INTRODUCTION

Analysis of the feature of controllable localization of electromagnetic waves plays an important role in designing nonlinear waveguide devices. Electromagnetic field excitation that propagate along waveguides and rapidly decay with increasing distance from the interface between the media are usually referred to as a nonlinear surface wave. Their theoretical investigation [1, 2] and experimental observation [3] have a quite long history. Nevertheless, analysis of nonlinear surface waves propagating over interfaces between dielectric media with different characteristics remains topical due to their wide application in optical systems of data storage [4].

Most theoretical publications are devoted to obtaining field distributions and dispersion relations for waves propagating over interfaces between a linear and nonlinear medium [5, 6] and three-layer structures [7]. Modifications of the models of interfaces between nonlinear crystals were considered in [8, 9]. Peculiarities of localization of nonlinear waves at the interface between nonlinear media with spatial dispersion were analyzed in [10]. The interaction near a defect of bound soliton states corresponding to different states of a two-level system were considered in [11]. The internal structure of a thin interlayer between nonlinear crystals was considered in [12].

The fields of nonlinear waves are calculated as a rule based on the nonlinear Schrödinger equation (NSE) which contains a cubic (relative to the sought field) term in the case of a medium with the Kerr effect [13]. It should be noted that the NSE will widely be used for analytic description of localization effects in fields of different physical origins including magnetic [14] and elastic [15].

Theoretical analysis of the interaction of nonlinear excitations with a planar or point defect is performed using mathematical models based on the NSE in which the potential describes the defect. In many cases, the problem can be reduced to a 1D problem; in this case, the defect is simulated by a short-range potential in form

$$U(x) = {{U}_{0}}\delta (x),$$
((1))

where δ(x) is the Dirac delta function and U0 is the intensity of interaction of the excitation with a defect located at the origin (sometimes, this quantity is referred to as defect “power”). The excitation is repelled by the defect for U0 > 0 and attracted to it for U0 < 0.

Clearly, a short-range potential of form (1) cannot describe comprehensively the effect of the properties of a thin defect layer on the effects of localization of excitations. For investigating the possibility of control over the propagation of waves in layered structures through interfaces between the media considering their intrinsic nonlinear properties, it was proposed that a potential with quadratic nonlinearity relative to the sought field be used [16–18]. In the case of weak coupling between plane-parallel waveguides, the field amplitude in them is much larger than the average field amplitude in the entire crystal; for this reason, it was proposed that nonlinear terms be considered only within waveguides themselves. Analysis of peculiarities of propagation of localized excitations in systems with such properties is topical in view of wide application of layered structures containing plane-parallel waveguides in nonlinear optics.

In this study, we propose a generalization of the model of a thin defect layer described in [16–18] and characterized by intrinsic Kerr nonlinearity to the case of contact between a linear medium and a crystal with Kerr nonlinearity. We will analyze the localization of excitations near a thin nonlinear defect layer possessing the waveguide properties and will show that there exist a few types of stationary states with localized components. We will consider the contact of a linear medium with nonlinear Kerr-type media with different signs of nonlinearity. The main goal of this study is the determination of energy of localized states of all types, which emerge in the given system, as well as conditions of their existence.

It should be noted that in [7], stationary nonlinear surface waves near the interface between media with different characteristics including the contact between a linear and nonlinear medium, were considered disregarding the peculiarities of the structure at the interface and its effect on localization of waves. On the other hand, the peculiarities of localization of excitations near the interface with nonlinear properties between nonlinear media, but with identical characteristics, were considered in [16, 17]. A distinguishing feature of this study is that we analyze localization of excitations near the interface with nonlinear properties between media with different characteristics (in particular, between a linear medium and a medium with cubic nonlinearity).

1. MODEL EQUATIONS

We assume that a thin interlayer separating crystals with harmonic (linear medium) and anharmonic (nonlinear medium) interactions of elementary excitations is located in the yz plane perpendicularly to the x axis. The interlayer thickness is much smaller than the characteristic distance of localization of excitations.

The linear (harmonic) crystal occupies half-space x < 0, while the nonlinear (anharmonic) crystal occupies half-space x > 0. Then, the nonlinearity parameter in the NSE has form

$$\gamma (x) = \left\{ \begin{gathered} 0,\quad x < 0, \hfill \\ \gamma ,\quad x > 0. \hfill \\ \end{gathered} \right.$$

The interface (being a planar defect) produces perturbation of the characteristics of the media, which are concentrated at distances much shorter than the localization width of the excitations in question.

