# On the Existence of Bright Solitons in Cubic-Quintic Nonlinear Schrödinger Equation with Inhomogeneous Nonlinearity

Authors: Juan Belmonte-Beitia
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By
using bifurcation theory, we prove that the norm of the positive solution goes to
zero as the parameterλ, called chemical potential in the Bose-Einstein
condensates' literature, tends to zero. Moreover, we solve the time-dependent cubic-quintic
nonlinear Schrödinger equation with inhomogeneous nonlinearities by using
a numerical method.


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## Body

## 1. Introduction

Nonlinear Schrödinger (NLS) equations appear in a great array of contexts [1], for example, in
semiconductor electronics [2, 3], optics in nonlinear media [4], photonics [5], plasmas [6], the fundamentation of
quantum mechanics [7], the dynamics of
accelerators [8], the
mean field theory of Bose-Einstein condensates [9, 10], or in biomolecule dynamics
[11]. In some of these
fields and in many others, the NLS equation appears as an asymptotic limit for
a slowly varying dispersive wave envelope propagating a nonlinear medium
[12].The study of these equations has
served as the catalyzer for the development of new ideas or even mathematical
concepts such as solitons [13] or singularities in partial differential equations
[14, 15].In the last years, there has been an increased
interest in a variant of the standard nonlinear Schrödinger equation, that is,
the so-called nonlinear  Schrödinger equation with inhomogeneous nonlinearity,
which is(1.1)iψt=−ψxx+g(x)|ψ|2ψ,with x∈ℝ,
where ψ(t,x) is a complex valued function and g(x) is a real function.This equation arises in different physical contexts
such as nonlinear optics and dynamics of Bose-Einstein condensates with
Feschbach resonance management [16–26]. Different aspects of the
dynamics of solitons in these contexts have been studied such as the emission
of solitons [16, 17] and the propagation of solitons when the space modulation
of the nonlinearity is a random [18], periodic [22], linear [19], or localized function [21]. Equation (1.1) admits
special solutions called standing waves, solitary waves, or bright solitons of
the form ψ(t,x)=u(x)eiλt,
where the profile u is time-independent. The function u satisfies(1.2)−uxx+λu+g(x)u3=0.The study of the existence of decaying solutions for
equations or systems like (1.2) has gained the interest of many mathematicians in
recent years. Many results are available for semilinear elliptic equations in ℝN.
Without being exhaustive, we refer to [27–35].At this point, it should be noticed that a way to get
compactness in semilinear elliptic problems in unbounded domains is to assume
the invariance of the coefficients under a compact group of symmetries. Indeed,
dealing with the following equation:(1.3)−Δu+a(x)u=b(x)|u|p−1uwith x∈ℝN,
Strauss radial compact imbedding (see, e.g., [36]) implies the existence of a
positive radial ground state as soon as a and b are radially symmetric, positive, and bounded.
More sophisticated conditions have been exploited, for example, in [37]. However, these results do
not apply to the one-dimensional case. Indeed, assuming radial symmetry means
that the coefficients a and b are even functions. One can therefore look for
even solutions, but H1(0,+∞) do not have better compactness properties than H1(ℝ).
Nevertheless, symmetry is always a simplifying condition and it has been
extensively used for finding connecting orbits in reversible Hamiltonian
systems [38]. In
[39], a unique
positive homoclinic solution is obtained for the model equation(1.4)−u′′+a(x)u=b(x)u3,assuming that a and b are even and bounded from below by a positive
constant such that xa′(x)>0 and xb′(x)<0,
for every x⁢≠0.
Analytical solutions of (1.4) have been calculated in [40, 41] for different functions a and b.
Finally,  we want to mention another approach to
the problem. In [42],
Torres, motivated by the study of the
propagation of electromagnetic waves through a multilayered optical medium,
proved the existence of two different kinds of homoclinic solutions to the
origin in a Schrödinger equation with a nonlinear term, by using a fixed-point
theorem in cones.Our purpose is to complete the mentioned bibliography
with a variant of the cubic-quintic nonlinear Schrödinger equation. The
cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearities
is(1.5)−u′′+λu=a(x)u3+b(x)u5.This equation can be seen as a
particular case of the so-called nonpolynomial Schrödinger equation (NPSE)
[43]. Really, in the
case of weak nonlinearity, the NPSE can be expanded, which leads to a
simplified 1D equation with a combination of cubic and
quintic terms [44]. In
this form, we can obtain (1.5).Equation (1.5) has a lot of applications to the mean
field theory for Bose-Einstein  condensates [45]
and nonlinear optics [4]. We want to remark that in the free
space, that is, a(x)=4 and b(x)=3σ,
where σ=±1 (see, e.g., [46]), (1.5) has a localized exact
solution:(1.6)u(x)=λ1/2(1+1+σλcosh(2λx))1/2.On the other hand, it is known
that for (1.5) some soliton solutions become bistable (i.e., singular solitons
with the same carried power but different propagation parameters) [47–50].An interesting problem arises when we consider the
time-dependent cubic-quintic nonlinear Schrödinger equation(1.7)iψt=−ψxx−g1(x)|ψ|2ψ−g2(x)|ψ|4ψsince the problem of collapse
appears. In the presence of the self-focusing quintic term, collapse is
inevitable, and it may affect the stability of solitons against small
perturbations.It is known that (1.7), for gi(x)≡Ci=constant,i=1,2 (homogeneous case), has no blowup solution in the
class(1.8){ψ∈H1(ℝ)∣∥ψ(0,x)∥L2<∥R∥L2},where R is the ground state of the
equation(1.9)−u′′+λu−C1u3−C2u5=0.In the class(1.10){ψ∈H1(ℝ)∣∥ψ(0,x)∥L2=∥R∥L2},one has that (1.7) has a unique
blowup solution (see, e.g., [14]).In any case, in this paper, we focus on the proof of
the existence of solutions of (1.5), and we will not consider the problem of
collapse and stability of the solutions of (1.7), which is an open problem in the
case of inhomogeneous nonlinearity and will be studied elsewhere.Thus, in this paper, we will prove the existence of
bright solitons for (1.5). To do it, we will use critical point theory (the
mountain pass theorem). Thus, we will use a variational approach to our
equation, and prove that it satisfies the conditions of the mountain pass
theorem. Moreover, using bifurcation theory, we will prove that when the
chemical potential λ→0+,∥uλ∥2→0.
Finally, we will solve this equation by using a numerical scheme, called
imaginary time method.The rest of the paper is organized as follows. In
Section2, we use a variational approach to the stationary cubic-quintic
nonlinear Schrödinger equation with inhomogeneous nonlinearity, and some
preliminary results are collected. Section 3 contains the main result about the
existence of positive solutions. Section 4 deals with a result about
bifurcation theory. Finally, Section 5 contains a numerical scheme to solve the
time-dependent cubic-quintic nonlinear Schrödinger equation with inhomogeneous
nonlinearity.


