On coupled non-linear Schrödinger systems with singular source term

1.   Introduction

This paper considers the Cauchy problem for the coupled non-linear Schrödinger system, shortened to CNLS,

Here and hereafter, for $j\in[1, n]$, $u_j$ is a complex valued function of the variable $(t, y)\in \mathbb{R}_+\times \mathbb{R}^N$, and $u: = (u_1, \dots, u_n)$. The constant $0\neq\tau\in \mathbb{C}$ and the coupling parameters satisfy:

The coupled Schrödinger system (CNLS) models many physical phenomena, such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optic and Bose-Einstein condensates ^[1,4,7], see also ^[2,3,5,6]. The passage of a light beam with two components along an optical fiber produces the decomposition of the ray into two. Then, the model of a scalar Schrödinger equation can be improved by a system of coupled Shrödinger equations ^[8]. As a consequence, new kinds of solutions appear, first observed by Manakov ^[9] in a Kerr medium.

In mathematical point of view, many authors focused their attention on coupled nonlinear equations of Schrödinger type. So, the list of references is necessarily incomplete. The local well-posedness in the energy space was proved in ^[10,11]. The existence of ground states was investigated in ^[12,13,14]. The scattering of defocusing global solutions was treated in ^[15]. The scattering in the repulsive regime with small data was obtained in ^[16,17]. See also ^{[18,19,20,21]} for the inhomogeneous case and ^[22] for the case of harmonic potential and ^[23] for the parabolic context.

To the author's knowledge, the above works treat only the case $\sigma\geq2$ in (CNLS). This is to avoid a singularity of the source term for $\sigma < 2$. The contribution of this paper is to try to fill in this gap in the literature.

The purpose of this paper is to prove the existence of a local solution to the coupled Schrödinger problem (CNLS) in the case $\frac32 < \sigma < 2$. Then, one establishes the existence of global solutions which scatter in some Sobolev spaces and the existence of non-global solutions with finite variance.

The paper is organized as follows: Section 2 contains the main results and some standard estimates needed in the sequel. Section 3 proves the local existence of solutions to (CNLS). Section 4 establishes the scattering of global solutions and the existence of non-global solutions to (CNLS).

Throughout this paper, we denote the spaces and norms

Lastly, $T^* > 0$ denotes the lifespan for an eventual solution to (CNLS).

2.   Background and main results

In this section, we give the main results and some standard estimates.

2.1.   Preliminary

Solutions of (CNLS) formally satisfy the conservation of the following real quantities, respectively the mass and the energy

Let the non-linear terms

Take for $s\in \mathbb{R}$,

Here and hereafter, one defines some real numbers

where $E[\cdot]$ refers to the integer part. Take the Banach space

Let the centered ball of radius $R > 0$, denoted by $B_T(R): = \{u\in C_T(\mathcal Y), \; \|u\|_{L^\infty_T(\mathcal Y)}\leq R\}$, and for $\nu > 0$, take

Finally, we define the vector Schrödinger group

2.2.   Main results

In what follows, we list the results proved in this paper. The first purpose of this note is to prove the existence of local solutions to (CNLS).

Theorem 2.1. Let $N\geq1$, ${\frac32} < \sigma < 2$, and $u_0\in\mathcal Y$, satisfying

Then, there are $T > 0$ and a unique solution to (CNLS) denoted by $u\in C_T(\mathcal Y)$. Moreover, the flow is locally continuous.

Remarks 2.1.

$1)$The condition $\sigma > \frac32$, which seems to be technical, is necessary in order to have (2.1);

$2)$ the continuity of the flow follows with standard arguments;

$3)$ assumption (2.5) avoids in some meaning the singularity of the source term for $\sigma < 2$;

$4)$ the proof follows ideas in ^[24], where the scalar case is treated;

$5)$ some tools needed in the proof are taken from ^[25];

$6)$ the proof is based on a Picard fixed point argument.

The second result is about the scattering of global solutions to (CNLS) in some Sobolev spaces.

