Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schrödinger equation

Applied Mathematics and Computation 301 (2017) 155–176

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Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schrödinger equation J. Escorcia a, E. Suazo b,∗ a

Department of Mathematics, University of Puerto Rico, P.O. Box 4010, Arecibo, PR, 00614-4010, United States School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 W. University Drive, Edinburg, TX 78539-2999, United States

b

a r t i c l e

i n f o

MSC: 81Q05 35C05 42A38 Keywords: Soliton-like equations Nonlinear Schrödinger like equations Fiber optics Gross–Pitaevskii equation Similarity transformations and Riccati–Ermakov systems

a b s t r a c t In this paper, by means of similarity transformations we study exact analytical solutions for a generalized nonlinear Schro¨ dinger equation with variable coefficients. This equation appears in literature describing the evolution of coherent light in a nonlinear Kerr medium, Bose–Einstein condensates phenomena and high intensity pulse propagation in optical fibers. By restricting the coefficients to satisfy Ermakov–Riccati systems with multiparameter solutions, we present conditions for existence of explicit solutions with singularities and a family of oscillating periodic soliton-type solutions. Also, we show the existence of bright-, dark- and Peregrine-type soliton solutions, and by means of a computer algebra system we exemplify the nontrivial dynamics of the solitary wave center of these solutions produced by our multiparameter approach. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The study of the nonlinear Schrödinger equation (NLS) with real potential V

1 2s iψt = − ψ + V (x,t )ψ + λ(x, t )|ψ | ψ , 2

ψ ( 0, x ) = ϕ ( x ),

x ∈ Rn ,

=

n 

∂x j x j

(1.1)

j=1

has been studied extensively not only for its role in physics, such as in Bose–Einstein condensates and nonlinear optics, but also for its mathematical complexity (for a review of the several results available see [1,3,4,12,13,18,19,27,32,33,47,51,52,58,63,68]). For the case λ = −1, V ≡ 0, and ns < 2 (subcritical case) Weinstein [65] proved that if ϕ ∈ H1 , then ψ exists globally in H1 . It is also known (see [13,21,58] for a complete review) that NLS for critical (ns = 2) and supercritical (ns > 2) cases present solutions that become singular in a finite time in Lp for some finite p. In [22] singular solutions of the subcritical NLS were presented in Lp . In [10] it was proved that if ϕ ∈  = { f ∈ H 1 (Rn ) : x → |x| f (x ) ∈ L2 (Rn )}, V(x,t) is real, locally bounded in time and subquadratic in space, and λ ∈ R, then the solution of the Cauchy initial value problem exists globally in  , provided that s < 2/n or s ≥ 2/n and λ ≥ 0. Also, in [10] it was shown that if V (x,t ) = b(t )x2j , b(t ) ∈ C (R; R ) in (1.1), then there exist blow-up solutions if λ < 0 and s = 2/n. The proof uses the generalized Melher’s formula introduced in [15]. In [44] and [59] a generalized pseudoconformal transformation (lens transform in optics [60]) was presented. In this paper, as a first ∗

Corresponding author. E-mail addresses: [email protected] (J. Escorcia), [email protected], [email protected] (E. Suazo).

http://dx.doi.org/10.1016/j.amc.2016.12.018 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

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main result we will use a generalized lens transformation to construct solutions with finite-time blow-up in Lp norm for 1 ≤ p ≤ ∞ of the general variable coefficient nonlinear Schrödinger:

iψt = −a(t )ψxx + (b(t )x2 − f (t )x + G(t ))ψ − ic (t )xψx − id (t )ψ + ig(t )ψx + h(t )|ψ |

2s

ψ.

(1.2)

In modern nonlinear sciences some of the most important models are the variable coefficient nonlinear Schrödinger-type ones. Applications include long distance optical communications, optical fibers and plasma physics, see [4,5,7–9,11,14,17,24– 26,29,39,43,45,46,49,54,55,62,64,67] and references therein. If we make a(t ) = /4π n0 ,  being the wavelength of the optical source generating the beam, and choose c (t ) = g(t ) = 0, then (1.2) models a beam propagation inside of a planar graded-index nonlinear waveguide amplifier with quadratic refractive index represented by b(t )x2 − f (t )x + G(t ), and h(t) represents a Kerr-type nonlinearity of the waveguide amplifier, while d(t) represents the gain coefficient. If b(t) > 0 [45] (resp. b(t) < 0, see [49]) in the low-intensity limit, the graded-index waveguide acts as a linear defocusing (focusing) lens. Depending on the selections of the coefficients in Eq. (1.2), the applications vary in very specific problems (see [62] and references therein): • Bose–Einstein condensates [27]: b(·) = 0, a, h constants and other coefficients are zero. • Dispersion-managed optical fibers and soliton lasers [29,54,55]: a(·), h(·), d(·) = 0 are respectively dispersion, nonlinearity and amplification, and the other coefficients are zero. a(·) and h(·) can be periodic as well, see [2,40]. • Pulse dynamics in the dispersion-managed fibers [39]: h(·) = 0, a is a constant and other coefficients are zero. In this paper to obtain the main results we use a fundamental approach consisting of the use of similarity transformations and the solutions of Riccati Ermakov systems with several parameters inspired by the work in [38]. Similarity transformations have been a very popular strategy in nonlinear optics since the lens transform presented by Talanov [60]; extensions and applications of this approach have been presented by Rypdal and Rasmussen [50–53], see also more recent contributions by [44,59]. Also for related work on transformations using Lie groups and Lie algebras we refer the reader to [6,23,42,48,66]. Applications include nonlinear optics, Bose–Einstein condensates, integrability of NLS and quantum mechanics, see for example [5,6,10,34] and references therein. Marhic in 1978 introduced (probably for the first time) a oneparameter {α (0)} family of solutions for the linear Schrödinger equation of the one-dimensional harmonic oscillator; the use of an explicit formulation (classical Melher’s formula [20,41]) for the propagator was fundamental. The solutions presented by Marhic constituted a generalization of the original Schrödinger wave packet with oscillating width. Also, in [15] a generalized Melher’s formula for a general linear Schrödinger equation of the one-dimensional generalized harmonic oscillator of the form (1.2) with h(t ) = 0 was presented. For the latter case in [31,35], multiparameter solutions in the spirit of Marhic in [38] have been presented. The parameters for the Riccati system arose originally in the process of proving convergence to the initial data for the Cauchy initial value problem (1.2) with h(t ) = 0 and in the process of finding a general solution of a Riccati system [56]. Ermakov systems with solutions containing parameters [31] have been used successfully to construct solutions for the generalized harmonic oscillator with a hidden symmetry [35], and they have also been used to present Galilei transformation, pseudoconformal transformation and others in a unified manner, see [35]. More recently they have been used in [36] to show spiral and breathing solutions and solutions with bending for the paraxial wave equation. In this paper, as a second main result we introduce a family of Schrödinger equations presenting periodic soliton solutions by using multiparameter solutions for Riccati–Ermakov systems. Further, as a third main result we show that these parameters provide a control on the dynamics of solutions for equations of the form (1.2). These results should deserve numerical and experimental studies. This paper is organized as follows: In Section 2, as an application of a generalized lens transformation and multiparameter solutions for Riccati systems we present conditions to obtain solutions with singularity in finite time in Lp norm, 1 ≤ p ≤ ∞ for (1.2). Also, we show that through this more general parameter approach we can obtain the same L∞ solutions with finite-time blow-up for standard NLS presented in [15] and finite-time blow-up for NLS with quadratic potential. In Section 3, we present a family of soliton solutions for (1.2) presenting bright- and dark-type solitons; this family includes the standard NLS models. This family has multiparameter solutions coming from solutions of a related Ermakov system, extending the results presented in [57], where a Riccati system was used. By the use of these parameters the dynamics of periodic solutions for (1.2) show bending properties, see Figs. 1 and 2. In Section 4, again, as an application of generalized lens transformations and an alternative approach to solve the Riccati system (A.1)–(A.6) we present how the parameters provide us with a control on the center axis of the solution of bright and dark soliton solutions for special coefficients in (1.2). Figs. 3 and 4 show the bending propagation of the solutions after introducing parameters, extending the results presented in [36,57] to (1.2). Also we show that it is possible to construct a transformation that reduces (1.2), with a(t ) = l0 = ±1 and G(t ) = 0, to standard NLS with convenient initial data (Lemma 4) in order to assure existence and uniqueness of classical solutions (Proposition 1). As an application we show how the dynamics of the Peregrine soliton solutions of the nonlinear Schrödinger equation consider change when the dissipation, d(t), and the nonlinear term, h(t) change, see Figs. 5–8. We have also prepared a Mathematica file as supplemental material where all the solutions for this section are verified. Finally, in Appendix A, we recall the main tools we have used for our results. These tools are a solution with multiparameters of the Riccati system (A.1)–(A.6) and a modification of the transformation introduced in [59]; we have introduced an extra parameter l0 = ±1 in order to use standard solutions for Peregrine-type soliton solutions. Also a 2D version of a generalized