## 2. The Variational Approach

In this paper,
we will study the cubic-quintic nonlinear Schrödinger equation with
inhomogeneous nonlinearitiesgi(x),i=1,2 (CQINLSE), on ℝ,
that is,(2.1)iψt=−ψxx−g1(x)|ψ|2ψ−g2(x)|ψ|4ψ,with gi:ℝ→ℝ,i=1,2,
which satisfies the following properties:(2.2)gi∈L∞(ℝ),gi(x)>0,lim⁡|x|→∞gi(x)=0,i=1,2.The solitary wave solutions of (2.1) are given by ψ(x,t)=eiλtu(x),
where u(x) is the solution of(2.3)−uxx+λu=g1(x)u3+g2(x)u5,which can be identified as
bright solitons due to their boundary conditions:(2.4)u(x)⟶0⁢asx⟶±∞.In this paper, we search for
positive solutions toλ>0.
Thus, the following theorem gives the existence of positive solitary waves, for λ>0.Theorem 2.1 (existence of a positive solution).
Whenλ>0,
(2.3) has a positive solution u∈H1(ℝ).In order to prove this theorem, we will introduce
a set of preparatory definitions and lemmas. Formally, (2.3) is the
Euler-Lagrange equation of the functional J:H1(ℝ)→ℝ,
defined by(2.5)J(u)=12∫ℝ[|ux|2+λ|u|2]dx−14∫ℝg1(x)|u|4dx−16∫ℝg2(x)|u|6dx.We define(2.6)∥u∥2=∫ℝ|ux|2+λ|u|2dx,Ψ1(u)=14∫ℝg1(x)|u|4dx,Ψ2(u)=16∫ℝg2(x)|u|6dx.We can therefore rewrite the
functional (2.5) in the following way:(2.7)J(u)=12∥u∥2−Ψ1(u)−Ψ2(u).Remark thatH1(ℝ)↪Lp(ℝ), p≥2,
and thus J is well defined on H1(ℝ) and is smooth. It is very easy to check that,
for each fixed λ>0, ∥⋅∥ is an equivalent norm to that which is usual
in H1(ℝ).
Clearly, J is of C2 class and its critical points give rise to
solutions of (2.3) such that lim⁡|x|→∞u=0.