Theorem 2.2. Let $N\geq2$ and $\max\{1+\frac1N, \frac32\} < \sigma < 2$. Take $v_0\in\mathcal Y$ satisfying (2.5), and $u_0: = e^{\frac{i\kappa t|x|^2}4}v_0$ for some real number $\kappa\gg 1$. Take $0\leq s < m-\frac{N}2$. Then, there is a unique global solution to (CNLS) in $C(\mathbb{R}_+, H^s)\cap \langle y\rangle^{-\frac N2}L^\infty(\mathbb{R}_+\times \mathbb{R}^N)$, which scatters in $H^s$.

Remarks 2.2.

$1)$The proof is based on the pseudo-conformal transformation ^[26] and Theorem 2.1;

$2)$ a similar result was first proved ^[24] in the scalar case;

$3)$ a part of the proof is omitted because it follows ^[25];

$4)$ the real number $\kappa$ depends on $\nu, \|v_0\|_{\mathcal Y}, N, m$.

Finally, one proves the existence of non-global solutions to (CNLS).

Proposition 2.1. Take the assumptions of Theorem 2.1, $m > 1+\frac1N$ and $\tau < 0$. Then, the solution to (CNLS) is non-global if one of the following statements holds:

$1)$$p\geq1+\frac2N$ and $E(u_0) < 0$;

$2)$ $E(u_0) < d$ and $I(u) < 0$.

Remarks 2.3.

$1)$The blow-up in the first case follows with classical variance method;

$2)$ the second case follows like [18,Theorem 2.8];

$3)$ $d$ denotes the ground state energy $d: = \inf_{0\neq u\in (H^1)^m}\{E(u), \quad I(u) = 0\}$, where $I(u): = BE(u)-(B-2)\|\nabla u\|^2$ and $B: = N(\sigma-1)$.

Some intermediate results are listed in what follows.

2.3.   Tools

In order to investigate the blow-up of solutions, one needs the next variance identity ^[27].

Proposition 2.2. Assume that ${u}\in C_T([H^1]^n)$ is a solution to (CNLS) for $\tau = -1$, and satisfies $x{u_j}(t)\in L^2$, for any $1\leq j\leq n$. Then,

The following estimate will be useful [25,Lemma 2.9].

Lemma 2.1. Let $\alpha\geq0$. Then, for all real number: $t$, we have

Recall also [25,Proposition 2.10].

Lemma 2.2. Let $\alpha\geq0$, $k, K\in \mathbb{N}$, and $s\geq 2k+K+3+E[\frac N2]+\alpha$. Then, for any $t\in \mathbb{R}$ and any $|\mu|\leq K$,

One recalls the nonlinear estimates [25,Proposition 3.1].

Lemma 2.3. Let $\alpha\geq0$, $u\in \mathcal B_{\nu, T}(R)$, and $r\in[1, \infty]$. Then, for all $|\mu|\leq -N+M_0+M$, we have

$1)$

$2)$

Finally, let us give some nonlinear estimates [25,Proposition 3.2].

Lemma 2.4. Let $\alpha\geq0$, $u, v\in \mathcal B_{\nu, T}(R)$, and $r\in[1, \infty]$. Then, for all $|\mu|\leq -N+M_0+M$, we have

$1)$

$2)$

3.   Local well-posedness

This section is devoted to Theorem 2.1, which deals with the existence of a unique solution of (CNLS) in $C_T(\mathcal Y)$. Take the function

where $u_0: = (u_{0, 1}, \dots, u_{0, n})$, and

Let us prove that $f(B_{\nu, T}(R))\subset B_{\nu, T}(R)$. By Lemma 2.2, for the choice

and for $|\alpha|\leq E[\frac N2]$, we have

Because $p\geq1$, by Lemma 2.3, for $|\gamma_1|\leq E[\frac N2]$, we have

Also, by Lemma 2.3, we get

Thus, with the Leibniz rule, if $|\gamma|\leq E[\frac N2]$, we get

Now, with the Leibniz rule via Lemma 2.3, for $E[\frac N2]\leq|\gamma|\leq M$, $\frac12 = \frac1{r_1}+\frac1{r_2}$, and $\alpha\in \mathbb{R}$,

Now, (2.2) enables to take $2-\sigma < \alpha < \sigma-\frac N{2m}$, $r_1 = 2$, and $r_2 = \infty$. This implies that $\langle y\rangle^{m(\alpha-\sigma)}\in L^2$. Thus, we have

