# Cloud-based Digital Twin for Cognitive Robotics
††thanks:  This research was partially funded by the German Research
Foundation DFG, as part of Collaborative Research Center
(Sonderforschungsbereich) 1320 Project-ID 329551904 “EASE - Everyday Activity
Science and Engineering”, University of Bremen (_http://www.ease-crc.org/_).
The research reported in this paper has been also partially supported by the
German Federal Ministry of Education and Research; Project-ID 16DHBKI047
”IntEL4CoRo”, University of Bremen.

1st Arthur Niedźwiecki  Institute for Artificial Intelligence
University of Bremen
Bremen, Germany
0009-0009-8894-3241  2nd Sascha Jongebloed  Institute for Artificial
Intelligence
University of Bremen
Bremen, Germany
0009-0006-3964-9597  3rd Yanxiang Zhan  Institute for Artificial Intelligence
University of Bremen
Bremen, Germany
0009-0007-3509-2661  4th Michaela Kümpel  Institute for Artificial
Intelligence
University of Bremen
Bremen, Germany
0000-0002-0408-3953  5th Jörn Syrbe  Institute for Artificial Intelligence
University of Bremen
Bremen, Germany
0009-0001-2308-9517  6th Michael Beetz  Institute for Artificial Intelligence
University of Bremen
Bremen, Germany
0000-0002-7888-7444

###### Abstract

The paper presents a novel cloud-based digital twin learning platform for
teaching and training concepts of cognitive robotics. Instead of forcing
interested learners or students to install a new operating system and bulky,
fragile software onto their personal laptops just to solve tutorials or coding
assignments of a single lecture on robotics, it would be beneficial to avoid
technical setups and directly dive into the content of cognitive robotics. To
achieve this, the authors utilize containerization technologies and Kubernetes
to deploy and operate containerized applications, including robotics
simulation environments and software collections based on the Robot operating
System (ROS). The web-based Integrated Development Environment JupyterLab is
integrated with RvizWeb and XPRA to provide real-time visualization of sensor
data and robot behavior in a user-friendly environment for interacting with
robotics software. The paper also discusses the application of the platform in
teaching Knowledge Representation, Reasoning, Acquisition and Retrieval, and
Task-Executives. The authors conclude that the proposed platform is a valuable
tool for education and research in cognitive robotics, and that it has the
potential to democratize access to these fields. The platform has already been
successfully employed in various academic courses, demonstrating its
effectiveness in fostering knowledge and skill development.

###### Index Terms:

Digital Twins, Remote Laboratories, Cognitive Robotics, Open Educational
Resources

##  I Introduction

The cognitive robotics research field incorporates engineering, computer
science, and psychology research to make a real robot move as intended.
Interpreting sensor data and actuating the motors in the intended way to
successfully perform tasks in the real world is a complex problem that needs
to be divided into multiple modules, for example, perception, motion planning,
reasoning over knowledge, navigation systems, trajectory calculation, and task
execution. There are expert researchers in all of these modules, and each
researcher, depending on the field, has a different perspective on how to
implement an intelligent robot. But to enable a cognitive holistic intelligent
robot it is necessary to integrate all these fields.

The complexity of such systems means that they are often difficult to
penetrate and can only be operated by experts at great expense. The
consequence of this is that there are only a few experts for cognitive robots,
usually specialized only in certain hardware, simply because the often
expensive hard- and software are not available to everyone. It is easy to
imagine that as diverse the researchers and research fields are, teaching and
training cognitive robotics is as complex as the diversity of the field and,
therefore, includes various frameworks.

Figure 1: Assignment on Pick&Place robots on the left, physics simulation on
the right, showing the robots believed world state.

We believe, to increase the number of people and collaboration in the field of
cognitive robotics, such frameworks need to be made more accessible to
research, industry and education. This includes lowering the inhibition
threshold by raising less hardware and software requirements as well as
providing intuitive tutorials that can be used by everybody.

We propose to use a _digital twin_ learning platform to teach & train concepts
of cognitive robotic frameworks. A _digital twin_ , _“the digital equivalent
to a physical product”_ [1] provides a digital equivalent of a real
laboratory, including a robot and its programmable frameworks. If such digital
twin learning platforms are made available through a cloud, the requirements
on hardware and software can be reduced tremendously, thus solving the
aforementioned problems. A digital twin also allows for a visualization of the
environments and robots. Such visualizations are essential on a robotics
learning platform, as it provides immediate visual feedback and an intuitive
and interactive way for learners to understand complex robotic behavior and
environmental interaction.

This article introduces a digital twin laboratory framework that can be
accessed by researchers and students through the web service JupyterHub [2], a
framework combining documentation and executable code for learning programming
skills. Through this programming interface, the learners interact with
leading-edge research software, pre-installed alongside the Robot Operating
System [3] (ROS). Through RVizWeb the programmer is able to visualize sensor
data like camera images and laser scans in real-time. Jupyter, ROS, AI-
applications, and RViz are wrapped in a Docker container, while Kubernetes [4]
orchestrates these containers like classroom sessions for multiple students.

In section II we describe and discuss similiar frameworks to ours. The
relevant terms and concepts are outlined in section III, while section IV is
dedicated to presenting our developed architecture. Section V focuses on
validating our contributions by applying the developed platform in educational
contexts and discussing the results. The paper concludes in section VI.

##  II Related Work

In the field of education and research, the utilization of platforms to
increase the accessibility of robotics in teaching and training garnered
considerable attention. One example for this are remote and virtual laboratory
platforms.

Remote laboratory platforms like REAL[5], SyRoTek[6] and ”Robot Programming
Network”[7] enable users to teleoperate robots either through specialized
software or via web interfaces, facilitating a hands-on learning experience
from a distance. Platforms like OpenUAV[8] or RobUALab[9] also offer a virtual
laboratory with the capability to run simulations, either locally on the
user’s machine or remotely on a server. While this conventional teleoperation
offers several advantages, our work focuses on simulations on the cloud, which
reduces the limitations imposed by physical hardware availability and
maintenance to enhance accessibility and scalability. Furthermore our platform
integrates widely recognized physics simulations, such as PyBullet, ensuring a
robust and realistic simulation environment.

Similar to our framework, the system developed by Avila et. al. also relies on
ROS as its core robotics framework and is designed for instruction in AI
software for robots[10]. Concurrently, Zhang et. al. have introduced an online
platform that facilitates learning by allowing the execution of code through
IPython kernels[11], an interactive Python interpreter. This platform,
although restricted to Python’s visualization capabilities, demonstrates a
platform similar to ours in educational settings. Lumpp et al. present a
methodology for programming robotic tasks using Docker and Kubernetes,
focusing on containerization and orchestration in ROS-based applications[12].
They successfully demonstrate the approach with a case study involving a
Robotnik RB-Kairos robot in an industrial setting.

The idea of using Digital Twins is well-established in the industry. For
example, Intrinsics Flowstate [13] is using Digital Twins to develop concepts
of industrial robots on a web-based construction platform with a graphical
programming interface.