J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

a

157

b

Fig. 1. (a) Solution for (3.41 ) with δ (0 ) = 0. (b) Solution for (3.41 ) with δ (0 ) = 1.

a

b

Fig. 2. (a) Solution for (3.45 ) with δ (0 ) = 0. (b) Solution for (3.45 ) with δ (0 ) = 1.

lens transformation is recalled. All the formulas from the appendix have been verified previously using computer algebra systems [28]. 2. Finite-time blow-up for nonautonomous nonlinear Schrödinger equations In this section as an application of the multiparameter solution for Riccati systems we present conditions needed in order to obtain solutions with singularities in finite time with L∞ norm for (1.2). We show that we can obtain the same L∞ solutions with finite-time blow-up for standard NLS presented in [15] and finite-time blow-up for the Gross–Pitaevskii equation. Also, as an application of a generalized lens transformation (see Section A.2) in this section we present conditions to obtain solutions with singularity in finite time with Lp norm for (1.2) for dimensions one and two. We present our first main result: Theorem 1 (Solutions with singularity in finite time with Lp norm, 1 ≤ p ≤ ∞). If the characteristic equation associated to (1.2), i.e.



μ

−

a

− 2c + 4d a

    d

d a

2 − μ + 4 ab − cd + d + μ = 0, 2

a

d

(2.1)

admits two standard solutions μ0 and μ1 subject to

μ0 (0 ) = 0, μ 0 (0 ) = 2a(0 ) = 0 a(t )β 2 (t )μ2s (t ),

μ1 (0 ) = 0, μ 1 (0 ) = 0,

(2.2)

and if we choose h(t ) = β and μ satisfies a solvable Riccati-type system (2.9)–(2.14) (with μ(0), β (0) = 0), then there exists an interval I of time such that if −α (0 ) ∈ γ0 (I ), then (1.2) presents a solution with finite-time blow-up in Lp norm at T ∗ = γ0−1 (−α (0 ) ) ∈ I. Further, solutions present the following explicit form:

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J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

(a) Bright soliton solution bended to (b) Centered bright soliton: v = 1, (c) Bright soliton solution bended to the left: v = 1, δ(0) = −30 and δ(0) = 16 and ε(0) = − 16 . the right: v = 1, δ(0) = 0 and ε(0) = ε(0) = 0. −10.

(d) Contour of the bright soliton solution bended to the left.

(e) Contour for the centered bright soliton.

(f) Contour of the bright soliton solution bended to the right.

Fig. 3. Control on the dynamics of the solution (4.3) for Eq. (4.1).

(a) Dark soliton solution bended to (b) Centered dark soliton: A = 2, (c) Dark soliton solution bended to the left: A = 2, δ(0) = −1 and δ(0) = 0, ε(0) = 0 the right: A = 2, δ(0) = 0 and ε(0) = 0. ε(0) = −2.

(d) Contour of the dark soliton solution bended to the left.

(e) Contour of the centered dark soliton.

(f) Contour of the dark soliton solution bended to the right.

Fig. 4. Control on the dynamics of the solution (4.6) for Eq. (4.4).

J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

(a) Solution with values v = −2 and κ(0) = 0.

159

(b) Contour of the solution.

Fig. 5. Dynamics of the solution (4.25) for Eq. (4.24) with d (t ) = sin t and h (t ) = −3e3−3 cos t .

(a) Solution with values v = −2 and κ(0) = 0.

(b) Contour of the solution.

Fig. 6. Dynamics of the solution (4.26) for Eq. (4.24) with d (t ) = (4t − sin t )/2 and h (t ) = −3e2t . 2

(i). If p = ∞, a solution for (1.2) is given explicitly by

ψy (x, t ) = 

1

μ(t )

eiSy (x,t ) ,

(2.3)

where y is a parameter and Sy (t, x ) = α (t )x2 + β (t )xy + γ (t )y2 + δ (t )x + ε (t )y + κ (t ) and α (t), β (t), γ (t), δ (t),  (t) and κ (t) satisfy the Riccati system (2.9)–(2.14). (ii). If 1 ≤ p < ∞, a solution for (1.2) is given explicitly by

ψ (x, t ) = 

1

μ(t )

2 ei(α (t )x +δ (t )x+κ (t )) χ (ξ , τ ),

ξ = β (t )x + ε (t ),

τ = γ (t ),

(2.4)

where α (t), β (t), γ (t), δ (t),  (t) and κ (t) are given by (A.10)–(A.22). Additionally, χ satisfies

iχτ + χξ ξ + |χ |2s χ = 0.

(2.5)

(iii). (2D case) The natural 2D version of (1.2), the nonlinear equation



iψt = −a(ψxx + ψyy ) + b x2 + y2

 ψ − ic(xψx + yψy ) − 2idψ

− (x f1 + y f2 )ψ + i(g1 ψx + g2 ψy ) − |ψ |

2s

ψ,

(2.6)

where a, b, c, d, f1, 2 and g1, 2 are real-valued functions of t, admits an explicit solution with finite-time blow-up of the form

ψ = μ−1 ei(α (x +y 2

2

) + ( δ1 x + δ 2 y ) + κ 1 + κ 2 )

χ ( ξ , η , τ ),

(2.7)

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J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

(a) Profile of Peregrine soliton at t = 0. (b) Profile of Peregrine soliton at x = 0.

(c) 3D view of Peregrine soliton. Fig. 7. Peregrine soliton solution for (4.29) with A = 0.5 and κ (0 ) = 0 for (4.27) with d (t ) = tanh(t ) − t and h (t ) = −8 cosh (t ). 2

(a) Profile of perturbated Peregrine (b) Profile of perturbated Peregrine solisoliton in the times t = 1 and t = 5. ton at x = 0.

(c) 3D view of perturbated Peregrine soliton. 2 Fig. 8. Dynamics of perturbated Peregrine soliton solution (4.30) with A = 0.5 and κ (0 ) = 0 for (4.27) with d (t ) = −(sin(2t ) − t ) and h (t ) = −2e−2 sin (t ) .

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161

where ξ = β (t )x + ε1 (t ), η = β (t )y + ε2 (t ), τ = γ (t ), h(t ) = a(t )β 2 (t )μ2s (t ), and α (t), β (t), γ (t), δ 1 (t), δ 2 (t), κ 1 (t), κ 2 (t), ε1 (t), ε2 (t) satisfy the given conditions in Lemma 4. Finally, χ is a solution of

iχτ + χξ ξ + χηη + |χ |2s χ = 0.

