiNLS) governing FNLS solitary standing waves with frequency . If is real-valued, , , and (with ) has exactly one negative eigenvalue (counting multiplicity), i.e. has Morse index equal to one, then we say that