Among existing platforms utilizing a digital-twin-like photo-realistic
simulation, ’The Construct’[14] stands out as the most similar to the in this
article proposed framework, offering a cloud-based learning environment
complete with visualization tools and simulation. However, contrary to ’The
Construct’, our platform is built on open-source software, providing
unrestricted and free access to our educational & training resources. This
openness is a fundamental aspect of our approach aimed at democratizing
learning in robotics. Furthermore, our approach exceeds the basics of ROS to
teach holistic, cognition-enabled autonomous systems.

##  III Background

The following chapter describes the terms and concepts used to explain the
proposed architecture.

### Semantic Digital Twin

In this study, we explore the application of _semantic Digital Twins_
(semDTs), which are semantically enhanced virtual models of real environments
and their associated entities [15], such as laboratories or production
settings that can autonomously be created by robots as described in [16].
Digital twins have garnered significant interest in the research community.
Our focus is on leveraging these virtual representations, particularly of
robotic laboratories, including the objects and robots within, for educational
purposes, like the pick & place lesson in figure 1. Such a setup democratizes
access to advanced robotics education, making it more inclusive and far-
reaching.

Figure 2: The cloud service framework built on BinderHub [17] initiates a new
docker container for each connecting student. The Figure 3 showcasing the
robot software framework within the container.

### Cognitive Robotics

Concepts of cognitive robots are best described by Sandini, Vernon and Sciutti
in [18]:
”The word cognition derives from the Latin verb cognoscere, a composition of
con (meaning related to) and gnoscere (to know). Cognitive robotics, then, is
the branch of robotics where knowledge plays a central role in supporting
action selection, execution, and understanding. […]”
The complexity of decision-making invites a broad field of research, from
computer science over (electrical) engineering to psychology and philosophy.

### Varying Environments and Robot Platforms

Robots and their environments are versatile. Since a mobile robot can be
applied in different environments, the _digital twin_ must be flexible too. A
robot focused on Human-Robot-Interaction (HRI) may be better suited for an
everyday environment than a search-and-rescue robot. Dynamically exchanging
the robot and its environment allows the researcher to tailor their
experiments to their needs.

### Cloud-Based

Cloud-based services leverage remote computing resources and storage, offering
significant reproducibility, flexibility, and scalability for education and
research. The rapid development of AI technologies like machine learning,
which require substantial computational resources, has given rise to open-
source cloud service solutions in recent years. These state-of-the-art
solutions are well-suited for building cloud-based robot simulation services
that equally require heavy computational power. Educators, students, and
researchers can rapidly build and test various robot algorithms and control
methods without the need to establish and maintain complex simulation
environments locally. In this study, we focus on offering a Platform-as-a-
Service (PaaS)[19] that allows educators and researchers to publish various
Software-as-a-Service (SaaS), such as tutorials, coding assignments for
courses, and robotics competition arenas.

##  IV Proposed Architecture

The following chapter explains the components of the cloud-based _digital
twin_ architecture.

Figure 3: To control a real-world robot, ROS gives commands to it. ROS is
running in the Docker and is loaded with AI Applications, e.g. the Cognitive
Robot Abstract Machine (CRAM). The Digital Twin Simulation, a 3D game engine,
is visualized in Jupyter via RVizWeb and/or a lightweight display for visual
applications. Jupyter Notebooks run code to control the AI Applications, while
a Terminal allows for direct access to the ROS processes.

### Containerized Application

Containerized applications use containerization technologies to package the
application software along with necessary runtime libraries and configuration
files into one executable process. The presented approach uses Docker [20].
This speeds up the deployment of software services across different computing
environments without concerning the underlying hardware and operating systems.
Each container is an isolated Linux environment, ensuring that each
application functions without being affected by others, and behaves
consistently.

Many open-source robot research software, such as ROS and Gazebo, are
developed based on Linux systems, leading to the widespread of
containerization solutions in the field of robotics research. Figure 3 depicts
a containerized application we constructed for robotics learning. It contains
Robot Operating System (ROS)[3], our research software, an web Integrated
Development Environment (JupyterLab [21]) and several visualization tools
(RvizWeb, XPRA desktop). With this containerized application, students can
launch a full-stack learning environment on their personal PC quickly and
control robots within the Local-Area-Network through the web-based interface.

Figure 4: ROSBoard [22] widget in Jupyter to visualize laser scan (top left),
the front camera feed (top mid) and the recorded map (top right). Gazebo
Physics Simulator (bottom left) and RViz visualization of the environment map
(bottom right) on the XPRA VNC virtual desktop, navigating the Tortugabot
through the _digital twin_ on the green path.

### Into the Cloud

The pre-built containerized application is ready to run locally as well as to
deploy on the cloud server. Our chosen cloud architecture is from the open-
source project BinderHub, an open source web service that lets users create
sharable, interactive, reproducible environments in the cloud [17]. The
underlying container orchestration system is Kubernetes, the de facto standard
for deploying and operating containerized applications. Therefore, this setup
can runs on major cloud platforms and self-hosted Linux servers. Additionally,
it provides way to adjust computational resources without disrupting running
services by adding or removing machines from the cluster.

BinderHub is a platform designed for open-source workflows. To create a
robotics containerized application, users only need to add an environment
configuration file, usually a ”Dockerfile,” to their publicly available git
repository. BinderHub builds a docker image accordingly and returns a URL of
the application. This is equivalent to deploying a Software-as-a-Service
(SaaS). Moreover, whenever there are updates in the code repository, BinderHub
automatically rebuilds the application. Users can also build applications for
different git branches or specific commits, thereby releasing different
versions. With returned URL, anyone can use the published containerized
application in a web browser. Each click on the URL creates a separate docker
container for the current user.

Figure 3 illustrates the usecase of creating a virtual classroom using this
cloud platform. The lecturer initially creates a GitHub repository to store
learning materials and a Dockerfile of the runtime environment. Submitting the
repository address to BinderHub’s home page generates a URL for the virtual
classroom. Simultaneously, the cloud server builds a docker image in the
background. Once the building is complete, the lecturer distrubute the URL to
the student. Hence learners can start the course immediately without
installing any special software.

### Visualization

Visualization is imperative when teaching robotics to enable learners’
volition through visual feedback. Live visualization of robot simulation in
the web browser is not trivial. To visualize a _digital twin_ , a PC shows a
3D simulation on the display. A website can not simply visualize these
programs, but there are ways to make it happen.

The most popular visualization software in ROS is Rviz, it shows camera
images, depth clouds, lasers, and most importantly, the current configuration
of the robot. Jupyter is extended with RvizWeb, a plugin that reimplements
Rviz for Jupyter and integrates into the web environment. ROSBoard is another
plugin to Jupyter that provides visualization for specific ROS datastructures,
like real-time laser scans, camera feeds and ground maps, as depicted in
figure 4 in the top.

A later addition to the Jupyter platform is a virtual display, such that
programs with a graphical user interface can be shown in the browser. For
physics simulation ROS offers Gazebo. In figure 4 both Gazebo and Rviz are
running in the online Docker container, visualized to the XPRA display via
x-forwarding.