(2.8)

Proof. To prove (i) we follow [15] and look for a solution of the form (2.3). After substituting on (1.2) we obtain the following Riccati system:

dα + b(t ) + 2c (t )α + 4a(t )α 2 = 0, dt

(2.9)

dβ + (c (t ) + 4a(t )α (t ) )β = 0, dt

(2.10)

dγ + a(t )β 2 (t ) = 0, dt

(2.11)

dδ + (c (t ) + 4a(t )α (t ) )δ = f (t ) + 2α (t )g(t ), dt

(2.12)

dε = (g(t ) − 2a(t )δ (t ) )β (t ), dt

(2.13)

h(t ) dκ = g(t )δ (t ) − a(t )δ 2 (t ) − s . dt μ (t )

(2.14)

Using (A.10)–(A.22) in the appendix, (2.9)–(2.13) can be solved, but (2.14) absorbs the nonlinearity and must be solved separately. Since there exists an interval J of time with μ0 (t) = 0 for all t ∈ J, and μ0 (t) and μ1 (t) have been chosen to be linearly independent on an interval, let’s say J , we observe that for t ∈ J ∩ J ≡ I, we get

γ0 (t ) =

W [μ0 (t ), μ1 (t )] = 0, 2μ20 (t )

and therefore from the general expression for μ given by (A.10), see [16], Eq. (2.3) will have finite-time blow-up at T ∗ = γ0−1 (−α (0 ) ) ∈ I. Now we proceed to prove (ii). Using the generalized lens transform, see Lemma 2, we can transform the nonautonomous and inhomogeneous nonlinear Schrödinger equation (1.2) into the standard one:

iχτ + χξ ξ + |χ |2s χ = 0,

(2.15)

recalling [21,33,58] that the autonomous focusing NLS (2.15) in dimension n allows solutions of the form χ = r = |ξ | and R is the solution of

R

(r ) +

n−1

R (r ) − R(r ) + R2s+1 (r ) = 0, r

R ( 0 ) = 0,

R ( ∞ ) = 0.

e iτ R ( r ) ,

where

(2.16)

In particular, for n = 1, a solution of (2.15) is given by −1/s

R(ξ ) = (s + 1 )1/2s cosh

(sξ ),

(2.17)

and for all t ∈ R and p ∈ [1, ∞] we have ||R||p < ∞. Therefore, we obtain a solution with time of the form

ψ (x, t ) = 

1

μ(t )

2 ei(α (t )x +δ (t )x+κ (t )) χ (ξ , τ ) =



1

μ(t )

Lp

finite-time blow-up for (1.2) in

2 ei(α (t )x +δ (t )x+κ (t )) eiτ R(ξ ).

(2.18)

Again, since μ(t ) = 2μ(0 )μ0 (t )(α (0 ) + γ0 (t ) ) we can predict a blow-up at T ∗ = γ0−1 (−α (0 )). To prove (iii), we consider the unique positive radial solution to

Q (ρ ) − Q (ρ ) + |Q (ρ )|1+4/n Q (ρ ) = 0,

(2.19)

usually referred to as the ground state. Q vanishes at infinity (see [30,61]). Similar to the one dimensional case, Lemma 3 provides a blow-up solution for (1.2) with 2s = 1 + 4/n given by

ψ (ρ , t ) = μ−1 (t )ei(α (t )(x +y 2

2

)+(δ1 (t )x+δ2 (t )y )+κ1 (t )+κ2 (t ))

Q ( ρ ).

This provides an example of an explicit blow-up solution ||ψ (t)||p → ∞ as t → T ∗ = γ0−1 (−α (0 )) for the nonautonomous nonlinear Schrödinger equation (1.2) in the two-dimensional case. 

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Theorem 1 above allows us to predict in an independent way the L∞ solution with finite-time blow-up for standard NLS found in [15] in 2008. Example 1. If we consider the equation

i

1 ∂ 2ψ ∂ψ 2s =− + h|ψ | ψ , ∂t 2 ∂ x2

h = constant,

s ≥ 0,

(2.20)

we can construct a solution with finite-time blow-up for the corresponding Riccati system (2.9)–(2.14). When we look for a solution of the form (2.3) it is easy to see that α0 (t ) = γ0 (t ) = 1/2t and β0 (t ) = −1/t, and by (A.10)–(A.16) the solution of the corresponding characteristic equation is given by μ(t ) = 2μ(0 )α (0 )t + μ(0 ). Further we obtain explicitly

α (t ) =

α (0 ) , 1 + 2α (0 )t

β (t ) =

β (0 ) , 1 + 2α (0 )t

β 2 (0 )t , 2(1 + 2α (0 )t )

γ (t ) = γ (0 ) −

δ (t ) =

ε (t ) = ε (0 ) −

δ (0 ) , 1 + 2α (0 )t

(2.21)

β (0 )δ (0 )t . 1 + 2α (0 )t

(2.22)

Eq. (2.14) must be solved separately, and κ (t) is given by

δ 2 (0 )t − 2(1 + 2α (0 )t )

κ (t ) = κ (0 ) − with

 ξs (t ) =

h

α (0 )

ξs (t )





1−s  1 1−s 1 1 + t α (0 ) − 2 , 2 (1 − s ) ln (1 + 2t α (0 ) ),

when s = 1,

(2.23)

when s = 1.

√ Now, choosing α (0 ) = 0, |ψ (t, x )| = 1/ 2 is bounded at all times. However, when α (0) = 0, one obtains

|ψ (x, t )| =  1 2

1 + t α (0 )

,

t ≥ 0,

(2.24)

which is bounded if α (0) > 0, and blows up at a finite time T ∗ = −1/2α (0 ) if α (0) < 0. As expected, this result agrees with the prediction of the theorem above, since γ0−1 (−α (0 )) = −1/2α (0 ). The following example shows blow-up for the Gross–Pitaevskii equation. Example 2. Let us consider the Gross–Pitaevskii equation

iψt = −(ψxx + ψyy ) +

2  (t ) 2 x j ψ + λ|ψ |2 ψ . 2

(2.25)

j=1

The characteristic equation associated to Gross–Pitaevskii equation is given by

μ

+ (t )μ = 0.

(2.26)

Assuming (t) is such that (2.26) allows two independent solutions μ0 (t) and μ1 (t) satisfying (2.2), then

α0 (t ) =

μ 0 (t ) , 2μ0 (t )

β0 (t ) = −

1

μ0 (t )

,

γ0 (t ) =

μ1 (t ) 2μ1 (0 )μ0 (t )

(2.27)

and

δ0 (t ) = ε0 (t ) = κ0 (t ) = 0.

(2.28)

By Theorem 1, and using (A.10)–(A.22) from the appendix, a solution for (2.15) is given by

ψ (ρ , t ) = μ−1 (t )ei(α (t )(x +y 2

2

)+(δ1 (t )x+δ2 (t )y )+κ1 (t )+κ2 (t ))

Q (ρ )

with μ(t ) = 2μ(0 )μ0 (t )(α (0 ) + γ0 (t ) ). Considering α (0 ) = 0, β (0 ) = 1, μ1 (0 ) = 1 and γ (0 ) = 0, then

τ = γ (t ) =

μ 0 (t ) , 2μ1 (t )

μ(t ) = μ(0 )μ1 (t ),

β (t ) =

−β (0 ) , μ1 (t )

α (t ) =

and considering κi (0 ) = 0 and for i = 1 and 2 we obtain

δi (t ) =

δi (0 ) , μ1 (t )

εi (t ) = −

β (0 )δi (0 )μ0 (t ) , μ1 (t )

κi (t ) = −

(δi (0 ) )2 μ0 (t ) . 2μ1 (t )

μ 1 (t ) , 2μ1 (t )

(2.29)

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163

Remark 1. Alternatively, we can use in this example the soliton solution u(x, t ) = eit Q (x ) to (2.15) so that after applying the pseudoconformal transform we can obtain solutions which blow up in finite time (see a nice discussion in [61]). Therefore, the following is a solution with blow up for (2.15) given by

χ (x, t ) =

1 t d/2

Q

x

t



e



x2 i | 2|t − ti

(2.30)

where Q is a ground state solution of (2.19). Further, the following is a solution for (2.25) given by:

ψ (x, t ) =

2 ei(α (t )x +δ (t )x+κ (t )) Q μ(t )τ

  2 ξ i |2ξ|τ − τi e , τ

(2.31)

where ξ = (β (t ) x+ ε(t ) )/τ , x = (x, y ) and ε(t ) = (ε1 (t ), ε2 (t ) ), making δi (0 ) = 0, and after simplification the module of (2.31) is

  |ψ (x, t )| = 

1

μ(0 )μ0 (t )



Q

−β (0 )x μ0 (t )

   

(2.32)

and then

lim ||ψ (t )|| p → ∞.