Following [17, 18], we will describe a planar defect with nonlinear properties using potential in form

$$U(x) = {{U}_{0}}\delta (x){\text{|}}\psi {{{\text{|}}}^{2}}.$$
((2))

We consider the interaction of nonlinear excitations localized near the thin layer with nonlinear properties based on the 1D NSE (we assume that \(\hbar \) = 1):

$$i\psi _{t}^{'} = - \frac{1}{{2m}}\psi _{{xx}}^{{''}} + \Omega (x)\psi - \gamma (x){\text{|}}\psi {{{\text{|}}}^{2}}\psi + U(x)\psi ,$$
((3))

where m is the effective mass of the excitation and

$$\Omega (x) = \left\{ \begin{gathered} {{\Omega }_{1}},\quad x < 0, \hfill \\ {{\Omega }_{2}},\quad x > 0, \hfill \\ \end{gathered} \right.$$

where Ω1, 2 are constant quantities.

The determination of stationary states with energy E can be reduced to the substitution of wavefunction ψ(x, t) = ψ(x)exp(–iEt) into NSE (3). As a result, we obtain from Eq. (3) time-independent NES

$$E\psi = - \frac{1}{{2m}}\psi _{{xx}}^{{''}} + \Omega (x)\psi - \gamma (x){\text{|}}\psi {{{\text{|}}}^{2}}\psi + U(x)\psi .$$
((4))

Obtaining the solution to NSE (4) with potential (2) is equivalent to the solution of NSE with zero potential,

$$\psi _{{xx}}^{{''}} + 2m(E - \Omega (x) + \gamma (x){\text{|}}\psi {\kern 1pt} {{{\text{|}}}^{2}})\psi = 0,$$
((5))

with two boundary conditions of conjugation at point x = 0 lying in the defect plane. The first boundary condition corresponds to the requirement of continuity of wavefunction

$$\psi ( - 0) = \psi ( + 0) = \psi (0).$$
((6))

The second boundary condition was obtained in [17, 18] by integrating both sides of Eq. (5) with potential (2) with respect to x on small interval [–ε; ε] and proceeding to limit ε → 0. This resulted in second boundary condition

$$\psi {\kern 1pt} '( + 0) - \psi {\kern 1pt} '( - 0) = 2m{{U}_{0}}{\text{|}}\psi (0){\kern 1pt} {{{\text{|}}}^{2}}.$$
((7))

In a linear medium without defect (γ(x) ≡ 0 and U(x) ≡ 0 everywhere), free waves with quadratic dispersion relation E = Ω + k2/2m propagate, where k is the wavenumber. In the presence of a simple defect described by short-range potential (1), a symmetric state localized on both sides of the defect exists in the linear medium. Such a state is described by wavefunction ψ(x) = Ae–q|x|, where q = –mU0, and exists only for attracting defect with U0 < 0. The energy of such a local level is E = Ω – m\(U_{0}^{2}\)/2.

2. LOCALIZATION OF EXCITATIONS AT THE INTERFACE BETWEEN A LINEAR CRYSTAL AND A SELF-FOCUSING MEDIUM

In the case of contact between a linear crystal and a self-focusing medium, which corresponds to positive nonlinearity (γ > 0) in Eq. (5), two types of states with a field profile asymmetric relative to the defect plane can appear depending on energy. The first type is described by the solutions to the NSE, which decay monotonically on both sides of the defect. The second type, known as quasi-local, consists of a state localized in the nonlinear medium and a standing wave in the linear medium.