##  V Application

With the opportunity to dive directly into a complicated architecture, it is
possible to teach how to control robots, visualize sensor data and learn about
general AI techniques in robotics. As the foundational platform we have used
this technology to teach robotics at the Fall School [23] since 2021, at IROS
2023 [24] and in academic courses.

### Knowledge Representation and Reasoning

In our research, we have evaluated the usage of our system in teaching
Knowledge Representation and Reasoning (KRR) for cognitive robots. This
evaluation was conducted through its implementation in various university
courses and Fall Schools. Our platform allows users to interact directly with
the system using a knowledge representation querying language. One example of
a KRR software we were teaching is KnowRob[25]. KnowRob is a knowledge
processing software for cognitive robots, which utilizes a Prolog-like
language for queries. To effectively teach this querying language, we
integrated our notebooks into courses such as ”KI basierte Robotersteuerung”
(”Ai-based robotic control”) in 2022 and 2023.

### Knowledge Acquisition and Retrieval

In addition to a basic KRR course to teach how knowledge can be represented
for the use in cognitive robots, we have also created a course on Web
Knowledge Acquisition and its Retrieval by robotic agents that has been
implemented in Fall Schools and will be extended and implemented in Workshop
Tutorials. The semDT platform allows users to directly access Web content and
use it to extend the robot knowledge base. The course also shows how robotic
agents can access the knowledge base and reason about the contained knowledge.
The notebooks will be integrated into courses in a new AI masters program.

### Task-Executives

Separate ROS applications can implement the functionality for navigation,
image processing, knowledge representation and motion planning. The Cognitive
Robot Abstract Machine (CRAM)[26] is given a goal, like preparing a breakfast
table and cleaning it up afterwards [27]. From this rather general description
of a goal, the task executive gathers information about the environment, uses
image processing to identify objects and queries the knowledge base for how to
pick up the desired object, and navigating to the breakfast table, to finally
acquire how and where to place it down without hitting any other items on the
table. Recurring lectures on Task Executives are found at the Fall School [23]
and the graduate course ”Robot Programming with ROS” [28].

### Discussion

Prior to the cloud-based technology we offered VirtualBox [29] virtual
machines (VM) with a full desktop environment, ROS and AI applications already
installed. They were too large to download quickly, but served the broadest
toolset for the individual student. Since we use RvizWeb, XPRA and ROSBoard we
have visualization of the Docker image, so it can run in the web. Kubernetes
was introduced to manage multiple clients. Running simulations requires a lot
of computation time: we guarantee 8 GB of RAM per client session, more is
needed for heavier digital twins, e.g. an Unreal Engine.

While hands-on tutorials with VirtualBox were unfeasible due to the VM size,
offering the web platform, our proposed architecture, kept most attendees
participating. At the Fall School 2023 we recorded the highest ratio in
participation to attendance over the past years, thanks to the ease of access.

### Future Work

To enable interested learners to get into the field of cognitive robotics we
are creating courses and tutorials. While our focus is currently on higher
education and in-service training, secondary education is planned.

While our server runs Kubernetes to orchestrate new students launching the
digital-twin environment, we were looking into two other platforms that
provide computational resources for open-source software. Github Codespaces
[30] fully supports running our robot application, but the cloud computing
resources are strictly limited. Google Colab [31] provides robust
computational resources, including GPUs, yet imposes restrictions on the
WebSocket protocol, which renders our visualization tool dysfunctional.

##  VI Conclusion

Building containerized applications is critical for cloud-based robot systems
and is a fundamental task of cloud solutions in many other research domains.
Within this setup, a significant challenge is the real-time rendering of 3D
robot digital twin environments and displaying them on a web-based client to
achieve platform independent accessibility.

Such a cloud-based and independent digital twin learning & training platforms
solves the aforementioned setup and hardware problems.

Tech giants like Amazon and NVIDIA have launched their robot simulation cloud
services, providing much more computational resources and cutting-edge
technologies. However, our focus remains on free and open-source solutions
tailored for the education and research domains, and to encourage more
researchers, developers, and educators to engage in robotics.