Assuming that $|\gamma_2|\leq N$, ${M\geq N+E[\frac N2]+1}$ gives

Now, we assume that $|\gamma_2|\geq N$, thus $|\gamma_1|\leq M-N\leq M-E[\frac N2]-1$. Since by (2.2) we have $m > \frac N{2(\sigma-1-\alpha)}$ and $\|\langle y\rangle^{m(1+\alpha-\sigma)}\| < \infty$. So, with Lemma 2.3, we get

So,

It follows that

We break down the next term as follows:

Now, with the Leibniz rule and Lemma 2.3, for $|\gamma|\leq -N+M_0+M$, we have

(2.2) gives $\langle y\rangle^{-m}\in L^2$, and

By Sobolev injection, we have

Thus, by (2.2), we get

Now, with Lemma 2.3, we have

Moreover, (2.2) gives $\langle y\rangle^{2m(1-\sigma)}\in L^2$, and so

Since $N+|\gamma_2|\leq|\gamma_1|+|\gamma_2|\leq -N+M_0+M$, we have $|\gamma_2|\leq M+M_0-2N$, and

Finally, assume that $M < |\gamma_1|\leq -N+M_0+M$. Thus, $|\gamma_2|\leq {M_0-N\leq M-E[\frac N2]-1}$. So, with Sobolev injections via Lemma 2.3, we have

Thus,

Collecting (3.7), (3.9), and (3.10), we get

Thus, with (3.1)–(3.3) and (3.11), we get

Now, by Lemma 2.1, (3.3), and (3.11), we write

Moreover, with Lemma 2.1 and (3.11), for $|\alpha|\leq -N+M_0+M$, yields

So, taking $R: = 2c\|u_0\|_{\mathcal Y}$, we get

Then, choosing $0 < T < < 1$, it follows that $f(B_T(R))\subset B_T(R)$. Now, we prove that

Using the time derivative identity $(e^{it\Delta})^{(k)} = (i\Delta)^ke^{it\Delta}$, we get

Thus, with Lemma 2.1 and Sobolev injections via (2.2), we write

Now, by Lemma 2.2, for $s = 5+m+5E[\frac N2]$ and $K = 2(1+E[\frac N2])$,

Now, since ${M\geq 4E[\frac N2]+5+m}$ and ${M_0-N > m > \frac N2}$, we get

Thus,

Now, with Lemma 2.2 with (3.13),

Thus, there is $C(T)\to0$ as $T\to0$ such that

So, taking $0 < T < < 1$, we get

Thus, $f(B_{\nu, T}(R))\subset B_{\nu, T}(R)$. Now, we prove that $f$ is a contraction. For $u, v\in C_T(\mathcal Y)$ and $w: = u-v$, we have

Let us control the three above terms. By Lemma 2.2 via (2.4), we have

Let $|\gamma|\leq E[\frac N2]$, and write

Taking account of Lemma 2.4, we have

Moreover, also with Lemma 2.4, we have

Now, by (3.1), it follows that

Let $E[\frac N2] < \gamma\leq M$. Then,

Now, by Lemma 2.4 and using (2.1), we get

Assume that $|\gamma_2|\leq N$, thus, ${M\geq N+E[\frac N2]+1}$ gives by Sobolev embedding and Lemma 2.3 via (2.1), we get

Moreover, with Lemma 2.4, we write

So, with Sobolev embeddings,

Finally, with Lemma 2.3, we have

Collecting the above estimates, we get

Now, assume that $M\geq|\gamma_2|\geq N$. Thus, arguing as in (3.3), we have

Moreover, since $|\gamma_1|\leq M-N\leq M-E[\frac N2]-1$, by (3.16) and (3.17), we write

Now, with the Leibniz rule via Lemma 2.3, for $|\gamma|\leq -N+M_0+M$, we have

Let us start with estimating the first term. By (3.6),

With Lemma 2.4, we get

Since ${\sigma > \frac32}$, we get

If $|\gamma_1| < N$, we write by Sobolev injections,

Thus,

If $|\gamma_1|\geq N$, it is sufficient to estimate the term

where we used (3.9). Moreover, Lemma 2.4 via (2.2) gives

Using the fact that

we have

Thus,

By (3.9), let us estimate the term

If $|\gamma_2|\leq N$, by a similar reasoning to (3.17), (for $\alpha = 3-\sigma$), we have