(2.33)

t→0

It is possible to predict finite-time blow-up for toy examples. Example 3. If we consider

iψt = −ψxx +

x2 2 [sin t − cos t ]ψ + ix sin t ψx − i sin t ψ − 3e(3−3 cos t ) |ψ |2 ψ , 4



(2.34)



the corresponding characteristic equation is given by μ − 6 sin t μ + (9 sin2 t − 3 cos t )μ = 0. The two fundamental solutions are given by μ0 (t ) = te3(1−cos t ) and μ1 (t ) = e3(1−cos t ) , and also by γ0 (t ) = 1/2t and μ(t ) = μ(0 )e3(1−cos t ) [2α (0 )t + 1]. The explicit solution of the form (2.3) will satisfy

|ψ (x, t )| =

1 , e3(1−cos t ) μ(0 )(2α (0 )t + 1 )

(2.35)

showing finite-time blow-up at T ∗ = −1/2α (0 ). Again, this result agrees with the prediction of the theorem above since γ0−1 (−α (0 )) = −1/2α (0 ). 3. Soliton solutions for a generalized variable-coefficient NLS using Ermakov’s system In this section we present a family of Schrödinger-type equations admitting soliton solutions for (1.2). Using a multiparameter solution for the Ermakov system, see Section 2, we present bright and dark-type solitons for (1.2), extending the results presented in [57] where a Riccati system was used. We will use Lemma 3. Further, by the use of these multiparameters, the solutions can be periodic with bending propagation as in [36] for the paraxial wave equation. We proceed to prove our second main result. Theorem 2 (Construction of solitons using Ermakov’s system). The nonlinear Schrödinger equation with variable coefficients of the form

iψt = −a(t )ψxx + B(t )x2 ψ − ic (t )xψx − id (t )ψ − M (t )xψ + ig(t )ψx + L(t )ψ + h(t )|ψ |

2

ψ

(3.1)

has a soliton-type solution of the form

ψy (t, x ) =

F (β (t )x + 2γ (t )y + ε (t )) i(α (t )x2 +β (t )xy+γ (t )y2 +δ (t )x+ε (t )y+κ (t )+ξ (t )) e ,  μ(t )

(3.2)

(y is a parameter) where F satisfies

F

= −ξ0 F + h0 F 3 ,

(3.3)

and the following balance between coefficients (using (A.32)–(A.38)) has been imposed:

B(t ) = b(t ) − c0 a(t )β 4 (t ),

(3.4)

M (t ) = f (t ) + 2c0 a(t )β 3 (t )ε (t ),

(3.5)

L(t ) = c0 a(t )β 2 (t )ε 2 (t ),

(3.6)

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h(t ) = h0 a(t )β 2 (t )μ(t ),

(3.7)

ξ (t ) = ξ0 (γ (t ) − γ (0 )).

(3.8)

Proof. We look for a solution of the form

ψy = Ay (x, t )eiSy (x,t )

(3.9)

with Sy (t, x ) = α (t ) + β (t )xy + γ (t ) + δ (t )x + ε (t )y + κ (t ) + ξ (t ), and y is a parameter (we omit the subindex y in calculations). Replacing (3.9) in (3.1) and assuming A ≥ 0, we obtain x2

y2

iAt − ASt = −a(t )Axx − 2ia(t )Ax Sx + a(t )Sx2 − ia(t )ASxx + b(t )x2 A − ic (t )xAx + c (t )xASx − id (t )A − f (t )xA + ig(t )Ax − g(t )ASx + h(t )A3 .

(3.10)

Taking the complex part, we obtain

At = −((4aα + c )x + 2aβ y + 2aδ − g)Ax − (2aα + d )A,

(3.11)

taking the real part and equating coefficients as in [15]. We thus obtain the nonlinear ODE

aAxx =

dξ A + h(t )A3 . dt

(3.12)

Now, using (3.4)–(3.6), we will obtain the Ermakov-type system from (3.10)

dα + b + 2cα + 4aα 2 = c0 aβ 4 , dt

(3.13)

dβ + (c + 4aα )β = 0, dt

(3.14)

dγ + aβ 2 = 0, dt

(3.15)

dδ + (c + 4aα )δ = f + 2cg + 2c0 aβ 3 ε , dt

(3.16)

dε = (g − 2aδ )β , dt

(3.17)

dκ = gδ − aδ 2 + c0 aβ 2 ε 2 , dt

(3.18)

α (t ) = −

1 μ (t ) d (t ) − . 4a(t ) μ(t ) 2a(t )

(3.19)

The solution for this system is given by Lemma 3 in the appendix. Therefore, we have obtained an explicit expression for Sy (x, t) in (3.9 ). We proceed to find an expression for A(x, t). Using (3.17), we can transform (3.11) into



 β˙ ε˙ μ˙ At + − x + 2aβ y − A + A=0 β β x 2μ

(3.20)

and look for a solution for (3.12) of the form

1 A(x, t ) = √ F (z ),

μ

z = C0 (t )x + C1 (t )y + C2 (t ).

(3.21)

Anzats (3.21) and Ermakov’s system guide us to choose C0 (t ) = β (t ), C1 (t ) = γ (t ) and C2 (t ) = ε (t ), and therefore, (3.12) becomes

Fzz =

ξ˙ h(t ) F+ F 3. β (t )a(t ) μ(t )β 2 (t )a(t ) 2

(3.22)

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From here we obtain conditions (3.7)–(3.8), and A would be given explicitly by

F (β (t )x + 2γ (t )y + ε (t ))



A(x, t ) =

μ(t )

. 

Remark 2. Eq. (3.3) presents the following classical nonlinear wave configurations, see [57] and references therein. If h0 < 0



F (z ) =

−ξ0 +



1 / 2

ξ02 − 2C0 h0

−h0

⎛

1 / 2 ⎞   2 1/4 −ξ0 + ξ02 − 2C0 h0 ⎠, × cn⎝ ξ0 − 2C0 h0 z,  2 ξ02 − 2C0 h0

(3.23)

then cn(u, k) is a Jacobi elliptic function. A familiar special case is obtained with C0 = 0, the bright soliton:



F (z ) =

−2ξ0 −h0

1

cosh



−ξ0 z

when cn(u, 1 ) = 1/ cosh u. If ξ 0 > 0



F (z ) =

ξ0 +



ξ02 − 2C0 h0 h0



1/2 ⎛ sn⎝

(3.24)

C0 h0

ξ0 +



ξ02 − 2C0 h0

−ξ0 −



ξ 2 − 2C0 h0 z,  0 −ξ0 + ξ02 − 2C0 h0

1 / 2 ⎞ ⎠,

(3.25)

then sn(u, k) is a Jacobi elliptic function. Another familiar case is obtained with C0 = ξ02 /(2h0 ), the dark soliton:



F (z ) =



ξ0 h0

tanh

ξ0 2

z

(3.26)

sn(u, 1 ) = tanh u. 3.1. Family of solutions The following family of equations (h(0), μ(0), α (0), β (0), γ (0), δ (0), ε (0) and κ (0) are parameters):

iψt = −

1 ψxx + (1 − β 4 )x2 ψ − 2β 3 (t )ε (t )xψ + β 2 (t )ε 2 (t )ψ + 2



h ( 0 )β 2 ( 0 )

β 4 (0 ) sin t + (2α (0 ) sin t + cos t )2 2

|ψ |2 ψ , (3.27)

with

β (t ) =  ε (t ) =

β (0 ) β (0 ) sin t + (2α (0 ) sin t + cos t )2 4

2

,

(3.28)

ε (0 )(2α (0 ) sin t + cos t ) − β (0 )δ (0 ) sin t ,  β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

(3.29)

admits a family of soliton solutions given by

ψy (x, t ) =

F (β (t )x + 2γ (t )y + ε (t )) i(α (t )x2 +β (t )xy+γ (t )y2 +δ (t )x+ε (t )y+κ (t )+ξ (t )) e  μ(t )

(3.30)

where F satisfies (3.3) and

 μ(t ) = μ(0 ) β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2 , α (t ) =

  α (0 ) cos 2t + sin 2t β 4 (0 ) + 4α 2 (0 ) − 1 /4 β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

γ (t ) = γ (0 ) −

1 β 2 (0 ) sin t arctan , 2 2α (0 ) sin t + cos t

(3.31)

,

(3.32)

(3.33)

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δ (t ) =

δ (0 )(2α (0 ) sin t + cos t ) + ε (0 )β 3 (0 ) sin t , β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

κ (t ) = κ (0 ) + sin2 t +

(3.34)

ε ( 0 )β 2 ( 0 ) ( α ( 0 )ε ( 0 ) − β ( 0 )δ ( 0 ) ) − α ( 0 )δ 2 ( 0 ) , β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

(3.35)

1 ε 2 ( 0 )β 2 ( 0 ) − δ 2 ( 0 ) sin 2t , 4 β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

ξ (t ) = ξ0 (γ (t ) − γ (0 )).