## References

  * [1] M. Grieves, _Virtually perfect: Driving innovative and lean products through product lifecycle management_. Space Coast Press Cocoa Beach, 2011, vol. 11.
  * [2] The Jupyter Trademark. Project Jupyter. Accessed 2024-03-13. [Online]. Available: https://jupyter.org
  * [3] M. Quigley, K. Conley, B. Gerkey, J. Faust, T. Foote, J. Leibs, R. Wheeler, A. Y. Ng, _et al._ , “Ros: an open-source robot operating system,” in _ICRA workshop on open source software_ , vol. 3, no. 3.2. Kobe, Japan, 2009, p. 5.
  * [4] The Kubernetes Authors. Kubernetes. Accessed 2024-03-13. [Online]. Available: https://kubernetes.io/
  * [5] E. G. Guimarães, E. G. Guimarães, A. Maffeis, J. A. L. Pereira, J. Pereira, B. Russo, E. N. Cardozo, E. Cardozo, M. Bergerman, M. F. Magalhães, and M. F. Magalhães, “Real: a virtual laboratory for mobile robot experiments,” _IEEE Transactions on Education_ , 2003.
  * [6] M. Kulich, M. Kulich, J. Chudoba, J. Chudoba, K. Košnar, T. Krajnik, J. Faigl, and L. Preucil, “Syrotek—distance teaching of mobile robotics,” _IEEE Transactions on Education_ , 2013.
  * [7] G. Casañ, E. Cervera, A. Abou Moughlbay, J. Alemany, and P. Martinet, “Ros-based online robot programming for remote education and training,” _Proceedings - IEEE International Conference on Robotics and Automation_ , vol. 2015, pp. 6101–6106, 06 2015.
  * [8] M. Schmittle, M. Schmittle, M. Schmittle, A. Lukina, A. Lukina, L. Vacek, L. Vacek, J. Das, C. P. Buskirk, S. Rees, J. Sztipanovits, R. Grosu, V. Kumar, and V. Kumar, “Openuav: a uav testbed for the cps and robotics community,” 2018.
  * [9] C. A. Jara, F. A. Candelas, F. A. Candelas, S. T. Puente, and F. Torres, “Hands-on experiences of undergraduate students in automatics and robotics using a virtual and remote laboratory,” 2011.
  * [10] E. A. Avila, D. P. Chapa, I. D. Arenas, and C. V. Hurtado, “A digital twin implementation for mobile and collaborative robot scenarios for teaching robotics based on robot operating system,” in _2022 IEEE Global Engineering Education Conference (EDUCON)_ , 2022, pp. 559–564.
  * [11] A. Zhang, Z. C. Lipton, M. Li, and A. J. Smola, “Dive into deep learning,” 2023\.
  * [12] F. Lumpp, M. Panato, F. Fummi, and N. Bombieri, “A container-based design methodology for robotic applications on kubernetes edge-cloud architectures,” _2021 Forum on specification & Design Languages (FDL)_, pp. 01–08, 2021.
  * [13] Intrinsic. Flowstate Intrinsic. Accessed 2024-03-13. [Online]. Available: https://intrinsic.ai/flowstate/
  * [14] R. A. Téllez, “A thousand robots for each student: Using cloud robot simulations to teach robotics,” in _Robotics in Education_ , 2016. [Online]. Available: https://api.semanticscholar.org/CorpusID:19140247
  * [15] M. Kümpel, C. A. Mueller, and M. Beetz, _Semantic Digital Twins for Retail Logistics_. Springer International Publishing, 2021, pp. 129–153. [Online]. Available: https://doi.org/10.1007/978-3-030-88662-2˙7
  * [16] M. Beetz, S. Stelter, D. Beßler, K. Dhanabalachandran, M. Neumann, P. Mania, and A. Haidu, _Robots Collecting Data: Modelling Stores_. Cham: Springer International Publishing, 2022, pp. 41–64. [Online]. Available: https://doi.org/10.1007/978-3-031-06078-6˙2
  * [17] B. Ragan-Kelley, C. Willing, F. Akici, D. Lippa, D. Niederhut, and M. Pacer, “Binder 2.0-reproducible, interactive, sharable environments for science at scale,” in _Proceedings of the 17th python in science conference_. F. Akici, D. Lippa, D. Niederhut, and M. Pacer, eds., 2018, pp. 113–120.
  * [18] G. Sandini, A. Sciutti, and D. Vernon, _Cognitive Robotics_. Berlin, Heidelberg: Springer Berlin Heidelberg, 2020, pp. 1–7. [Online]. Available: https://doi.org/10.1007/978-3-642-41610-1˙198-1
  * [19] F. Doelitzscher, A. Sulistio, C. Reich, H. Kuijs, and D. Wolf, “Private cloud for collaboration and e-learning services: from iaas to saas,” _Computing_ , vol. 91, pp. 23–42, 2011.
  * [20] D. Merkel _et al._ , “Docker: lightweight linux containers for consistent development and deployment,” _Linux j_ , vol. 239, no. 2, p. 2, 2014.
  * [21] J. M. Perkel, “Why jupyter is data scientists’ computational notebook of choice,” _Nature_ , vol. 563, no. 7732, pp. 145–147, 2018.
  * [22] W. Vollprecht. Jupyter-ROS. Accessed 2024-03-13. [Online]. Available: https://jupyter-ros.readthedocs.io
  * [23] EASE, University of Bremen. EASE Fall School 2023. Accessed 2024-03-13. [Online]. Available: https://ease-crc.org/fall-school-2023/
  * [24] ——. Everyday Activity robot manipulation in an interactive learning environment. Accessed 2024-03-13. [Online]. Available: https://ease-crc.org/teaching-cognition-enabled-cognitive-robotics-in-an-integrated-learning-environment
  * [25] M. Beetz, D. Beßler, A. Haidu, M. Pomarlan, A. K. Bozcuoğlu, and G. Bartels, “Know rob 2.0 — a 2nd generation knowledge processing framework for cognition-enabled robotic agents,” in _2018 IEEE International Conference on Robotics and Automation (ICRA)_ , 2018, pp. 512–519.
  * [26] M. Beetz, G. Kazhoyan, and D. Vernon, “The cram cognitive architecture for robot manipulation in everyday activities,” 2023.
  * [27] G. Kazhoyan, S. Stelter, F. K. Kenfack, S. Koralewski, and M. Beetz, “The robot household marathon experiment,” in _IEEE International Conference on Robotics and Automation (ICRA)_ , 2021, accepted for publication. [Online]. Available: https://arxiv.org/abs/2011.09792
  * [28] A. Niedzwiecki. Course: Robot Programming with ROS. Accessed 2024-03-13. [Online]. Available: https://ai.uni-bremen.de/teaching/cs-ros-ws23
  * [29] Oracle. Oracle VM VirtualBox. Accessed 2024-03-13. [Online]. Available: https://www.virtualbox.org/
  * [30] GitHub Inc. GitHub Codespaces. Accessed 2024-03-13. [Online]. Available: https://github.com/features/codespaces
  * [31] Google. Welcome to Colaboratory. Accessed 2024-03-13. [Online]. Available: https://colab.research.google.com/


$\displaystyle=\begin{cases}-\frac{1}{2}\alpha-\frac{1}{4}\left(2+\beta-\alpha\right)t,&\xi\leq
0,\\\ \frac{1}{2}\xi
t-\frac{1}{2}\alpha-\frac{1}{4}\left(2+\beta-\alpha\right)t,&0<\xi\leq\beta,\\\
\frac{1}{4}\left(t+2\right)\xi-\frac{1}{2}(\beta+\alpha)-\frac{1}{4}(2-\alpha)t,&\beta<\xi\leq
2+\beta,\\\
\frac{1}{4}\left(1-\alpha\right)\left(t-2\right)\xi+\frac{1}{2}(4+\beta-3\alpha-\alpha\beta)\\\
\quad-\frac{1}{4}(2-3\alpha-\alpha\beta)t,&2+\beta<\xi\leq 4+\beta,\\\
\frac{1}{2}\alpha+\frac{1}{4}\left(2+\beta-\alpha\right)t,&4+\beta<\xi.\end{cases}$

Thus, we find for $t\geq 2$ the following Eulerian solution

$\displaystyle u(t,x)$
$\displaystyle=\begin{cases}-\frac{1}{2}\alpha-\frac{1}{4}\left(2+\beta-\alpha\right)t,&x\leq
x_{1}(t),\\\ \frac{4x-2\alpha+\alpha t}{2t},&x_{1}(t)<x\leq x_{2}(t)\\\
\frac{4x-4\alpha-(2-\beta-2\alpha)t}{2(t+2)},&x_{2}(t)<x\leq x_{3}(t),\\\
\frac{4x-8-(2+\beta)t}{2(t-2)},&x_{3}(t)<x\leq x_{4}(t),\\\
\frac{1}{2}\alpha+\frac{1}{4}(2+\beta-\alpha)t,&x_{4}(t)<x,\end{cases}$
$\displaystyle F(t,x)$ $\displaystyle=\begin{cases}0,&x\leq x_{1}(t),\\\
\frac{8x-4\alpha+4\alpha t+(2+\beta-\alpha)t^{2}}{2^{2}},&x_{1}(t)<x\leq
x_{2}(t),\\\
\frac{8x+4(2\beta-\alpha)+4(2\beta+\alpha)t+(2+\beta-\alpha)t^{2}}{2\left(t+2\right)^{2}},&x_{2}(t)<x\leq
x_{3}(t),\\\
\frac{8x+4(2\beta-\alpha)-4(4+2\beta-\alpha)t+(2+\beta-\alpha)t^{2}}{2\left(t-2\right)^{2}},&x_{3}(t)<x\leq
x_{4}(t),\\\ 2+\beta-\alpha,&x_{4}(t)<x,\end{cases}$

where we introduced

$\displaystyle x_{1}(t)$ $\displaystyle=\frac{1}{2}\alpha-\frac{1}{2}\alpha
t-\frac{1}{8}(2+\beta-\alpha)t^{2},$ $\displaystyle x_{2}(t)$
$\displaystyle=\frac{1}{2}\alpha-\frac{1}{2}\alpha
t-\frac{1}{8}(2-\beta-\alpha)t^{2},$ $\displaystyle x_{3}(t)$
$\displaystyle=\frac{1}{2}(2+\alpha)+\frac{1}{2}(2-\alpha)t+\frac{1}{8}(\beta+\alpha)t^{2},$
$\displaystyle x_{4}(t)$
$\displaystyle=\frac{1}{2}(4-\alpha)+\frac{1}{2}\alpha
t+\frac{1}{8}(2+\beta-\alpha)t^{2}.$

###  A.2. Cusp initial data

Consider the initial data given in Example 5.2. We observe that
$u_{0,x}(x)\rightarrow-\infty$ as $x\uparrow 0$. Thus wave breaking occurs at
$x=0$ initially. However, there is only an infinitesimal amount of energy
concentrating.