2.1. Local States

If the excitation energy lies in interval E < min{Ω1, Ω2}, NSE (5) has a solution in form

$$\psi (x) = \left\{ \begin{gathered} {{\psi }_{{0c}}}\exp ({{q}_{1}}x),\quad x < 0, \hfill \\ {{A}_{c}}{\text{/}}\cosh {{q}_{2}}(x - {{x}_{0}}),\quad x > 0. \hfill \\ \end{gathered} \right.$$
((8))

Parameters of this solution are determined after its substitution into Eq. (5) and continuity condition (6):

$$q_{{1,2}}^{2} = 2m({{\Omega }_{{1,2}}} - E),$$
((9))
$$A_{c}^{2} = q_{2}^{2}{\text{/}}(m\gamma ),$$
((10))
$${{\psi }_{{0c}}} = {{q}_{2}}{\text{/}}\{ {{(m\gamma )}^{{1/2}}}\cosh {{q}_{2}}{{x}_{0}}\} .$$
((11))

Parameter x0 characterizes the position of the soliton “center” in the nonlinear crystal on the right of the defect. It is connected with the excitation localization energy that is determined from the dispersion relation obtained after the substitution of solution (8) into boundary condition (7):

$${{q}_{2}}\tanh {{q}_{2}}{{x}_{0}} - {{q}_{1}} = 2{{U}_{0}}q_{2}^{2}(1 - {{\tanh }^{2}}{{q}_{2}}{{x}_{0}}){\text{/}}\gamma .$$
((12))

This relation can be used for determining one of the wavenumbers (either q1 or q2 since they are coupled), which makes it possible to determine energy as a function of parameters: E = E(m, U0, γ, x0). The position x0 of the soliton center is a free parameter. Analysis of dispersion relation (12) will be performed in different particular cases that permit the obtaining of its solution in explicit form.

In the case of identical values of Ω1 = Ω2 = Ω, relation (9) implies that q1 = q2 = q. Then, the dispersion relation assumes form

$$\gamma = - 2{{U}_{0}}q(1 + \tanh q{{x}_{0}})$$
((13))

(we do not consider below the trivial case when tanhqx0 = 1). Relation (13) can be used for obtaining energy in explicit form for the localized state, the position of its center coinciding with the plane of the defect (i.e., x0 = 0). In this case, damping decrement can be determined from expression (13): q = –γ/2U0. Since q > 0 and γ > 0 in the medium with self-focusing, we obtain U0 < 0. In other words, the localized state of the type considered here exists only for the attracting thin defect layer with nonlinear properties, which is described by potential (2) precisely as in the case of a simple defect described by potential (1). The energy of the local state in this case is determined by expression

$$E = \Omega - {{\gamma }^{2}}{\text{/8}}mU_{0}^{2}.$$
((14))

The amplitude of vibrations of the defect layer can be determined from relation (11):

$${{\psi }_{{0c}}} = - {{(\gamma {\text{/}}m)}^{{1/2}}}{\text{/}}2{{U}_{0}}.$$
((15))

In the long-wave approximation for qx0 ≪ 1, we can obtain the damping decrement from relation (13):

$$q = \{ - 1 \pm (1 - 2\gamma {{x}_{0}}{\text{/}}U_{0}^{{1/2}})\} {\text{/}}{{x}_{0}}.$$
((16))

The sign in this expression is chosen depending on the sign of x0 under the requirement q > 0.

It should be noted that the long-wave approximation condition qx0 ≪ 1 indicates the closeness of excitation energy to the spectral edge, where condition |Ω – E | ≪ (1/2)m\(x_{0}^{2}\) holds.

Considering relations (16) and (9), we can obtain the energy of the localized state in explicit form:

$$E = \Omega - {{\{ - 1 \pm {{(1 - \gamma {{x}_{0}}{\text{/}}{{U}_{0}})}^{{1/2}}}\} }^{2}}{\text{/}}8mx_{0}^{2}.$$
((17))

Such a local states exists when x0 < U0/2γ.

If we now assume that Ω1 ≠ Ω2, dispersion relation (12) for x0 = 0 assumes form

$${{q}_{1}} = - 2{{U}_{0}}q_{2}^{2}{\text{/}}\gamma .$$
((18))

This relation shows that since q1 > 0 and γ > 0 in the medium with self-focusing, such states are localized near the attracting defect with U0 < 0.

Relation (18) combined with (9) gives the energy of localized state in explicit form:

$$E = {{\Omega }_{2}} - {{\Omega }_{{c0}}}\{ 1 \pm {{[1 + 2({{\Omega }_{1}} - {{\Omega }_{2}}){\text{/}}{{\Omega }_{{c0}}}]}^{{1/2}}}\} ,$$
((19))

where Ωc0 = γ2/16m\(U_{0}^{2}\). The existence of such a local state with energy (19) requires that condition Ω2 < Ω1 + γ2/32m\(U_{0}^{2}\) must hold.