(3.36)

Now we show that this family contains classic soliton examples. Example 4. (Classical bright soliton) Eq. (3.27) with δ (0 ) = 0, h(0 ) = −2, β (0 ) = 1, μ(0 ) = 1 and α (0 ) = γ (0 ) = ε (0 ) = κ (0 ) = 0 becomes the classical NLS

iψt = −

1 ψxx − 2|ψ |2 ψ , 2

(3.37)

which admits the bright soliton of the form (y is velocity)

|ψy (x, t )|2 = sech2 (x − ty ).

(3.38)

Example 5. (Classical dark soliton) Eq. (3.27) with δ (0 ) = 0, h(0 ) = 2, β (0 ) = 1, μ(0 ) = 1 and α (0 ) = γ (0 ) = ε (0 ) = κ (0 ) = 0 becomes the classical NLS

iψt = −

1 ψxx + 2|ψ |2 ψ , 2

(3.39)

which admits the dark soliton of the form (y is velocity)

|ψy (x, t )|2 = tanh2 (x − ty ).

(3.40)

The following examples show periodic solutions with bending propagation. Example 6. (Bright-type soliton) Eq. (3.27) with δ (0) a parameter, h(0 ) = −2, β (0 ) = 2/3, μ(0 ) = 1 and α (0 ) = γ (0 ) = ε (0 ) = κ (0 ) = 0 becomes

iψt = −

1 ψxx + (1 − β 4 )x2 ψ − 2β 3 (t )ε (t )xψ + β 2 (t )ε 2 (t )ψ − 2

8

 16 81

9

2

sin t +

|ψ |2 ψ ,

(3.41)

cos2 t

with

2

β (t ) =  3

ε (t ) =  3

16 81

−2δ (0 ) sin t 16 81

,

(3.42)

.

(3.43)

2

sin t + cos2 t

2

sin t + cos2 t

(3.41) admits a bright-type soliton solution with absolute value of the form

|ψy (x, t )|2 = 

sech (β (t )x + 2γ (t )y + ε (t )) 2

β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

.

(3.44)

We observe that δ (0) produces a bending effect, see Fig. 1. Example 7. (Dark-type soliton) Eq. (3.27) with δ (0) a parameter, h(0 ) = 2, β (0 ) = 2/3, μ(0 ) = 1 and α (0 ) = γ (0 ) = ε (0 ) = κ (0 ) = 0 becomes

iψt = −

1 ψxx + (1 − β 4 )x2 ψ − 2β 3 (t )ε (t )xψ + β 2 (t )ε 2 (t )ψ + 2

8

 9

16 81

2

|ψ |2 ψ

(3.45)

sin t + cos2 t

and admits a dark-type soliton solution with absolute value of the form

|ψy (x, t )|2 = 

tanh (β (t )x + 2γ (t )y + ε (t )) 2

β 4 (0 ) sin2 t + (2α (0 ) sin t + cos t )2

.

We observe again that δ (0) produces a bending effect, see Fig. 2.

(3.46)

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4. Dynamics of explicit solutions through parameters In this section using Lemmas 1 and 2, we will give examples of nonautonomous nonlinear Schrödinger equations with multiparameter solutions. These toy examples show the control of the dynamics of the solutions as an application of our multiparameter approach. We have prepared a Mathematica file as supplemental material for this section. In this file all the solutions for this section are verified. Also, all the formulas from the appendix have been verified previously in [28]. In the following two examples we use Lemma 2 from the appendix. 4.1. Dynamics of the bright soliton: bending propagation Consider the nonautonomous nonlinear Schrödinger equation

iψt +

1 e2 sinh t sech t ψxx + ix tanh t ψx + i cosh t ψ + |ψ|2 ψ = 0, 2 2 sinh t + cosh t

x ∈ R,

t > 0.

(4.1)

Then, the characteristic equation and its solution are given by

μ

− (4 cosh t − 2 tanh t )μ + (4 cosh2 t − 6 sinh t )μ = 0. μ(t ) = (2 + coth t )e2 sinh t tanh t . By Lemma 2 (4.1) can be reduced to:

iuτ − uξ ξ − 2|u|2 u = 0,

ξ = β (t )x + ε (t ), τ = γ (t ),

(4.2)

where the general solution of the Riccati system associated is:

α (t ) = δ (t ) =

csch t sech t , 2 + coth t

δ (0 ) csch t 2 + coth t

,

β (t ) =

csch t , 2 + coth t

ε (t ) = ε (0 ) −

γ (t ) = γ (0 ) −

δ (0 ) 2 + coth t

,

1 , 4 + 2 coth t

κ (t ) = κ (0 ) −

δ ( 0 )2 4 + 2 coth t

.

In order to use the similarity transformation method we proceed to use the familiar solution for (4.2): u(τ , ξ ) = √ √ v sech( vξ ) exp(−ivτ ), v > 0. Therefore, a solution for the Schro¨ dinger equation (4.1) is given by:



ψ (x, t ) =

v coth t 2 + coth t





sech

  x csch t − δ (0 ) v + ε (0 )

√

2 + coth t

2x csch t sech t + 2δ (0 )x csch t − δ (0 )2 + v × exp i 4 + 2 coth t 2



× exp [i(κ (0 ) − vγ (0 ) ) − sinh t ].

(4.3)

The dynamics of the solution (4.3) are shown in Fig. 3, where it was possible to produce a change in the central axis of the bright soliton for certain values δ (0) and ε (0). 4.2. Dynamics of the dark soliton: bending propagation Consider the nonautonomous nonlinear Schrödinger equation

iψt = −

1 1 4 cosh t cosh t ψxx + cosh t ψ (x2 − i ) − ix cosh t ψx + |ψ|2 ψ , 2 2 1 + sinh t

x ∈ R,

t > 0;

(4.4)

then, the characteristic equation and its solution are respectively

μ

− tanh t μ = 0, μ(t ) = 1 + sinh t. By Lemma 2 (4.4) can be reduced to

iuτ + uξ ξ − 2|u|2 u = 0,

ξ = β (t )x + ε (t ), τ = γ (t ).

The general solution of the corresponding Riccati system is given by

α (t ) = −

1 , 2 + 2 csch t

β (t ) =

1 , 1 + sinh t

γ (t ) =

2 − γ ( 0 ), 1 + csch t

(4.5)

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δ (t ) =

δ (0 ) 1 + sinh t

ε (t ) = ε (0 ) −

,

2δ ( 0 ) , 1 + csch t

κ (t ) = κ (0 ) −

δ 2 (0 ) 2 + 2 csch t

.

In order to use the similarity transformation method we proceed to use the familiar solution for (4.5): u(ξ , τ ) = A tanh (Aξ ) exp(−2iA2 τ ), A ∈ R. As a consequence, a solution for the Schro¨ dinger equation (4.4) has the following form:

 

A

ψ (x, t ) = √

1 + sinh t

tanh A



2x csch t − 2δ (0 ) + ε (0 ) csch t + 1

−x2 + 2δ (0 )x csch t − δ 2 (0 ) − 8A2 × exp i 2 + 2 csch t







 κ ( 0 ) + 2A2 γ ( 0 ) .