It follows that

If $|\gamma_2|\geq N$, it is sufficient to estimate the term

Moreover, with Lemma 2.4 via (2.2), we write

Since $N+|\gamma_1|\leq|\gamma_1|+|\gamma_2|\leq -N+M_0+M$, we have $|\gamma_1|\leq M+M_0-2N$. So, arguing as in (3.6), we get

Then,

Taking (3.15), (3.18), (3.19), (3.21), (3.22), (3.23), and (3.24), we get

Now, taking Lemma 2.1, (3.2), (3.3), and (3.11) via the estimates of $(I)$, we write

Moreover, taking Lemma 2.1, and the estimate of $(I_3)$, for $|\alpha|\leq -N+M_0+M$, we get

Finally, $f$ is a contraction of $B_T(R)$ for small $T > 0$, and the result follows with a classical Picard argument.

4.   Global and non-global solutions

In this section, we prove Theorem 2.2 and Proposition 2.1.

4.1.   Global solutions and scattering

Let us start with the next auxiliary result.

Proposition 4.1. Let $\max\{1+\frac1N, \frac32\} < \sigma\leq2$, $\kappa\gg 1$, and $v_0\in\mathcal Y$ such that $\inf_{ \mathbb{R}^N}|v_0(x)|\geq\nu$. Then, there is a unique $v\in C([0, \frac1\kappa], \mathcal Y)$ solution to

Proof of Proposition 4.1: For simplicity and without loss of generality, let us fix $\tau = 1$. One applies a Picard fixed point argument. Let the function

on the space $B_{\nu, \frac1\kappa}(R)$. Taking (3.1), (3.3), and (3.11), via $\sigma > 1+\frac1N$, we write

Moreover, arguing as in (3.14), we write

Moreover, arguing as in (3.25), and the last lines of the previous section, we get

This implies that, for large $\kappa\gg 1$, an application of the Picard Theorem finishes the proof. □

Take the pseudo conformal transformation ^[26],

Here, $0 < T < \frac1\kappa$, and $v$, given by Proposition 4.1, is a solution to (4.1). Thus, $u$ resolves (CNLS). Now, let define $u_j^+: = e^{\frac{i\kappa|x|^2}4-\frac i\kappa\Delta}v_j(\frac1\kappa, \cdot)$. So, following [25,Section 4], we have $u_j\in C(\mathbb{R}_+, H^s)\cap L^\infty(\mathbb{R}_+\times \mathbb{R}^N, \langle y\rangle^\frac{N}2dy\, dt)$, and

4.2.   Non-global solutions

Here, one proves Proposition 2.1. Let $u\in C_T(\mathcal Y)$ be a solution to (CNLS). Thus, $m > 1+\frac N2$ implies that $\langle y\rangle^{1-m}\in L^2$, and

This implies that the solution has a finite variance $\int_{ \mathbb{R}^N}|u(t, y)|^2|x|^2\, dy.$ Let us check that the energy is well-defined. Indeed,

Thus, one can apply [18,Theorem 2.8].

5.   Conclusions

This work examines the singular coupled non-linear Schrödinger system (CNLS) with three main objectives. First, it investigates the local existence of solutions. Second, it establishes the existence of global solutions that scatter in certain Sobolev spaces. Finally, it demonstrates the existence of non-global solutions. The primary difficulty arises from the condition $\sigma < 2$, which introduces a singularity in the term $|u_j|^{\sigma-2}$ near zero. This singularity renders the classical contraction method in the energy space ineffective. This paper aims to address this gap in the literature by leveraging ideas from ^[24]. The approach highlights that the singularity issue is localized near zero, requiring the solution to avoid this region, see assumption (2.5). This is challenging because the Schrödinger equation lacks a maximum principle. The global solutions that scatter are obtained through a pseudo-conformal transformation based on the local solutions. Lastly, the existence of non-global solutions is demonstrated using the classical variance method.

Author contributions

Saleh Almuthybri and Tarek Saanouni: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest. No data-sets were generated or analyzed during the current study.