Applying $L$ from Definition 2.3, we find the following Lagrangian initial
data

$\displaystyle y_{0}(\xi)$ $\displaystyle=\begin{cases}\xi,&\xi\leq-1,\\\
\bar{y}(\xi),&-1<\xi\leq\frac{11}{3},\\\
\xi-\frac{8}{3},&\frac{11}{3}<\xi,\end{cases}$ $\displaystyle U_{0}(\xi)$
$\displaystyle=\begin{cases}1,&\xi\leq-1,\\\
\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{2}{3}},&-1<\xi\leq\frac{11}{3},\\\
1,&\frac{11}{3}<\xi,\end{cases}$ $\displaystyle V_{0}(\xi)$
$\displaystyle=\begin{cases}0,&\xi\leq-1,\\\
\frac{4}{3}\left(\operatorname{sgn}(\bar{y}(\xi))\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+1\right),&-1<\xi\leq\frac{11}{3},\\\
\frac{8}{3},&\frac{11}{3}<\xi,\end{cases}$

where $\bar{y}(\xi)$ is implicitly given by

(A.1)
$\bar{y}(\xi)+\frac{4}{3}\left(1+\operatorname{sgn}(\bar{y}(\xi))\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}\right)=\xi.$

In particular, $\bar{y}$ is continuous and strictly increasing on
$[-1,\frac{11}{3}]$ and

$\bar{y}(-1)=-1,\quad\bar{y}\left(\frac{4}{3}\right)=0,\quad\text{~and
}\quad\bar{y}\left(\frac{11}{3}\right)=1.$

The wave breaking function, given by (2.4), reads

(A.2) $\displaystyle\tau(\xi)$
$\displaystyle=-2\frac{y_{0,\xi}}{U_{0,\xi}}(\xi)=\begin{cases}3\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}},&\xi\in(-1,\frac{4}{3}],\\\
\infty,&\text{otherwise.}\end{cases}$

Note that to each $t\in(0,3]$, there exists exactly one
$\xi(t)\in(-1,\frac{4}{3}]$ such that $\tau(\xi)=t$, since $\bar{y}$ is
strictly increasing on $[-1,\frac{4}{3}]$. Furthermore, setting $t=\tau(\xi)$
and rewriting (A.2), we find

(A.3) $\displaystyle y_{0}\left(\xi(t)\right)$
$\displaystyle=\bar{y}\left(\xi\left(t\right)\right)=-\left(\frac{t}{3}\right)^{3}.$

Thus, combining (A.1) and (A.3), the wave breaking curve $\xi(t)$ in
Lagrangian coordinates is given by

(A.4) $\displaystyle\xi(t)$
$\displaystyle=-\left(\frac{t}{3}\right)^{3}+\frac{4}{3}\left(1-\frac{t}{3}\right).$

We now compute $V(t,\xi)$ for $0\leq t\leq 3$. The energy is unchanged for the
characteristics labeled by $\xi<\xi(t)$, as along these no wave breaking has
occurred yet. In particular, this means that $V(t,\xi)=V_{0}(\xi)$ for
$\xi\in(-\infty,\xi(t))$.

For $\xi\in[\xi(t),\frac{4}{3}]$, we find, using (2.3c),

$\displaystyle V(t,\xi)$
$\displaystyle=V_{0}(\xi)-\alpha\int_{\xi(t)}^{\xi}V_{0,\xi}(\eta)d\eta$
$\displaystyle=V_{0}(\xi)-\alpha\int_{\xi(t)}^{\xi}\frac{4}{9}\bigl{|}\bar{y}(\eta)\bigr{|}^{-\frac{2}{3}}\bar{y}_{\xi}(\eta)d\eta$
$\displaystyle=\frac{4}{3}\bigg{(}-\big{(}1-\alpha\big{)}\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+1-\frac{1}{3}\alpha
t\bigg{)}.$

Since $V(t,\frac{4}{3})=V_{0}(\frac{4}{3})-\frac{4}{9}\alpha t$ and no wave
breaking occurs for $\xi\in(\frac{4}{3},\infty)$, one has
$V(t,\xi)=V_{0}(\xi)-\frac{4}{9}\alpha t$ for $\xi\in(\frac{4}{3},\infty)$. To
summarize, for $0\leq t\leq 3$,

(A.5) $\displaystyle V(t,\xi)$ $\displaystyle=\begin{cases}0,&\xi\leq-1\\\
\frac{4}{3}\big{(}-|\bar{y}(\xi)|^{\frac{1}{3}}+1\big{)},&-1<\xi<\xi(t),\\\
\frac{4}{3}\big{(}-(1-\alpha)\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+1-\frac{1}{3}\alpha
t\big{)},&\xi(t)\leq\xi\leq\frac{4}{3},\\\
\frac{4}{3}\big{(}|\bar{y}(\xi)|^{\frac{1}{3}}+1-\frac{1}{3}\alpha
t\big{)},&\frac{4}{3}<\xi\leq\frac{11}{3},\\\
\frac{4}{3}\big{(}2-\frac{1}{3}\alpha t\big{)},&\frac{11}{3}<\xi.\end{cases}$

Next we compute $U(t,\xi)$ and $y(t,\xi)$ for $t\in[0,3]$, using (2.3a) and
(2.3b). We only show the details for $\xi\in(-1,\frac{4}{3}]$. For
$\xi\in(-1,\xi(t)]$, combining (2.3) and (A.5) yields

(A.6) $\displaystyle U(t,\xi)$
$\displaystyle=\left(-\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+\frac{t}{3}\right)^{2}-\frac{1}{18}\left(2-\alpha\right)t^{2},$

and

(A.7) $\displaystyle y(t,\xi)$
$\displaystyle=\left(-\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+\frac{t}{3}\right)^{3}-\frac{1}{54}\big{(}2-\alpha)t^{3}.$

For $\xi\in[\xi(t),\frac{4}{3}]$ wave breaking has already occured. Thus,
instead of integrating (2.3) from $0$ to $t$, we integrate from $\tau(\xi)$ to
$t$. Furthermore, as $U$ and $y$ are continuous in time, we get from (A.6) and
(A.7) that

