# On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Setting—Nonlinear Star Graphs

#### 1.2. The Nodal Structure

## 2. Statement of Main Theorems

**Theorem**

**1.**

- 1.
- the number of edges E is odd, or
- 2.
- E is even and$$\sqrt{\frac{{m}_{+}}{{m}_{-}}}\frac{1+{m}_{-}}{1+{m}_{+}}>\frac{E}{E-2},$$$$\begin{array}{cc}\hfill K\left({m}_{+}\right)\sqrt{1+{m}_{+}}=& \frac{\pi}{2}\phantom{\rule{0.166667em}{0ex}}\frac{{\ell}_{\frac{E}{2}+2}}{{\ell}_{1}},\hfill \\ \hfill K\left({m}_{-}\right)\sqrt{1+{m}_{-}}=& \frac{\pi}{2}\phantom{\rule{0.166667em}{0ex}}\frac{{\ell}_{\frac{E}{2}+1}}{{\ell}_{1}},\hfill \end{array}$$$$K\left(m\right)={\int}_{0}^{1}\frac{1}{\sqrt{1-{u}^{2}}\sqrt{1-m{u}^{2}}}du$$

**Theorem**

**2.**

**Corollary**

**1.**

- 1.
- E is odd, or
- 2.
- E is even and the fractions ${\tilde{\ell}}_{e}={\ell}_{e}/{n}_{e}$ ($e=1,\cdots ,E$) satisfy the condition (5),

**Corollary**

**2.**

## 3. General Background on the Solutions of Nonlinear Quantum Star Graphs

#### 3.1. The Nonlinear Interval - Solutions and Spectral Curves

**Proposition**

**1.**

- 1.
- All solutions are periodic ${\chi}_{m,k}^{(\pm )}\left(x\right)={\chi}_{m,k}^{(\pm )}(x+{\Lambda}^{(\pm )}(m,k))$ with a nonlinear wavelength$$\begin{array}{cc}\hfill {\Lambda}^{(+)}(m,k)=& \frac{4\sqrt{1+m}K\left(m\right)}{k}\hfill \\ \hfill {\Lambda}^{(-)}(m,k)=& \frac{4\sqrt{1-2m}K\left(m\right)}{k},\hfill \end{array}$$
- 2.
- For $m\to 0$ one regains the standard relation ${\Lambda}^{(\pm )}(0,k)=\frac{2\pi}{k}$ for the free linear Schrödinger equation. In the repulsive case ${\Lambda}^{(+)}(m,k)$ is an increasing function of m (at fixed k) that increases without bound as $m\to 1$. In the attractive case ${\Lambda}^{(-)}(m,k)$ is a decreasing function of m (at fixed k) with ${\Lambda}^{(-)}\left(\frac{1}{2},k\right)=0$.
- 3.
- The nodal points are separated by half the nonlinear wavelength. Namely, ${\chi}_{m,k}^{(\pm )}(n{\Lambda}^{(\pm )}(m,k)/2)=0$ for $n=0,1,,\cdots $.
- 4.
- The solutions are anti-symmetric around each nodal point and symmetric around each extremum, i.e., it has the same symmetry properties as a sine function.
- 5.
- As $\mathrm{sn}\left(K\left(m\right),m\right)=1$ and $\mathrm{dn}\left(K\left(m\right),m\right)=\sqrt{1-m}$ the amplitude$${A}^{(\pm )}(k,m)=max{\left({\chi}_{m,k}^{(\pm )}\left(x\right)\right)}_{x\ge 0}={\chi}_{m,k}^{(\pm )}\left(\frac{{\Lambda}^{(\pm )}(m,k)}{4}\right)$$$$\begin{array}{cc}\hfill {A}^{(+)}(m,k)=& k\sqrt{\frac{2m}{1+m}}\hfill \\ \hfill {A}^{(-)}(m,k)=& k\sqrt{\frac{2m}{1-2m}}.\hfill \end{array}$$
- 6.
- As $m\to {0}^{+}$ the amplitude of the solutions also decreases to zero ${A}^{(\pm )}(0,k)=0$ for both the repulsive and the attractive case. In this case the effective strength of the nonlinear interaction becomes weaker and the oscillations are closer. In the repulsive case the amplitude remains bounded as $m\to 1$ with ${A}^{(+)}(1,k)=k$. In the attractive case ${A}^{(-)}(m,k)$ grows without bound as $m\to \frac{1}{2}$.

**Proposition**

**2.**

**Corollary**

**3.**

- $\left[K\left(m\right)-E(1,m)\right]$ and $\frac{1}{m}\left[\Pi (1,m,m)-K\left(m\right)\right]$ are increasing functions of m. This follows from their integral representations (see Appendix A). Explicitly, writing each expression as an integral, the corresponding integrands are positive and pointwise increasing functions of m.
- $K\left(m\right)$ and $m(1-m)$ are also positive increasing functions of m in the relevant intervals.