In the long-wave approximation for q2x0 ≪ 1, we can obtain from relation (12) the energy of the localized state in explicit form

$$E = {{\Omega }_{2}} - {{\Omega }_{{cx}}}\{ 1 \pm {{[1 + 2({{\Omega }_{1}} - {{\Omega }_{2}}){\text{/}}{{\Omega }_{{cx}}}]}^{{1/2}}}\} ,$$
((20))

where Ωcx = γ2/4m(γx0 – 2U0)2.

2.2. Quasi-Local States

If the excitation energy lies in interval Ω1 < E < Ω2, NSE (5) has a solution in form

$$\psi (x) = \left\{ \begin{gathered} {{B}_{c}}\cos (kx + \varphi ),\quad x < 0, \hfill \\ {{A}_{c}}{\text{/}}\cosh {{q}_{2}}(x - {{x}_{0}}),\quad x > 0. \hfill \\ \end{gathered} \right.$$
((21))

A solution of this type exists if inequality Ω1 < Ω2 holds. The parameters of solution (21) are determined by substituting it into Eq. (5) and continuity condition (6). The value of q2 is determined by expression (9); amplitude Ac is defined by expression (10), and characteristics of the wave in the linear crystal have form

$${{k}^{2}} = 2m(E - {{\Omega }_{1}}),$$
((22))
$${{B}_{c}} = {{\psi }_{{0c}}}{\text{/}}\cos \varphi .$$
((23))

Substituting solution (21) into boundary condition (7), we obtain dispersion relation

$$k\tan \varphi + {{q}_{2}}\tanh {{q}_{2}}{{x}_{0}} = 2{{U}_{0}}q_{2}^{2}(1 - {{\tanh }^{2}}{{q}_{2}}{{x}_{0}}){\text{/}}\gamma .$$
((24))

From this relation, we can determine one of the wavenumbers (either k or q2 since they are coupled), which makes it possible to determine energy as a function of parameters E = E(m, U0, γ, φ, x0). The position of the soliton center x0 and phase φ are now free parameters.

Solution (24) describes the state in which a linear wave decays after passing through the thin defect layer towards the bulk of the anharmonic crystal (i.e., wave localization occurs). Since the energy of such a stationary state is in the spectrum of linear waves, and the excitation is localized on one side of the defect plane, the states of this type can be referred to as quasi-local states.

If we consider solution (21) for which x0 = 0, the energy of such a quasi-local state is determined from solution (24):

$$E = {{\Omega }_{2}} - \Omega _{{c0}}^{\varphi }\{ - 1 \pm {{[1 - 2({{\Omega }_{2}} - {{\Omega }_{1}}){\text{/}}\Omega _{{c0}}^{\varphi }]}^{{1/2}}}\} ,$$
((25))

where \(\Omega _{{c0}}^{\varphi }\) = Ωc0tan2φ. The state under consideration as well as the states that will be obtained below exist not for all values of phase φ. For the existence of such a local state with energy (25), the following condition must hold for the phase: tan2φ > 32m\(U_{0}^{2}\)(Ω2 – Ω1)/γ2.

In the long-wave approximation for q2x0 ≪ 1, solution (24) gives the energy of quasi-local state in explicit form

$$E = {{\Omega }_{2}} - \Omega _{{cx}}^{\varphi }\{ - 1 \pm {{[1 - 2({{\Omega }_{2}} - {{\Omega }_{1}}){\text{/}}\Omega _{{cx}}^{\varphi }]}^{{1/2}}}\} ,$$
((26))

where \(\Omega _{{cx}}^{\varphi }\) = Ωcxtan2φ. Such a local state with energy (26) exists when the following condition holds for the phase: tan2φ > 8m(Ω2 – Ω1)(γx0 – 2U0)2/γ2.

3. LOCALIZATION OF EXCITATIONS AT THE INTERFACE BETWEEN A LINEAR CRYSTAL AND A DEFOCUSING MEDIUM

In the case when a linear crystal is in contact with a defocusing medium (i.e., crystal with negative nonlinearity, γ < 0), local and quasi-local states can appear in Eq. (5) depending on energy. In contrast to the contact with a medium with self-focusing, for the contact with a defocusing medium, there appear two types of local states and two types of quasi-local states differing in the localization profile of the field amplitude and in the energy range in which they exist.