## 3. Existence of a Positive Solution

In order to obtain critical points ofJ,
we will use the mountain pass theorem [51]. This theorem deals with the existence of critical
points of a functional J∈(C1(E),ℝ),
where E is a Hilbert space (although, in general, E
can be a Banach space), which satisfies the following two "geometric"
assumptions.(MP1)
There existr,ρ>0 such that J(u)≥ρ,
for all u∈E,
with ||u||=r.(MP2)
There existsv∈E, ||v||>r,
such that J(v)≤0=J(0).
Moreover, it
assumes the compactness condition (PS)c,
called the Palais-Smale condition at level c.Every sequenceun such that(1)
J(un)→c,(2)
J′(un)→0
has a
converging subsequence. J′(u) is called the derivative of J at u,
which exists by the Riesz theorem and is given by the following
expression:(3.1)(J′(u)∣ζ)=(u∣ζ)−(Ψ1′(u)∣ζ)−(Ψ2′(u)∣ζ),∀ζ∈H1(ℝ),where(3.2)(Ψ1′(u)∣ζ)=∫ℝg1(x)u3ζ,(Ψ2′(u)∣ζ)=∫ℝg2(x)u5ζ.The sequences satisfying
(1),(2) are called(PS)c sequences.Consider the class of all the paths joiningu=0 and u=v:(3.3)Γ={γ∈C([0,1],E):γ(0)=0,γ(1)=v},and set(3.4)c=inf⁡γ∈Γ⁢max⁡t∈[0,1]⁢J(γ(t)),with t being the variable of the curve γ.
For reader's convenience, we will enunciate a simplified version of the
mountain pass theorem (see [51] for the general setting).Theorem 3.1 (mountain pass theorem).
IfJ∈C1(E,ℝ) satisfies the geometric conditions (1.1) and (1.2)
and the (PS)c Palais-Smale condition holds, then c is a positive critical level for J.
Precisely, there exists z∈E such that J(z)=c>0 and J′(z)=0.
In particular, z≠0 and z≠v.Consider the following
lemma.Lemma 3.2.
The functionalJ satisfies  the geometric assumptions of the
mountain pass theorem.Proof.
(1) From the definition ofJ,
the hypothesis (2.2) on gi,⁢i=1,2,
and the Sobolev embedding H1(ℝ)↪Lp(ℝ),p≥2 (see, e.g., [36]), we obtain(3.5)J(u)=12∥u∥2−Ψ1(u)−Ψ2(u)≥12∥u∥2−C∥u∥4−M∥u∥6,where C and M are positive constants. As a consequence,
there exist r,ρ>0 such that(3.6)J(u)≥ρ,∀u∈H1(ℝ),with∥u∥=r,which proves that J verifies (MP1).
(2) Considerv0∈H1(ℝ)∖{0},
and let s be a parameter such that for s>0,(3.7)Jλ(sv0)=s22∥v0∥λ2−s4Ψ1(v0)−s6Ψ2(v0)↘−∞as⁢s↗∞. As a consequence, taking v=s0v0 with s0≫1,
we obtain Jλ(v)<0=Jλ(0).
It follows that J(sv)→−∞ as s⁢→+∞.In order to apply the mountain pass theorem (2), we
have to study(PS)c sequences.
Lemma 3.3.
Palais-Smale sequences are bounded.Proof.
From condition (1) of the definition of (PS)
sequences,J(un)≤k and we obtain(3.8)∥un∥2≤2k+2Ψ1(un)+2Ψ2(un).
FromJ′(un)→0 and using the definition of J′,
we infer(3.9)|∥un∥2−4Ψ1(un)−6Ψ2(un)|=|(J′(un)∣un)|≤∥J′(un)∥∥un∥=ϵ∥un∥,for ϵ>0.