$U(\tau(\xi),\xi)=-\frac{1}{18}(2-\alpha)\tau^{2}(\xi),\quad\text{ and }\quad
y(\tau(\xi),\xi)=-\frac{1}{54}(2-\alpha)\tau^{3}(\xi).$

Consequently, for $\xi\in[\xi(t),\frac{4}{3}]$, we find

$\displaystyle U(t,\xi)$
$\displaystyle=-\frac{1}{18}(2-\alpha)\tau^{2}(\xi)-\frac{2}{3}\int_{\tau(\xi)}^{t}\left((1-\alpha)\left|\bar{y}(\xi)\right|^{\frac{1}{3}}+\frac{1}{6}\alpha
s\right)ds$
$\displaystyle=\frac{1}{9}(1-\alpha)\left(\tau(\xi)-t\right)^{2}-\frac{1}{18}(2-\alpha)t^{2},$

where we used that
$\left|\bar{y}(\xi)\right|^{\frac{1}{3}}=\frac{1}{3}\tau(\xi)$ by (A.2). For
$y(t,\xi)$, we obtain

$\displaystyle y(t,\xi)$
$\displaystyle=-\frac{1}{54}(2-\alpha)\tau^{3}(\xi)+\int_{\tau(\xi)}^{t}\left(\frac{1}{9}(1-\alpha)\left(\tau(\xi)-s\right)^{2}-\frac{1}{18}(2-\alpha)s^{2}\right)ds$
$\displaystyle=-\frac{1}{54}\left(2-\alpha\right)t^{3}-\frac{1}{27}(1-\alpha)\left(\tau(\xi)-t\right)^{3}.$

Following the same lines for all remaining $\xi$, yields for $t\in[0,3)$,

(A.8a) $\displaystyle U(t,\xi)$
$\displaystyle=\begin{cases}1-\frac{2}{3}t+\frac{1}{18}\alpha
t^{2},&\xi\leq-1,\\\
\left(-\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+\frac{t}{3}\right)^{2}-\frac{1}{18}\left(2-\alpha\right)t^{2},&-1<\xi<\xi(t),\\\
\frac{1}{9}(1-\alpha)\left(3\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}-t\right)^{2}-\frac{1}{18}(2-\alpha)t^{2},&\xi(t)\leq\xi\leq\frac{4}{3},\\\
\left(\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+\frac{t}{3}\right)^{2}-\frac{1}{18}\left(2+\alpha\right)t^{2},&\frac{4}{3}<\xi\leq\frac{11}{3},\\\
1+\frac{2}{3}t-\frac{1}{18}\alpha t^{2},&\frac{11}{3}<\xi,\end{cases}$ (A.8b)
$\displaystyle y(t,\xi)$
$\displaystyle=\begin{cases}\xi+t-\frac{1}{3}t^{2}+\frac{1}{54}\alpha
t^{3},&\xi\leq-1,\\\
\left(-\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+\frac{t}{3}\right)^{3}-\frac{1}{54}\left(2-\alpha\right)t^{3},&-1<\xi<\xi(t),\\\
-\frac{1}{27}\left(1-\alpha\right)\left(3\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}-t\right)^{3}-\frac{1}{54}(2-\alpha)t^{3},&\xi(t)\leq\xi\leq\frac{4}{3},\\\
\left(\bigl{|}\bar{y}(\xi)\bigr{|}^{\frac{1}{3}}+\frac{t}{3}\right)^{3}-\frac{1}{54}(2+\alpha)t^{3},&\frac{4}{3}<\xi\leq\frac{11}{3},\\\
\xi-\frac{8}{3}+t+\frac{1}{3}t^{2}-\frac{1}{54}\alpha
t^{3},&\frac{11}{3}<\xi,\end{cases}$

where $\xi(t)$ is given by (A.4).

A closer look at the above expressions reveals that it is possible to compute
$u(t,x)$ and $F(t,x)$ with the help of Definition 2.4. We will only detail
this step for $\xi\in(-1,\frac{4}{3}]$.

For $\xi\in(-1,\xi(t)]$, we find by rearranging, (A.8b)

$-\left|\bar{y}(\xi)\right|^{\frac{1}{3}}+\frac{t}{3}=\left(y(t,\xi)+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{1}{3}},$

which implies, recalling (A.8a) and (A.5),

$\displaystyle u(t,y(t,\xi))$
$\displaystyle=U(t,\xi)=\left(y(t,\xi)+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{2}{3}}-\frac{1}{18}(2-\alpha)t^{2},$
$\displaystyle F(t,y(t,\xi))$
$\displaystyle=V(t,\xi)=\frac{4}{3}\left(\left(y(t,\xi)+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{1}{3}}+1-\frac{t}{3}\right).$

Thus,

$\displaystyle u(t,x)$
$\displaystyle=\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{2}{3}}-\frac{1}{18}(2-\alpha)t^{2},$
$\displaystyle F(t,x)$
$\displaystyle=\frac{4}{3}\left(\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{1}{3}}+1-\frac{t}{3}\right),$

for $x\in(y(t,-1),y(t,\xi(t))]=(-1+t-\frac{1}{3}t^{2}\frac{1}{54}\alpha
t^{3},-\frac{1}{54}(2-\alpha)t^{3}]$, where we inserted (A.3) into (A.8b) to
determine the right endpoint of this interval.

For $\xi\in(\xi(t),\frac{4}{3}]$, we obtain, using (A.8b),

$\displaystyle(\tau(\xi)-t)$
$\displaystyle=-3\left(\frac{1}{\left(1-\alpha\right)}\left(y(t,\xi)+\frac{1}{54}(2-\alpha)t^{3}\right)\right)^{\frac{1}{3}},$

which yields, following the same lines as before,

$\displaystyle u(t,x)$
$\displaystyle=\left(1-\alpha\right)^{\frac{1}{3}}\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{2}{3}}-\frac{1}{18}(2-\alpha)t^{2},$
$\displaystyle F(t,x)$
$\displaystyle=\frac{4}{3}\left(\big{(}1-\alpha)^{\frac{2}{3}}\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{1}{3}}+1-\frac{t}{3}\right),$

for
$x\in(y(t,\xi(t)),y(t,\frac{4}{3})]=(-\frac{1}{54}(2-\alpha)t^{3},-\frac{1}{54}\alpha
t^{3}]$.