× exp i

(4.6)

Fig. 4 describes the evolution in time of the solution (4.6). Again for this case we have bending propagation. The difficulty of applying Lemma 2 is solving the Riccati system. Next, we present an alternative approach to deal with the Riccati system and see how the dynamics of the solutions change with multiparameters. 4.3. An alternative method to solve the coupled Riccati system and applications to soliton solutions Assume the following conditions on the Riccati system (A.1)–(A.6): a(t ) = −l0 with l0 = ±1, β (t ) = 1, τ (t ) = t and ε (t ) = 0. Under this hypothesis one obtains the explicit formulas α (t ) = l0 c (t )/4, δ (t ) = −l0 g(t )/2 and h(t ) = −l0 λμ(t ), where the last expression shows the unique dependence of the coefficient of the nonlinearity in terms of the characteristic function μ. Furthermore one obtains the particular Riccati system

dc + c 2 + 4l0 b = 0, dt

(4.7)

dg + 2l0 f + cg = 0, dt

(4.8)

dκ l0 + g2 = 0, dt 4

(4.9)

dμ = ( 2 d − c ) μ. dt

(4.10)

The solution of this system is given by

dc + c 2 + 4l0 b = 0, dt

α (t ) = l0

c (t ) , 4

κ (t ) = κ (0 ) −

(4.11)

δ (t ) = −l0 l0 4



t 0

g(t ) , 2

h(t ) = −l0 λμ(t ),

(4.12)

g2 (z )dz,

(4.13)

 t  μ(t ) = μ(0 )exp (2d (z ) − c(z ))dz , μ(0 ) = 0, 0

 

g(t ) = g(0 ) − 2l0 exp −

t 0

c (z )dz





t

z

exp 0

0

(4.14)

 c (y )dy f (z )dz.

(4.15)

Further, with these restrictions we have a way to construct transformations that allow us to prove uniqueness of the solutions. Lemma 1. Suppose that h(t ) = −l0 λμ(t ) with λ ∈ R, l0 = ±1 and that c(t), α (t), δ (t), κ (t), μ(t) and g(t) satisfy Eqs. (4.11)– (4.15). Then

ψ (x, t ) = 

1

μ(t )

2 ei(α (t )x +δ (t )x+κ (t )) u(x, t )

(4.16)

is a solution to the Cauchy problem for the nonautonomous nonlinear Schro¨ dinger equation

iψt = l0 ψxx + b(t )x2 ψ − ic (t )xψx − id (t )ψ − f (t )xψ + ig(t )ψx + h(t )|ψ |2 ψ ,

(4.17)

ψ (x, 0 ) = ψ0 (x )

(4.18)

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if and only if u(x, t) is a solution of the Cauchy problem for the standard nonlinear Schro¨ dinger equation

iut − l0 uxx + l0 λ|u|2 u = 0.



u(x, 0 ) =

(4.19)

μ(0 )e−i(α (0)x +δ (0)x+κ (0))ψ0 (x ). 2

(4.20)

The following proposition establishes the uniqueness of classical solutions for the nonautonomous nonlinear Schro¨ dinger equation (4.17). Proposition 1. Assume that Eqs. (4.11)–(4.15) are satisfied (corresponding to l0 = −1). If c(t), d(t) and f (t ) ∈ C 1 ([−T , T ] ) for some T > 0 and h(t ) = λμ(t ) with λ ∈ R, then the Cauchy problem for the nonlinear nonautonomous Schro¨ dinger equation

iψt = −ψxx + b(t )x2 ψ − ic (t )xψx − id (t )ψ − f (t )xψ + ig(t )ψx + h(t )|ψ|2 ψ

(4.21)

ψ (x, 0 ) = ψ0 (x )

(4.22)

has a unique classical solution in the space

q Lt∞ Lx (R

× [−T , T ] ) for q = 2, ∞.

2 (R × [−T , T ] ) classical solutions for the Cauchy problem (4.21)–(4.22) in the space Proof. Let us consider ψ 1 , ψ 2 ∈ Ct,x q ∞ Lt Lx (R × [−T , T ] ) for q = 2, ∞. By Lemma 1 and the conditions in the coefficients c(t), d(t), f(t) and h(t), we have

u j (x, t ) =



2 μ(t )e−i(α (t )x +δ (t )x+κ (t ))ψ j (x, t ) ∈ Ct,x (R × [−T , T ] ) 2

with

j = 1, 2,

(4.23)

which are classical solutions for the Cauchy problem (4.19)–(4.20) on [−T , T ], and initial condition

u j (x, 0 ) =



μ(0 )e−i(α (0)x +δ (0)x+κ (0))ψ0 (x ) 2

for

j = 1, 2.

Therefore, for each j = 1, 2 we have

u j Lt∞ Lqx (R×[−T,T ]) ≤ M ψ j Lt∞ Lqx (R×[−T,T ]) ,

q = 2 , ∞,

where M > 0 is the maximum for μ(t) in the interval [−T , T ]. Using the classical uniqueness result given  in [61], we have 2 u1 = u2 , and so, the final result is obtained by multiplying equation (4.23) by the factor ei(α (t )x +δ (t )x+κ (t )) / μ(t ).  Now we want to see how the dynamics of the solutions of the nonlinear Schro¨ dinger equation (4.17) change when the parameters of dissipation, d(t), and the nonlinear term, h(t) change. We will use the alternative Riccati system (4.7)–(4.10) and therefore Lemma 1. 4.4. Perturbations of the bright soliton: competition between dissipation and nonlinearity Let us consider the nonautonomous nonlinear Schro¨ dinger equation

iψt = −ψxx +

 x2  2 sin t − cos t ψ + ix sin t ψx − id (t )ψ + h(t )|ψ |2 ψ , 4

t, x ∈ R.

(4.24)

Then the functions α , δ and κ are respectively:

α (t ) =

sin t , 4

δ (t ) = 0,

κ (t ) = κ (0 ).

We will construct explicit solutions for (4.24) using Lemma 1 and one of the solutions for the NLS equation

iut + uxx + 3|u|2 u = 0, that is given by

u(x, t ) =

 −

t, x ∈ R,

√ 2v sech( −vx ) exp(−vit ), 3

v < 0.

4.4.1. Periodic solutions for (4.24) with d (t ) = sin t and h(t ) = −3e3−3 cos t In this case one obtains a solution of the following form:





3 x2 ψ (x, t ) = exp (cos t − 1 ) + i sin t + κ (0 ) − vt 2 4 see Fig. 5.



−

√ 2v sech( −vx ), 3

(4.25)

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4.4.2. Solutions with fast decay (4.24) with d (t ) = (4t − sin t )/2 and h(t ) = −3e2t The choice of the parameters allows us to construct a solution with fast decay for large values of time:



ψ (x, t ) =



−

√ 2v x2 sech( −vx ) exp i sin t + κ (0 ) − vt 3 4





− t2 ,

(4.26)

see Fig. 6. 4.5. Perturbations of the Peregrine soliton We are interested to see how the parameters change the dynamics of solutions for (4.17), and for this end we consider



iψt = −ψxx + x2 t 2 − 1/2

 2 2 ψ + 2it ψx (x + et ) − id (t )ψ − et xψ + h(t )|ψ |2 ψ , t, x ∈ R.

(4.27)

As before we can find explicitly

√  √ 2 2 α (t ) = t/2, δ (t ) = tet , κ (t ) = κ (0 ) + e2t 2t − 2D( 2t ) /8,

where D(t ) = e−t it to

2

t 0

2

ez dz is the Dawson function. We will construct explicit solutions for (4.27) using Lemma 2 to reduce

iut + uxx + 2|u|2 u = 0,

t, x ∈ R,

(4.28)

and one of the solutions for (4.28) is





3 + 16iA2 t − 16A4 t 2 − 4A2 x2 u(x, t ) = A exp(2iA t ) , 1 + 16A4 t 2 + 4A2 x2 2

A ∈ R. 2

4.5.1. Peregrine-type soliton for (4.27) with d (t ) = tanh t − t and h(t ) = −8 cosh t The correct choice of the parameters d(t) and h(t) allows us to construct solutions with properties similar to those of the classical Peregrine soliton, as can be seen in Fig. 7. The solution for this case will be given by

 t √ √   1 2 2 ψ (x, t ) = exp i x2 + tet x + κ (0 ) + e2t 2t − 2D( 2t ) 2 8   2 A 3 + 16iA t − 16A4 t 2 − 4A2 x2 × exp(2A2 t ) sech t . 4 2 2 2 2

(4.29)

1 + 16A t + 4A x

2

4.5.2. Dynamics of the solution for (4.27) with d (t ) = −(sin 2t + t ) and h(t ) = −2e−2 sin t Here we see how the solutions are perturbated, see Fig. 8 in the appendix. The solution is given by

 t 2 ψ (x, t ) = A exp i x2 + tet x + κ (0 ) + 2  2 4 2 2 ×



√ √  1 2t 2  e 2t − 2D( 2t ) 8

3 + 16iA t − 16A t − 4A x2 1 + 16A4 t 2 + 4A2 x2





2



exp 2iA2 t + sin t .