Thus,(3.10)∫ℝg1(x)un4+∫ℝg2(x)un6≤∥un∥2+ϵ∥un∥.
Using (3.8), we
obtain(3.11)∥un∥2≤2k+12∫ℝg1(x)un4+13∫ℝg2(x)un6≤2k+12[∫ℝg1(x)un4+∫ℝg2(x)un6]≤2k+12∥un∥2+ϵ2∥un∥,and thus, for all n and some constant k,
we deduce that(3.12)12∥un∥2≤2k+ϵ2∥un∥and the boundedness of  (PS) sequences follows.Lemma 3.4.
Ψi is weakly continuous and Ψi′ is compact, for each i=1,2.Proof.
Letun⇀u weakly in H1(ℝ).
As any weakly convergent sequence is bounded, that is, there exists a constant K>0 such that ∥un∥H1(ℝ)≤K,
then, according to the Sobolev imbedding theorem, constants C,M>0 must exist such that ∥un∥L4(ℝ)<C and ∥un∥L6(ℝ)<M.
So, given ϵ>0,
from condition (2.2), it follows that there exist R1,R2>0 such that(3.13)∫|x|≥R1g1(x)(|un|4−|u|4)dx≤ϵ,∫|x|≥R2g2(x)(|un|6−|u|6)dx≤ϵ.
On the other hand, letBR1 and BR2 be the open balls of
radii R1 and R2,
respectively. Since H1(BR1) is compactly embedded in L4(BR1) and H1(BR2) is also compactly embedded in L6(BR2),
we have that un→u strongly in L4(BR1) and L6(BR2),
respectively. Moreover, there exists a constant M>0  such that(3.14)|[∫|x|≤R1g1(x)|un|4dx]1/4−[∫|x|≤R1g1(x)|u|4dx]1/4|=|∥g11/4un∥L4(BR1)−∥g11/4u∥L4(BR1)|≤|∥g11/4(un−u)∥L4(BR1)|≤M∥un−u∥L4(BR1)≤ϵ,for ϵ>0 and a sufficiently large n.
It is easy to check that a similar inequality exists for g2(x).
Putting together the two preceding inequalities, it follows that Ψi,⁢i=1,2, is weakly continuous.
The proof thatΨi′,i=1,2, is compact is similar. Let(3.15)∥Ψ1′(un)−Ψ1′(u)∥=sup⁡∥φ∥≤1{∫g1(x)(|un|3−|u|3)φ⁢dx},for φ∈H1(ℝ).
Using the Holder inequality, we obtain(3.16)∥Ψ1′(un)−Ψ1′(u)∥≤∥g1(x)(|un|3−|u|3)∥Lp(ℝ)∥φ∥Lq(ℝ),with 1/p+1/q=1,p≥2,⁢q<∞.
Using the arguments previously exposed, we immediately show
that(3.17)∥g1(x)(|un|3−|u|3)∥Lp(ℝ)≤ϵ,for n≫1 and ϵ>0.
This shows that Ψ1' is a compact operator. The proof for Ψ2' is similar.We are now prepared to prove Theorem2.1.Letun be a sequence that verifies the Palais-Smale
conditions (1) and (2). Since ∥un∥≤K,
we have that un⇀u weakly in H1(ℝ).
According to Lemma 3.4, Ψi′, i=1,2,
is compact. Therefore, there exists a subsequence, still denoted as un,
such that Ψi′(un)→Ψi′(u).
On the other hand, we know that(3.18)J′(u)=u−Ψ1′(u)−Ψ2′(u). Hence, we deduce
that(3.19)un=J′(un)+Ψ1′(un)+Ψ2′(un).AsJ′(un)→0,
since un is a Palais-Smale sequence, we
obtain(3.20)un⟶Ψ1′(u)+Ψ2′(u),proving that (PS)c holds for every c.We can thus apply the mountain pass theorem (2) since
the conditions of this theorem are satisfied, and there existsuλ∈H1(ℝ) such that J(uλ)=c and J′(uλ)=0.
The positivity is clear, using the maximum principle.