In analysis of stationary states in the defocusing medium, we set for convenience g = –γ > 0.

3.1. Local States

When the excitation energy is in the range E < min{Ω1, Ω2}, NSE (5) has a solution of form

$$\psi (x) = \left\{ \begin{gathered} {{\psi }_{{0s}}}\exp ({{q}_{1}}x),\;x < 0, \hfill \\ {{A}_{s}}{\text{/}}\sinh {{q}_{2}}(x - {{x}_{0}}),\quad x > 0. \hfill \\ \end{gathered} \right.$$
((27))

For this solution to be bounded, condition x0 < 0 must hold. Parameters of solution (27) are determined after its substitution into Eq. (5) and continuity condition (6). Quantities q1, 2 are defined by expressions (9), and amplitudes have form

$$A_{s}^{2} = q_{2}^{2}{\text{/}}(mg),$$
((28))
$${{\psi }_{{0s}}} = - {{q}_{2}}{\text{/}}\{ {{(mg)}^{{1/2}}}\sinh {{q}_{2}}{{x}_{0}}\} .$$
((29))

Substituting solution (27) into boundary condition (7), we obtain dispersion relation

$${{q}_{2}}\coth {{q}_{2}}{{x}_{0}} - {{q}_{1}} = 2{{U}_{0}}q_{2}^{2}({{\coth }^{2}}{{q}_{2}}{{x}_{0}} - 1){\text{/}}g.$$
((30))

In the long-wave approximation for q2x0 ≪ 1, we can obtain from relation (30) the damping decrement of the wave in the linear medium:

$${{q}_{1}}(1 - 2{{U}_{0}}{\text{/}}g{{x}_{0}}){\text{/}}{{x}_{0}}.$$
((31))

This expression implies that since q1 > 0 and g > 0, such states are localized both near the attracting defect and near the repelling defect when one of the following conditions holds: (i) x0 > 0 and U0 < gx0/2 or (ii) x0 < 0 and U0 > gx0/2. The energy of such a state is determined after the substitution of expression (31) into (9):

$$E = {{\Omega }_{1}} - {{(1 - 2{{U}_{0}}{\text{/}}g{{x}_{0}})}^{2}}{\text{/}}2mx_{0}^{2}.$$
((32))

Using relation (29), we can determine the amplitude of vibrations of the defect layer in the long-wave approximation for q2x0 ≪ 1: ψ0s = –1/x0(mg)1/2.

If the excitation energy lies in the interval Ω2 < E < Ω1, which is possible when the bottom of the continuous spectrum band in the linear defocusing medium is lower than that in the linear medium (Ω2 < Ω1), NSE (5) has a different solution:

$$\psi (x) = \left\{ \begin{gathered} {{\psi }_{{0t}}}\exp ({{q}_{1}}x),\quad x < 0, \hfill \\ {{A}_{t}}\tanh {{q}_{t}}(x - {{x}_{0}}),\quad x > 0. \hfill \\ \end{gathered} \right.$$
((33))

The parameters of this solution are determined after its substitution into Eq. (5) and continuity condition (6). Quantity q1 is defined by expression (9), while remaining characteristics have form

$$q_{t}^{2} = m(E - {{\Omega }_{1}}),$$
((34))
$$A_{t}^{2} = q_{t}^{2}{\text{/}}(mg),$$
((35))
$${{\psi }_{{0t}}} = - {{q}_{t}}\tanh {{q}_{t}}{{x}_{0}}{\text{/}}{{(mg)}^{{1/2}}}.$$
((36))

After the substitution of solution (33) into boundary condition (7), we obtain dispersion relation

$$2{{q}_{t}}{\text{/}}\sinh 2{{q}_{t}}{{x}_{0}} + {{q}_{1}} = - 2{{U}_{0}}q_{t}^{2}{{\tanh }^{2}}{{q}_{t}}{{x}_{0}}{\text{/}}g.$$
((37))

In the long-wave approximation for qt  x0 ≪ 1, relation (37) gives the energy of such a state:

$$E = {{\Omega }_{2}} + {{\Omega }_{{1t}}}\{ - 1 \pm {{(1 + {{\Omega }_{{2t}}}{\text{/}}{{\Omega }_{{1t}}})}^{{1/2}}}\} ,$$
((38))

where Ω1t = g/4mx0U0 and Ω2t = 2(Ω1 – Ω2) – 1/2m\(x_{0}^{2}\). The plus sign in relation (38) is chosen for Ω1t > 0 and the minus sign, for Ω1t < 0.