(4.30)

5. Final remarks In this work, inspired by the work of Mahric on multiparameter solutions for the linear Schrödinger equation with quadratic potential, we have established a relationship between solutions with parameters of Riccati–Ermakov systems with the dynamics of nonlinear Schrödinger equations with variable coefficients of the form (1.2). We have shown that for special coefficients of (1.2) it is possible to find explicit solutions presenting blow up, periodic soliton solutions with bending properties and more. This work should motivate further analytical and numerical studies looking to clarify the connections on the dynamics of variable-coefficient NLS with the dynamics of Riccati–Ermakov systems. Acknowledgment The second named author was supported by the Simons Foundation #316295. The authors thank Drs. S. K. Suslov, A. Mahalov and S. Roudenko for many different valuable discussions of this project and valuable mentorship. Appendix A. Nonlinear coupled Riccati–Ermakov systems and similarity transformations All the formulas from this appendix have been verified previously in [28].

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A.1. Modified Riccati system and a similarity transformation In Section 4 we need a slightly modified nonlinear coupled Riccati system that includes for convenience of our results a parameter l0 = ±1 (the case l0 = 1 has already been considered in [15,56,59]):

dα + b(t ) + 2c (t )α + 4a(t )α 2 = 0, dt

(A.1)

dβ + (c (t ) + 4a(t )α (t ))β = 0, dt

(A.2)

dγ + l0 a(t )β 2 (t ) = 0, dt

(A.3)

l0 = ±1,

dδ + (c (t ) + 4a(t )α (t ))δ = f (t ) + 2α (t )g(t ), dt

(A.4)

dε = (g(t ) − 2a(t )δ (t ))β (t ), dt

(A.5)

dκ = g(t )δ (t ) − a(t )δ 2 (t ). dt

(A.6)

Considering the standard substitution

α=

d (t ) 1 μ (t ) − , 4a(t ) μ(t ) 2a(t )

(A.7)

it follows that the Riccati equation (A.1) becomes:

μ

− τ (t )μ + 4σ (t )μ = 0,

(A.8)

with

a

τ (t ) = − 2c + 4d, a

d σ (t ) = ab − cd + d + 2 2



d

a

− a d

 .

(A.9)

We will refer to (A.8) as the characteristic equation of the Riccati system. Here a(t), b(t), c(t), d(t), f(t) and g(t) are real value functions depending only on the variable t. A solution of the Riccati system (A.1)–(A.6) with multiparameters is given by the following expressions (with the respective inclusion of the parameter l0 ) [15,56,59]:

μ(t ) = 2μ(0 )μ0 (t )(α (0 ) + γ0 (t )), α (t ) = α0 (t ) − β (t ) = −

(A.10)

β02 (t ) , 4(α (0 ) + γ0 (t ) )

β (0 )β0 (t ) = 2(α (0 ) + γ0 (t ) )

γ (t ) = l0 γ (0 ) −

(A.11)

β (0 )μ(0 ) w(t ), μ(t )

l0 β 2 ( 0 ) , 4(α (0 ) + γ0 (t ) )

l0 = ±1,

(A.12)

(A.13)

δ (t ) = δ0 (t ) −

β0 (t )(δ (0 ) + ε0 (t ) ) , 2(α (0 ) + γ0 (t ) )

(A.14)

ε (t ) = ε (0 ) −

β (0 )(δ (0 ) + ε0 (t ) ) , 2(α (0 ) + γ0 (t ) )

(A.15)

κ (t ) = κ (0 ) + κ0 (t ) −

(δ (0 ) + ε0 (t ) )2 , 4(α (0 ) + γ0 (t ) )

(A.16)

subject to the initial arbitrary conditions μ(0), α (0), β (0) = 0, γ (0), δ (0), ε (0) and κ (0). α 0 , β 0 , γ 0 , δ 0 , ε 0 and κ 0 are given explicitly by

α0 (t ) =

d (t ) 1 μ 0 (t ) − , 4a(t ) μ0 (t ) 2a(t )

(A.17)

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J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

 

w(t ) β0 (t ) = − , μ0 (t )

w(t ) = exp −

γ0 (t ) =

d (0 ) 1 μ1 (t ) + , 2a ( 0 ) 2μ1 (0 ) μ0 (t )

δ0 (t ) =

w(t ) μ0 (t )

ε0 (t ) = −

κ0 (t ) =



t



0

f (s ) −

2a(t )w(t ) δ (t ) + 8 μ 0 (t ) 0

a(t )μ0 (t ) 2 δ (t ) − 4 μ 0 (t ) 0

t 0

 (c (s ) − 2d (s ) ) ds ,

(A.18)

(A.19)

  g( s )

ds μ0 ( s ) , μ0 ( s ) + 2a ( s ) w (s )

d (s ) g( s ) a (s ) 



t

0

t 0

a(s )σ (s )w(s )



2 μ 0 (s )

(μ0 (s )δ0 (s ) ) ds + 2

a(s )σ (s )

2  2 (μ0 (s )δ0 (s ) ) ds − 2 μ0 ( s )



t 0



(A.20) t

0





d (s ) a(s )w(s ) f (s ) − g( s ) μ 0 (s ) a (s )



ds,

(A.21)



d (s ) a (s ) g( s ) (μ (s )δ0 (s ) ) f (s ) − μ 0 (s ) 0 a (s )

ds, (A.22)

with δ0 (0 ) = g0 (0 )/(2a(0 ) ), ε0 (0 ) = −δ0 (0 ), κ0 (0 ) = 0. Here μ0 and μ1 represent the fundamental solution of the characteristic equation subject to the initial conditions μ0 (0 ) = 0, μ 0 (0 ) = 2a(0 ) = 0 and μ1 (0) = 0, μ 1 (0 ) = 0. Using the system (A.11)–(A.16), in [59] we see a generalized lens transformation is presented. Next we recall this result (here we present a slight perturbation introducing the parameter l0 = ±1 in order to use Peregrine-type soliton solutions): Lemma 2 (l0 = 1, [59]). Assume that h(t ) = λa(t )β 2 (t )μ(t ) with λ ∈ R. Then the substitution

ψ (x, t ) = 

1

μ(t )

2 ei(α (t )x +δ (t )x+κ (t )) u(ξ , τ ),

(A.23)

where ξ = β (t )x + ε (t ) and τ = γ (t ), transforms the equation

iψt = −a(t )ψxx + b(t )x2 ψ − ic (t )xψx − id (t )ψ − f (t )xψ + ig(t )ψx + h(t )|ψ|2 ψ into the standard Schro¨ dinger equation

iuτ − l0 uξ ξ + l0 λ|u|2 u = 0,

l0 = ±1,

(A.24)

as long as α , β , γ , δ , ε and κ satisfy the Riccati system (A.1)–(A.6) and also Eq. (A.7). A.2. Ermakov system and a similarity transformation We recall the following useful results for Sections 2 and 3. Lemma 3 ([31,34,35]). The following nonlinear coupled system (Ermakov system):

dα + b + 2cα + 4aα 2 = c0 aβ 4 , dt

(A.25)

dβ + (c + 4aα )β = 0, dt

(A.26)

dγ + aβ 2 = 0, dt

(A.27)

dδ + (c + 4aα )δ = f + 2cg + 2c0 aβ 3 ε , dt

(A.28)

dε = (g − 2aδ )β , dt

(A.29)

dκ = gδ − aδ 2 + c0 aβ 2 ε 2 dt

(A.30)

α (t ) = −

1 μ (t ) d (t ) − 4a(t ) μ(t ) 2a(t )

(A.31)

J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

173

admits the following multiparameter solution given explicitly by:

 μ(t ) = μ0 (t )μ(0 ) 4(γ0 (t ) + α (0 ) )2 + β 4 (0 ), α (t ) = α0 (t ) − β (t ) = − 

β02 (t )(γ0 (t ) + α (0 ) ) , 2 4(γ0 (t ) + α (0 ) ) + β 4 (0 ) β (0 )β0 (t )