## 4. Bifurcation from the Positive Solution

By using the bifurcation theory (see, e.g., [52, 53] for an introduction to
bifurcation theory), we can investigate the behavior of the positive solution uλ when λ↘0.Letc(λ) denote the mountain pass critical level of J for the positive solution uλ.
We suppose that gi(x),i=1,2,
moreover to verify condition (2.2), satisfies the
following: there exist Ci>0,Ki>0,i=1,2,
and τ1>1 and τ2∈(0,1) such that(4.1)gi(x)≥Ki|x|−τi,∀|x|≥Ci,⁢i=1,2. Thus, we are able to prove the
following lemma.
Lemma 4.1.
Ifgi(x),i=1,2, satisfies conditions (2.2) and (4.1), then c(λ)→0 as λ→0+.Proof.
Fix the functionΦ(x)=|x|e−|x| and set uα(x)=Φ(αx).
There hold(4.2)∥uα′∥L2(ℝ)2=αA1,A1=∫ℝ|Φ′|2dx,∥uα∥L2(ℝ)2=α−1A2,A2=∫ℝΦ2dx. One thus finds
that(4.3)∥uα∥2=12∥uα′∥L2(ℝ)2+λ∥uα∥L2(ℝ)2=12A1α+λA2α−1.
If we takeλ=α2,
then we obtain ∥uα∥2=A3α,⁢λ>0,
for A3=1/2A1+A2>0.
Moreover, by using (4.1), we deduce(4.4)Ψ1(uα)=14∫ℝg1(x)|uα(x)|4dx≥14K1∫|x|≥C1|x|−τ1|uα(x)|4dx,Ψ2(uα)=16∫ℝg2(x)|uα(x)|6dx≥16K2∫|x|≥C2|x|−τ2|uα(x)|6dx. By changing variable y=αx,
we discover that(4.5)∫|x|≥C1|x|−τ1|Φ(αx)|4dx=∫|y|≥C1α|yα|−τ1|Φ(y)|4α−1dy≥ατ1−1∫|y|≥C1|y|−τ1|Φ(y)|4dx,∫|x|≥C2|x|−τ2|Φ(αx)|6dx=∫|y|≥C2α|yα|−τ2|Φ(y)|6α−1dy≥ατ2−1∫|y|≥C2|y|−τ2|Φ(y)|6dx,where the domain of integration
can be changed since 0<α≤1.
Therefore, there exist A5>0 and A6>0 such that Ψ1(uα)≥K1ατ1−1A5/4 and Ψ2(uα)≥K2ατ2−1A6/6. Putting together all the preceding
estimates, we obtain(4.6)J(uα)=12∥uα∥2−Ψ1(uα)−Ψ2(uα)≤12A3α−K1A54ατ1−1−K2A66ατ2−1.
Recall that(4.7)c(λ)=inf⁡γ∈Γ⁢max⁡0≤t≤1⁢J(γ(t)),where Γ is the class of all paths joining 0 and v, J(v)≤0.
Let us consider the following continuous curve:(4.8)γ:[0,1]⟶H1(ℝ),t⟶tMuα,where M is a constant, M≫1.
It is clear that γ is a path in Γ since it joins 0 and vα=Muα and J(Muα)≤0,
for sufficiently large M.
Then,(4.9)c(λ)≤max⁡0≤t≤1⁢J(tMuα)(λ=α2). We now use the above estimate to
evaluate max⁡0≤t≤1J(tMuα).
There holds(4.10)J(tMuα)≤β(t)=12A3t2M2α−14K1A5ατ1−1t4M4−16K2A6ατ2−1t6M6. The maximum of β is achieved at(4.11)tmax⁡=±M−K1A5ατ1−τ22K2A6+K12A52α2τ1−2τ24K22A62+α1−τ2K2A6. Therefore,(4.12)c(λ)≤12A3tmax⁡2M2α−14K1A5ατ1−1tmax⁡4M4−16K2A6ατ2−1tmax⁡6M6,with λ=α2.
Since τ1>1 and τ2∈[0,1),
we find that c(λ)→0 as λ=α2→0+.We have proved that the MP critical pointuλ satisfies J(uλ)=c(λ)→0 as λ→0+.Moreover, multiplying (2.3) by uλ,
using the fact that u′(±∞)=0,
since u(±∞)=0 and by integration, we obtain(4.13)∫ℝ|uλ|2dx+λ∫ℝuλ2dx=∫ℝg1(x)uλ4dx+∫ℝg2(x)uλ6dxor(4.14)J(uλ)=Ψ1(uλ)+2Ψ2(uλ). As J(uλ)→0 as λ→0+,
it must be verified that Ψ1(uλ)→0 and Ψ2(uλ)→0 as λ→0+.
Thus, ∥uλ∥→0 as λ→0+.We have proved that whenλ→0+, ∥uλ∥2→0.
In fact, we can prove that, for λ=0,
if there exist solutions which are different from the trivial
ones, these solutions would have an infinite amount
of nodes. To do so, we will first prove the nonexistence of positive solutions
of the equation(4.15)−uxx−g1(x)u3−g2(x)u5=0,u(±∞)=0.Letu be a strict positive solution of (4.15).
Then,(4.16)uxx=−g1(x)u3(x)−g2(x)u5<0,∀x∈ℝ.Letx0 be a global
maximum point of the solution u,
that is, u(x0)=max⁡x∈ℝu(x)>0.
This maximum point clearly exists owing to the boundary conditions. Then, u′(x0)=0 and, moreover, u′′(x)<0,
for all x∈ℝ. Then, from [x0,+∞), u must be decreasing and therefore u′(x)<0,
for all x∈[x0,+∞).
Then, u must cross the x-axis since u′′(x)<0,
for all x∈ℝ,
which contradicts the initial hypothesis.As(4.17)uxx=−g1(x)u3(x)−g2(x)u5(x)<0if⁢u>0,uxx=−g1(x)u3(x)−g2(x)u5(x)>0if⁢u<0,it is clear that the solution
has an infinite amount of nodes.