The amplitude of vibrations of the defect layer in the long-wave approximation for qt  x0 ≪ 1 is determined from relation (36), ψ0t = –\(q_{t}^{2}\)x0/(mg)1/2.

3.2. Quasi-Local States

If a linear crystal contacts a crystal with negative nonlinearity (γ = –g < 0) in the energy range Ω1 < E < Ω2, NSE (5) has a solution of form

$$\psi (x) = \left\{ \begin{gathered} {{B}_{s}}\cos (kx + \varphi ),\quad x < 0, \hfill \\ {{A}_{s}}{\text{/}}\sinh {{q}_{2}}(x - {{x}_{0}}),\quad x > 0. \hfill \\ \end{gathered} \right.$$
((39))

Parameters of this solution are determined by substituting it into Eq. (5) and continuity condition (6). Quantity q2 is defined by expression (9), amplitude As is defined by (28), and wavenumber k, by (22), while the wave amplitude in the linear crystal has form

$${{B}_{s}} = {{\psi }_{{0s}}}{\text{/}}\cos \varphi .$$
((40))

Substituting solution (39) into boundary condition (7), we obtain dispersion relation

$$k\tan \varphi + {{q}_{2}}\coth {{q}_{2}}{{x}_{0}} = 2{{U}_{0}}q_{2}^{2}({{\coth }^{2}}{{q}_{2}}{{x}_{0}} - 1){\text{/}}g.$$
((41))

For such a quasi-local state, position x0 of the soliton center (which must be negative for the boundedness of solution (39)) and phase φ are free parameters.

In the long-wave approximation for q2x0 ≪ 1, relation (41) can be used for obtaining energy of quasi-local state in explicit form

$$E = {{\Omega }_{1}} + {{(1 - 2{{U}_{0}}{\text{/}}g{{x}_{0}})}^{2}}{{\cot }^{2}}\varphi {\text{/}}2mx_{0}^{2}.$$
((42))

In energy range E > max{Ω1, Ω2} NSE (5) for the contact of a linear crystal with a crystal with negative nonlinearity, NSE (5) has a different solution

$$\psi (x) = \left\{ \begin{gathered} {{B}_{t}}\cos (kx + \varphi ),\quad x < 0, \hfill \\ {{A}_{t}}\tanh {{q}_{t}}(x - {{x}_{0}}),\quad x > 0. \hfill \\ \end{gathered} \right.$$
((43))

Parameters of this solution are determined by substituting it into Eq. (5) and continuity condition (6). Quantity qt is defined by expressions (34), amplitude At is defined by (35), and wavenumber k, by (22), while the wave amplitude in the linear crystal has form

$${{B}_{t}} = {{\psi }_{{0t}}}{\text{/}}\cos \varphi .$$
((44))

Substituting solution (43) into boundary condition (7), we obtain dispersion relation

$$k\tan \varphi - 2{{q}_{t}}{\text{/sinh}}2{{q}_{t}}{{x}_{0}} = 2{{U}_{0}}q_{t}^{2}{{\tanh }^{2}}{{q}_{1}}{{x}_{0}}{\text{/}}g.$$
((45))

In the long-wave approximation for qtx0 ≪ 1, relation (45) can be used for obtaining energy of quasi-local state

$$E = {{\Omega }_{2}} + \Omega _{{1t}}^{\varphi }\{ - 1 \pm {{(1 + \Omega _{{2t}}^{\varphi }{\text{/}}\Omega _{{1t}}^{\varphi })}^{{1/2}}}\} ,$$
((46))

where \(\Omega _{{1t}}^{\varphi }\) = Ω1t tan2φ, \(\Omega _{{2t}}^{\varphi }\) = 2(Ω1 – Ω2) – cot2φ/m\(x_{0}^{2}\). The plus sign in expression (46) is chosen for \(\Omega _{{1t}}^{\varphi }\) > 0 and the minus sign, for \(\Omega _{{1t}}^{\varphi }\) < 0. For the existence of such a state, condition U0 < gtan4φ/{8x0[m(Ω2 – Ω1)tan2φ – 1/2\(x_{0}^{2}\)]} must hold. Clearly, this condition can be satisfied for an attracting as well as for a repelling defect.