4(γ0 (t ) + α (0 ) ) − β 4 (0 ) 2

γ (t ) = γ (0 ) −

(A.32)

(A.33)

,

(A.34)

β 2 (0 ) 1 arctan 2 2(γ0 (t ) + α (0 ))

(A.35)

and

δ (t ) = δ0 (t ) − β0 (t ) ε (t ) =

ε (0 )β 3 (0 ) + 2(γ0 (t ) + α (0 ))(ε0 (t ) + δ (0 )) , 4(γ0 (t ) + α (0 ))2 + β 4 (0 )

−β (0 )(δ (0 ) + ε0 (t ) ) + 2ε (0 )(γ0 (t ) + α (0 ) )



4(γ0 (t ) + α (0 ) ) + β 4 (0 ) 2

(A.36)

,

(A.37)

  β 3 (0 )ε (0 )(ε0 (t ) + δ (0 )) (γ0 (t ) + α (0 )) β 2 (0 )ε 2 (0 ) − (ε0 (t ) + δ (0 ))2 κ (t ) = κ0 (t ) + κ (0 ) − + , 4(γ0 (t ) + α (0 ))2 + β 4 (0 ) 4(γ0 (t ) + α (0 ))2 + β 4 (0 )

(A.38)

subject to arbitrary initial data μ(0), α (0), β (0) = 0, γ (0), δ (0), ε (0), κ (0) where α 0 (t), β 0 (t), γ 0 (t), δ 0 (t),  0 (t) and κ 0 (t) are given by (A.17)–(A.22). We will also need a 2D version of the results above for the blow-up results of Section 2: Lemma 4. [42] The nonlinear equation



iψt = −a(ψxx + ψyy ) + b x2 + y2

 ψ − ic(xψx + yψy ) − 2idψ

− (x f1 + y f2 )ψ + i(g1 ψx + g2 ψy ) + h|ψ |

2s

ψ,

(A.39)

where a, b, c, d, f1, 2 and g1, 2 are real-valued functions of t, can be transformed to

iχτ − l0 (χξ ξ + χηη ) = −l0 h0 |χ |2s χ

(l0 = ±1 )

(A.40)

by the ansatz

ψ = μ−1 ei(α (x +y 2

2

) + ( δ1 x + δ 2 y ) + κ 1 + κ 2 )

χ ( ξ , η , τ ),

where ξ = β (t )x + ε1 (t ), η = β (t )y + ε2 (t ), τ = γ (t ), h(t ) = h0

(A.41) a(t )β 2 (t )μ2s (t )

(h0 is a constant), provided that

dα + b + 2cα + 4aα 2 = 0, dt

(A.42)

dβ + (c + 4aα )β = 0, dt

(A.43)

dγ + aβ 2 = 0, dt

(A.44)

dδ1,2 + (c + 4aα )δ1,2 = f1,2 + 2gα , dt

(A.45)

d ε1,2 = (g − 2aδ1,2 )β , dt

(A.46)

dκ1,2 = gδ1,2 − aδ12,2 . dt

(A.47)

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Here,

α=

d 1 μ

− , 4a μ 2a

(A.48)

and solutions of the system (A.42)–(A.47) are given by (A.10)–(A.16). Further, we can establish a n-dimensional generalized Lens transformation: Proposition 2. The nonautonomous NLS

iψt = −a(t )ψ + b(t )|x|2 ψ + h0 a(t )β 2 (0 )μ2 (0 )μ(t )n(q−1)/2−2 |ψ |q−1 ψ ,

(A.49)

with t ∈ R and x ∈ Rn can be transformed into the standard NLS

iuτ = l0 u + c0 |ξ | u − l0 h0 |u|q−1 u, 2

l0 = ±1,

(A.50)

by the transformation

ψ (t, x ) = 

1

μ(t )

ei (

n

i=1

α (t )x2i +δi (t )xi +κi (t )) u (τ , ξ ),

ξi = β (t )xi + εi (t ),

τ = γ (t )

(A.51)

where

dα + b(t ) + 4a(t )α 2 = c0 aβ 4 , dt

(A.52)

dβ + 4a(t )α (t ))β = 0, dt

(A.53)

dγ + l0 a(t )β 2 (t ) = 0, dt

l0 = ±1,

(A.54)

dδi + 4a(t )α (t )δi = 2c0 aβ 3 εi , dt

(A.55)

d εi = −2a(t )β (t )δi (t ), dt

(A.56)

dκi = −a(t )δi2 (t ) + c0 aβ 2 εi2 . dt

(A.57)

i = 1, . . . , n. Considering the standard substitution

α=

d (t ) 1 μ (t ) − , 4a(t ) μ(t ) 2a(t )

(A.58)

it follows that the Riccati equation (A.1) becomes

μ

− τ (t )μ + 4σ (t )μ = 0,

(A.59)

with

τ (t ) =

a

− 2c + 4d, a

σ (t ) = ab − cd + d2 +

d 2



d

a

− a d

 .

(A.60)

We will refer to (A.8) as the characteristic equation of the Riccati system. Here a(t), b(t), c(t), d(t), f(t) and g(t) are real value functions depending only on the variable t. We observe that for the critical case n(q − 1 )/2 = 2 and the right choice of parameters values we would obtain an invariant transformation. For completion of this presentation we explain how the choice of h(t ) = h0 a(t )β 2 (0 )μ2 (0 )μ(t )n(q−1 )/2−2 it is useful to obtain the following familiar tranformations for NLS; the linear case was presented in [35].

J. Escorcia, E. Suazo / Applied Mathematics and Computation 301 (2017) 155–176

175

• Galilei transformation Using Proposition 2 with the coefficients a = 1, b = c = d = f = g = 0 with h(t ) = h0 a(t )β 2 (t )μ(t ) and α (0 ) = κ (0 ) = 0;

β (0 ) = μ(0 ) = 1 the transformation (A.23) becomes

ψ (x, t ) = ei(δ (0)x−δ

2

(0 )t )

χ (x − 2δ (0 )t + ε (0 ), t − γ (0 ) ),

(A.61)

where χ solves the free Schrödinger equation (l0 = −1)

iχτ = −χξ ξ + h0 |χ | p−1 χ .

(A.62)



 The Galilei transformation ψ (x, t ) = exp i V x/2 − V 2 t/4 χ (x − V t + x0 , t − t0 ) when choosing γ (0 ) = t0, ε (0 ) = x0 and δ (0 ) = V/2. • Pseudoconformal transformation

x 1 1 2 ψ (x, t ) = √ e(ix /4t ) χ − , − , 2t

2t

4t

when choosing δ (0 ) = κ (0 ) = ε (0 ) = 0, β (0 ) = 1, μ(0 ) = 0 and μ (0 ) = 2. • Lens transformation It is also possible obtain a transformation from the harmonic oscillator to the free particle known as the Lens transform. Indeed, taking a = b = 1 , c = d = f = g = 0 in (1.2) and by letting μ(0 ) = β (0 ) = 1 and α (0 ) = γ (0 ) = δ (0 ) = κ (0 ) = ε (0 ) = 0, we would obtain the lens transform





e−(i/2)x tan 2t x tan 2t χ , . √ cos 2t 2 cos 2t 2

ψ (x, t ) =

It is easy to see that with the tansformation from Proposition 2 we can find the traditional multiparameter solution for the standard NLS [58] that combines different transformatios including space translation, phase shift and galilean invariance. Example 8. Taking a(t ) = 1, b(t ) = c (t ) = d (t ) = f (t ) = g(t ) = 0 in (1.2) with h(t ) = λa(t )β 2 (t )μ(t ) and choosing α (0 ) = 0, by (A.10), we obtain μ(t ) = μ(0 ) and (1.2) becomes

iψt = −ψxx − h0 β (0 )μ(0 )

p−1 2 −1

|ψ |2s ψ .

Further, a solution of the form (A.23) is given by

ψ (x, t ) =

2 2 ei(2δ (0)x+κ (0)−2δ (0)t ) ei(−γ (0)+β (0)t ) R μ (0 )

where R(x ) = (s + 1 )1/2s cosh

−1/s



β (0 ) 2

 x + ε (0 ) − 2β (0 )δ (0 )t ,

(sx ) is a solution of

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