## 5. Numerical Method

In this section, we will solve (2.1) by using an
imaginary time method. This method allows us to calculate the positive
solution, or ground state, of (2.3).First, we have the changet→−it in (2.1). Thus, we obtain(5.1)ut=uxx+g1(x)|u|2+g2(x)|u|4u.We must study (5.1) numerically.
To this end, we have developed a Fourier pseudospectral scheme for the
discretization of the spatial derivatives combined with a split-step scheme to
compute the time evolution. Split-step schemes are based on the observation
that many problems may be decomposed into exactly solvable parts and on the
fact that the full problem may be approximated as a composition of the
individual problems. For instance, the solution of partial differential
equations of the type ∂tu(x,t)=N(t,x,u,∂x,…)u=(A+B)u can be approximated from the exact solutions
of the problems ∂tu=Au and ∂tu=Bu.Let us decompose the evolution operator in (5.1) by
taking [54, 55](5.2)A=∂xx,B=g1(x)|u|2+g2(x)|u|4.To proceed with split-step-type
methods, it is necessary to compute the explicit form of the operatorse(t−t0)A and e(t−t0)B.
To obtain the action of the operators, we solve the subproblems(5.3)∂tu=∂xxu,∂tu=g1(x)|u|2u+g2(x)|u|4u.Thus, after some algebra and
defining the time stepτ as τ=(t−t0)/c,⁢c∈R,
after a suitable renaming, we obtain the explicit form of the
operators(5.4)e(t−t0)A≡ecτA=ℱ−1e−k2cτℱ,e(t−t0)B≡ecτB=e(g1(x)|u|2+g2(x)|u|4)(t−t0),where ℱ denotes the spatial Fourier transform. As we
now have the explicit form of the solutions of subproblems (5.3), we
are able to obtain the positive solution of (2.1) to any degree of accuracy. We
have used the second-order splitting classical method whose equation
is(5.5)u(x,t+τ)=eτA/2eτBeτA/2u(x,t)+O(τ3).
This scheme has many advantages.
First, it is more accurate than the numerical methods based
on finite difference [56]. Second, from the practical
point of view, the calculation of the Fourier transform, which is the most
computer time-consuming step in the calculations, may be done by using the fast
Fourier transform (FFT). Thus, the computational cost of the method is of order O(N2log⁡N),
with N being the points' number in each spatial
direction of the grid which is quite acceptable. The use of discrete transforms
to represent the continuous Fourier transform in (5.5) implicitly imposes
periodic boundary conditions on u.
However, since u is expected to be negligible on the boundaries
(otherwise the computational domain must be enlarged), this is not an essential
point. Another convenient property of this scheme is
its preservation of the L2-norm of the solutions.Thus, the pictures in Figure1 show the positive
solutions of (2.1) for various g1(x),g2(x) functions, which satisfy condition (2.2). For
the first picture (Figure 1(a)),(5.6)g1(x)=C11+x2,g2(x)=C21+x4,with C1=−1 and C2=−1.
For the second picture (Figure 1(b)),(5.7)g1(x)=C1e−x2,g2(x)=C21+2x2,with C1=−1 and C2=−1.Some stationary solutions of (2.1) for
(a) g1(x),g2(x) functions given
by (5.6), and (b) g1(x),g2(x) functions given
by (5.7) (see text).
(a)(b)


## 6. Conclusions

In this paper, we have proved the existence of bright
soliton or positive solutions of the cubic-quintic
nonlinear Schrödinger equations with inhomogeneous nonlinearities. In order to
do this, we have used a variational approach and a critical point theory.
Moreover, by using bifurcation theory, we have proved that the norm of the
positive solution goes to zero as the parameterλ tends to zero. Finally, by using an imaginary
time method, we have numerically solved the inhomogeneous nonlinear Schrödinger
equation for different nonlinearities.



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*Source: 935390-2008-08-20.xml*