Considering all $\xi\in\mathbb{R}$, the exact solution in Eulerian coordinates
for $t\in[0,3)$ is given by

$\displaystyle u(t,x)$
$\displaystyle=\begin{cases}1-\frac{2}{3}t+\frac{1}{18}\alpha t^{2},&x\leq
x_{1}(t),\\\
\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{2}{3}}-\frac{1}{18}(2-\alpha)t^{2},&x_{1}(t)<x\leq
x_{2}(t),\\\
\left(1-\alpha\right)^{\frac{1}{3}}\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{2}{3}}-\frac{1}{18}(2-\alpha)t^{2},&x_{2}(t)<x\leq-\frac{1}{54}\alpha
t^{3},\\\
\left(x+\frac{1}{54}(2+\alpha)t^{3}\right)^{\frac{2}{3}}-\frac{1}{18}(2+\alpha)t^{2},&-\frac{1}{54}\alpha
t^{3}<x\leq x_{3}(t),\\\ 1+\frac{2}{3}t-\frac{1}{18}\alpha
t^{2},&x_{3}(t)<x,\end{cases}$ $\displaystyle F(t,x)$
$\displaystyle=\begin{cases}0,&x\leq x_{1}(t),\\\
\frac{4}{3}\left(\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{1}{3}}+1-\frac{t}{3}\right),&x_{1}(t)<x\leq
x_{2}(t),\\\
\frac{4}{3}\left((1-\alpha)^{\frac{2}{3}}\left(x+\frac{1}{54}(2-\alpha)t^{3}\right)^{\frac{1}{3}}+1-\frac{t}{3}\right),&x_{2}(t)<x\leq-\frac{1}{54}\alpha
t^{3},\\\
\frac{4}{3}\left(\left(x+\frac{1}{54}(2+\alpha)t^{3}\right)^{\frac{1}{3}}+1(1+\alpha)\frac{t}{3}\right),&-\frac{1}{54}\alpha
t^{3}<x\leq x_{3}(t),\\\ \frac{8}{3}-\frac{4}{9}\alpha
t,&x_{3}(t)<x,\end{cases}$

where we introduced

$\displaystyle x_{1}(t)$
$\displaystyle=-1+t-\frac{1}{3}t^{2}+\frac{1}{54}\alpha t^{3},~$
$\displaystyle x_{2}(t)$ $\displaystyle=-\frac{1}{54}(2-\alpha)t^{3},$
$\displaystyle x_{3}(t)$
$\displaystyle=1+t+\frac{1}{3}t^{2}-\frac{1}{54}\alpha t^{3}.$

It is possible to compute the solution for $t>3$ as well, but this is left to
the interested reader. Since no more wave breaking happens for $t>3$,
$V(t,\xi)=V(3,\xi)$ for all $\xi\in\mathbb{R}$. Furthermore, $y(3,\xi)$ and
$U(3,\xi)$ can be computed using (A.8b) and (A.8a). It then remains to solve
(2.3).

###  A.3. Cosine initial data

Consider the initial data in Example 5.3. We map the data to Lagrangian
coordinates by applying $L$ from Definition 2.3, resulting in

$\displaystyle y_{0}(\xi)$ $\displaystyle=\begin{cases}\xi,&\xi\leq 0,\\\
\bar{y}(\xi),&0<\xi\leq 4+2\pi^{2},\\\
\xi-2\pi^{2},&4+2\pi^{2}<\xi,\end{cases}$ $\displaystyle U_{0}(\xi)$
$\displaystyle=\begin{cases}1,&\xi\leq 0,\\\ \cos(\pi\bar{y}(\xi)),&0<\xi\leq
4+2\pi^{2},\\\ 1,&4+2\pi^{2}<\xi,\end{cases}$ $\displaystyle V_{0}(\xi)$
$\displaystyle=H_{0}(\xi)=\begin{cases}0,&\xi\leq 0,\\\
\frac{\pi}{4}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))&0<\xi\leq
4+2\pi^{2},\\\ 2\pi^{2},&4+2\pi^{2}<\xi,\end{cases}$

where $\bar{y}(\xi)$ is implicitly defined by

(A.9)
$\frac{1}{2}(2+\pi^{2})\bar{y}(\xi)-\frac{\pi}{4}\sin(2\pi\bar{y}(\xi))=\xi.$

Thus, $\bar{y}$ is continuous and strictly increasing on $[0,4+2\pi^{2}]$ and

$\bar{y}\big{(}k+\tfrac{k}{2}\pi^{2}\big{)}=k,\hskip
5.69046ptk\in\\{0,1,2,3,4\\}.$

We compute the wave breaking function using (2.4), yielding

(A.10)
$\displaystyle\tau(\xi)=-2\frac{y_{0,\xi}}{U_{0,\xi}}(\xi)=\begin{cases}\frac{2}{\pi\sin\left(\pi\bar{y}(\xi)\right)},&\xi\in\big{(}0,1+\tfrac{1}{2}\pi^{2}\big{)}\cup\big{(}2+\pi^{2},3+\tfrac{3}{2}\pi^{2}\big{)},\\\
\infty,&\text{otherwise}.\end{cases}$

Consequently, there is no wave breaking before $t=\frac{2}{\pi}$, and we can
therefore compute the solution for $t<\frac{2}{\pi}$ by solving (2.3) with
$V(t,\xi)=V_{0}(\xi)$. Hence,

$\displaystyle y(t,\xi)$
$\displaystyle=\begin{cases}\xi+t-\frac{\pi^{2}}{4}t^{2},&\xi<0,\\\
\bar{y}(\xi)+\cos(\pi\bar{y}(\xi))t\\\
\quad+\frac{\pi}{16}\left(2\pi\bar{y}(\xi)-4\pi-\sin(2\pi\bar{y}(\xi))\right)t^{2},&0\leq\xi<4+2\pi^{2},\\\
\xi-2\pi^{2}+t+\frac{\pi^{2}}{4}t^{2},&4+2\pi^{2}\leq\xi,\end{cases}$
$\displaystyle U(t,\xi)$
$\displaystyle=\begin{cases}1-\frac{\pi^{2}}{2}t,&\xi<0,\\\
\cos(\pi\bar{y}(\xi))\\\
\quad+\frac{\pi}{8}\left(2\pi\bar{y}(\xi)-4\pi-\sin(2\pi\bar{y}(\xi))\right)t,&0\leq\xi<4+2\pi^{2},\\\
1+\frac{\pi^{2}}{2}t,&4+2\pi^{2}\leq\xi,\end{cases}$ (A.11) $\displaystyle
V(t,\xi)$ $\displaystyle=H(t,\xi)=V_{0}(\xi).$

At $t=\frac{2}{\pi}$ wave breaking occurs at the two points
$\xi_{1}=\frac{1}{4}(2+\pi^{2})$ and $\xi_{2}=\frac{5}{4}(2+\pi^{2})$.
Thereafter, for every $t\in(\frac{2}{\pi},\infty)$ there are precisely four
values of $\xi$ such that $\tau(\xi)=t$.