**Proposition**

**3.**

#### 3.2. Nonlinear Quantum Star Graphs

#### 3.3. Nodal Edge Counting and Central Dirichlet Solutions

## 4. Proofs of Main Theorems

**Remark**

**1.**

#### 4.1. The Repulsive Case $g=1$:

**Proof**

**of**

**Theorem**

**1.**

- One may choose more negative signs than positive signs so that ${\sum}_{e}{\sigma}_{e}<0$. Then ${lim}_{k\to \infty}{f}_{\sigma}\left(k\right)=\frac{1}{2}{\sum}_{e=1}^{E}{\sigma}_{e}$ is trivially negative. The difficulty here is in showing that such a choice is consistent with ${f}_{\sigma}\left(\frac{\pi}{{\ell}_{1}}\right)>0$. This generally leads to some conditions which the edge lengths should satisfy.
- One may choose as many positive as negative signs, which makes it easier to satisfy ${f}_{\sigma}\left(\frac{\pi}{{\ell}_{1}}\right)>0$ (i.e., the conditions on the edge lengths are less restrictive). Yet, the difficulty here lies in ${lim}_{k\to \infty}{f}_{\sigma}\left(k\right)=0$, which means that one needs to show that this limit is approached from the negative side (i.e., find the conditions on the edge lengths which ensures this).

#### 4.2. The Attractive Case $g=-1$:

**Proof**

**of**

**Theorem**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Elliptic Integrals and Jacobi Elliptic Functions

## References

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**Figure 2.**The shaded regions indicate choices of relative edge lengths $1<\frac{{\ell}_{\frac{E}{2}+1}}{{\ell}_{1}}<\frac{{\ell}_{\frac{E}{2}+2}}{{\ell}_{1}}$ that satisfy the condition (5) of Theorem 1. The dashed lines indicate the boundary of regions for a star graph with E edges (where $E=4,10,50,200$). Condition (5) is satisfied below the dashed lines.

**Figure 3.**Spectral curves ${k}_{n,\ell}^{(\pm )}\left(m\right)$ for the repulsive (

**a**) and attractive case (

**b**). The n-th curve is obtained from the curve for $n=1$ by rescaling ${k}_{n,\ell}^{(\pm )}\left(m\right)=n{k}_{1,\ell}^{(\pm )}\left(m\right)$.

**Figure 4.**The spectral curves ${k}_{n,\ell}^{(\pm )}\left(N\right)$ (full green lines) in the repulsive (

**a**) and attractive case (

**b**). The n-th curve is obtained from the curve for $n=1$ by scaling ${k}_{n,\ell}^{(\pm )}\left(N\right)=n{k}_{1,\ell}^{(\pm )}(N/{n}^{2})$. The (blue) dashed lines indicate trajectories of the flow (17). The deformation parameter m is constant along the flow.

**Figure 5.**Spectral curves $k\left(N\right)$ for a nonlinear star graph with $E=3$ edges and edge lengths ${\ell}_{1}=1$, ${\ell}_{2}=\sqrt{2}$, and ${\ell}_{3}=\sqrt{3}$ with repulsive (

**a**) and attractive (

**b**) nonlinear interaction. The spectral curves have been obtained by numerically solving the matching conditions using a Newton-Raphson method. For $N\to 0$ one obtains the spectrum of the corresponding linear star graph. Apart from one curve, all shown curves are connected to the linear spectrum this way. In the attractive case one spectral curve (shown in blue) is not connected to the linear spectrum. Such curves can sometimes be found by coincidence, e.g., if one is close to a bifurcation and numerical inaccuracy allows to jump from one solution branch to another (and this is indeed how we found it). In the repulsive case there is one spectral curve that has a sharp cusp. This indicates that there may be a bifurcation nearby that has additional solution branches that have not been found. In general it is a non-trivial numerical task to ensure that a diagram of spectral curves is complete. Here, completeness has not been attempted as the picture serves a mainly illustrative purpose.

**Figure 6.**Upper panel (

**a**): Two spectral curves (green and blue) of the star graph described in the caption of Figure 5 with attractive nonlinear interaction. The yellow and pink dots indicate positions that correspond to central Dirichlet solutions. The nodal edge structure $\mathbf{n}$ is indicated for each part of the curve. The latter is constant along spectral curves apart from jumps at the positions that correspond to central Dirichlet solutions. Lower panel (

**b**): The three diagrams show how the nodal points move through the centre while N is increased through a central Dirichlet point on a spectral curve (the green curve in the upper panel). Only some nodal points close to the centre are shown. In the left diagram the three large dots are the closest to the centre and the arrows indicate how they move when N is increased. The numbers give the number of nodal domains on each edge. Increasing N further two nodal points on different edges merge at the centre as shown in the middle diagram. On the corresponding edges one nodal domain disappears. Further increasing N the nodal point moves from the centre into the remaining edge where the number of nodal domains is increased by one.

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**MDPI and ACS Style**

Band, R.; Gnutzmann, S.; Krueger, A.J.
On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs. *Symmetry* **2019**, *11*, 185.
https://doi.org/10.3390/sym11020185

**AMA Style**

Band R, Gnutzmann S, Krueger AJ.
On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs. *Symmetry*. 2019; 11(2):185.
https://doi.org/10.3390/sym11020185

**Chicago/Turabian Style**

Band, Ram, Sven Gnutzmann, and August J. Krueger.
2019. "On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs" *Symmetry* 11, no. 2: 185.
https://doi.org/10.3390/sym11020185