CONCLUSIONS

It is shown that near a thin interlayer with nonlinear properties, which separates a linear and nonlinear medium, stationary states of several types can exist. Such states are generated by different types of soliton solutions to the NSE.

In the mathematical formulation of the model for describing a planar defect with nonlinear properties, we used a potential with a term quadratic in the sought field, the form of which was proposed in [17, 18]. The solution of the NSE with such a potential is reduced to obtaining the solution to the NSE without the potential with nonlinear boundary conditions. We have obtained the solutions to the formulated contact boundary value problem with such conditions. The expressions for energy have been obtained in explicit analytic form. It is shown that when nonlinear properties of the defect are considered, the nonlinear profile of localized excitations and the domain of their existence are modified.

It has been established that two types of stationary states can appear at the interface with nonlinear properties between linear and nonlinear crystals. One of these states can be described by a monotonically decaying field profile (i.e., the profile localized on both sides of the defect) and is therefore referred to as a localized state. The other type of stationary states is described by a field profile decaying monotonically only on one side of the defect plane (where the nonlinear crystal is located), while the field on the other side of the defect plane, where the linear crystal is located, has the form of a standing linear wave. The second type can be called the quasi-local state.

For each type of stationary states, we have obtained several forms of field distribution, which are determined by the sign of nonlinearity of the medium and by the range of possible excitation energy.

For the contact of a linear crystal with a self-focusing Kerr medium (i.e., crystal with positive nonlinearity, γ > 0), when the excitation energy is in the range E < min{Ω1, Ω2}, there exists a local state with an asymmetric profile, which is generated by the soliton solution to the NSE with hyperbolic cosine on the semiaxis in the region of the nonlinear crystal and an exponentially decaying solution on the semiaxis in the region of the linear crystal. In this case, there also appears a quasi-local state with energy in range Ω1 < E < Ω2 (i.e., when the bottom of the band of the continuous spectrum in the nonlinear crystal is higher than in the linear crystal). In the region of the nonlinear crystal, the field profile is described by the soliton solution to the NSE with the hyperbolic cosine, while in the region of the linear crystal, it is described by a standing linear wave.

In the case of the contact of a linear crystal with a defocusing Kerr medium (i.e., crystal with negative nonlinearity, γ < 0), there exist two types of local and two types of quasi-local states. Localized states with an asymmetric profile of one type are realized in excitation energy range E < min{Ω1, Ω2}. These states are generated by the soliton solution to the NSE with hyperbolic sine on the semiaxis in the region of the nonlinear crystal and an exponentially decaying solution on the semiaxis in the region of the linear crystal.

Localized states with the asymmetric profile of the other type are realized in excitation energy range Ω2 < E < Ω1 (i.e., when the bottom of the band of the continuous spectrum in the linear crystal is higher than in the nonlinear crystal). Such states are generated of the soliton solution to the NSE with hyperbolic tangent on the semiaxis in the region of the nonlinear crystal and an exponentially decaying solution on the semiaxis in the region of the linear crystal.

In this case also, two types of quasi-local states appear. One type exists in excitation energy range Ω1 < E < Ω2 (i.e., when the bottom of the band of the continuous spectrum in the nonlinear crystal is higher than in the linear crystal). This type of states in the region of the nonlinear crystal is described by the soliton solution to the NSE with hyperbolic sine, while in the region of the linear crystal, it is described by a standing linear wave. The other type of quasi-local states is realized when E > max{Ω1, Ω2}. The field profile in it in the region of the nonlinear crystal is described by the soliton solution to the NSE with hyperbolic tangent, while in the region of the linear crystal, it is described by a standing linear wave.

Thus, by controlling the excitation energy, we can obtain different field localization profiles on different sides of the planar defect.

Analysis of the model of a thin defect layer with nonlinear properties, which is described by potential (2), imparts new features of the spectral structure of localized states in contrast to the model of a simple defect described by potential (1). The main difference lies in the dispersion equations and, as a consequence, in energy levels and domains of existence of states.

The results of this study can be treated as an extension of analysis of peculiarities in localization of nonlinear excitations in media with nonlinear defects to the case of the interface between linear and nonlinear media, which was performed in [16–18].