The wave breaking curve starting at $\frac{1}{4}(2+\pi^{2})$, splits into two
parts, one moving to the left ending up at $\xi=0$ as $t\rightarrow\infty$,
and one curve moving to the right tending towards
$\xi=(1+\tfrac{1}{2}\pi^{2})$ as $t\rightarrow\infty$. Setting $t=\tau(\xi)$,
denoting these two curves in Lagrangian coordinates by $\xi_{1,\textrm{l}}(t)$
and $\xi_{1,\textrm{r}}(t)$, respectively, and rewriting (A.10), leads to

$\displaystyle\bar{y}\left(\xi_{1,\textrm{l}}(t)\right)$
$\displaystyle=\frac{1}{\pi}\arcsin\left(\frac{2}{\pi t}\right),$ (A.12)
$\displaystyle\bar{y}\left(\xi_{1,\textrm{r}}(t)\right)$
$\displaystyle=1-\frac{1}{\pi}\arcsin\left(\frac{2}{\pi t}\right).$

Analogously, by symmetry, the wave breaking curve emanating from
$\xi=\frac{5}{4}(2+\pi^{2})$ splits into two parts. Denote the corresponding
Lagrangian curves by $\xi_{2,\textrm{l}}(t)$ and $\xi_{2,\textrm{r}}(t)$,
respectively, then

$\displaystyle\bar{y}\left(\xi_{2,\textrm{l}}(t)\right)$
$\displaystyle=2+\frac{1}{\pi}\arcsin\left(\frac{2}{\pi t}\right),$
$\displaystyle\bar{y}\left(\xi_{2,\textrm{r}}(t)\right)$
$\displaystyle=3-\frac{1}{\pi}\arcsin\left(\frac{2}{\pi t}\right).$

Combining (A.12) and (A.9), yields an explicit expression for
$\xi_{1,\textrm{l}}(t)$, in particular,

$\xi_{1,\textrm{l}}(t)=\frac{1}{2\pi}\left(2+\pi^{2}\right)\arcsin\left(\frac{2}{\pi
t}\right)-\frac{\pi}{4}\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right),\hskip 5.69046ptt\geq\frac{2}{\pi}.$

Explicit expressions for the other Lagrangian wave breaking curves can be
obtained completely analogously.

Now we compute $V(t,\xi)$ for $t\geq\frac{2}{\pi}$. The energy is unchanged
for characteristics with labels $\xi<\xi_{1,\textrm{l}}(t)$. Thus,
$V(t,\xi)=V_{0}(\xi)$ for $\xi\in(-\infty,\xi_{1,\textrm{l}}(t))$.

For $\xi\in[\xi_{1,\textrm{l}}(t),\xi_{1,\textrm{r}}(t)]$, using (A.12), we
obtain

$\displaystyle V(t,\xi)$
$\displaystyle=V_{0}(\xi)-\frac{\pi^{2}}{2}\alpha\int_{\xi_{1,\mathrm{l}}(t)}^{\xi}\left(1-\cos(2\pi\bar{y}(\xi))\right)\bar{y}_{\xi}(\xi)d\xi$
$\displaystyle=V_{0}(\xi)-\frac{\pi^{2}}{2}\alpha\int_{\bar{y}\left(\xi_{1,\textrm{l}}(t)\right)}^{\bar{y}(\xi)}\left(1-\cos(2\pi
z)\right)dz$
$\displaystyle=\frac{\pi}{4}\left(1-\alpha\right)\left(2\pi\bar{y}(\xi)-\sin\left(2\pi\bar{y}(\xi)\right)\right)$
$\displaystyle\qquad-\frac{\pi}{4}\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi t}\right)\right)\\!.$

Inserting $\xi=\xi_{1,\mathrm{r}}(t)$ and using (A.12), we find that the total
accumulated energy loss at $\xi_{1,\textrm{r}}(t)$ equals

$V_{1,\textrm{d}}(t)=\frac{\pi}{2}\alpha\left(\pi+\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi t}\right)\right).$

For $\xi\in(\xi_{1,\textrm{r}}(t),\xi_{2,\textrm{l}}(t))$, no wave breaking
occurs, but one has to take the total accumulated energy loss up to the point
$\xi_{1,\textrm{r}}(t)$ into consideration. Therefore, for
$\xi\in(\xi_{1,\textrm{r}}(t),\xi_{2,\textrm{l}}(t))$, we have

$V(t,\xi)=V_{0}(\xi)-V_{1,\textrm{d}}(t).$

The remaining cases for $\xi\geq\xi_{2,\textrm{l}}(t)$ follows by symmetry. To
summarize, for $t\geq\frac{2}{\pi}$ we have

(A.13) $\displaystyle\begin{aligned} V(t,\xi)&=\begin{cases}0,&\xi\leq 0,\\\
~\frac{\pi}{4}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi))),&0<\xi<\xi_{1,\textrm{l}}(t),\\\
\frac{\pi}{4}(1-\alpha)(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))\\\
\quad-\frac{\pi}{4}\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi
t}\right)\right),&\xi_{1,\textrm{l}}(t)\leq\xi\leq\xi_{1,\textrm{r}}(t),\\\
\frac{\pi}{4}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))-\frac{\pi^{2}}{2}\alpha\\\
\quad-\frac{\pi}{2}\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi
t}\right)\right),&\xi_{1,\textrm{r}}(t)<\xi<\xi_{2,\textrm{l}}(t),\\\
\frac{\pi}{4}(1-\alpha)(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))+\frac{\pi^{2}}{2}\alpha\\\
\quad-\frac{3\pi}{4}\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi
t}\right)\right),&\xi_{2,\textrm{l}}(t)\leq\xi\leq\xi_{2,\textrm{r}}(t),\\\
\frac{\pi}{4}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))-\pi^{2}\alpha\\\
\quad-\pi\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi
t}\right)\right),&\xi_{2,\mathrm{r}}(t)<\xi\leq 4+2\pi^{2},\\\
\pi^{2}(2-\alpha)\\\ \quad-\pi\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi
t}\right)\right),&4+2\pi^{2}<\xi.\end{cases}\end{aligned}$

The total Eulerian energy can be read off from (A.11) and (A.13), in
particular,

$\displaystyle F_{\infty}(t)$ $\displaystyle=\begin{cases}2\pi^{2},&0\leq
t<\frac{2}{\pi},\\\
\pi^{2}(2-\alpha)-\pi\alpha\left(\sin\left(2\arcsin\left(\frac{2}{\pi
t}\right)\right)-2\arcsin\left(\frac{2}{\pi
t}\right)\right),&\frac{2}{\pi}\leq t.\end{cases}$

It remains to compute $U(t,\xi)$ and $y(t,\xi)$ for $t~\geq\frac{2}{\pi}$. We
only carry out the details for $U(t,\xi)$ in the interval
$(0,\xi_{2,\textrm{l}}(t))$. To compute $U(t,\xi)$ the following definite
integrals are needed,

(A.14) $\int_{\frac{2}{\pi}}^{t}\arcsin\left(\frac{2}{\pi
s}\right)ds=-1+t\arcsin\left(\frac{2}{\pi
t}\right)+\frac{2}{\pi}\operatorname{arcosh}\left(\frac{\pi}{2}t\right),$

and

$\displaystyle\int_{\frac{2}{\pi}}^{t}\sin\left(2\arcsin\left(\frac{2}{\pi
s}\right)\right)ds$
$\displaystyle=-\frac{4}{\pi}\ln\left(\frac{2}{\pi}\right)-\frac{4}{\pi}\sqrt{1-\left(\frac{2}{\pi
t}\right)^{2}}$ (A.15)
$\displaystyle\qquad+\frac{4}{\pi}\ln\left(\sqrt{t^{2}-\left(\frac{2}{\pi}\right)^{2}}+t\right).$

Thus, by combining (A.13) with (A.14) and (A.15), we define

$\displaystyle b(t)$ $\displaystyle:=\int_{\frac{2}{\pi}}^{t}V_{\infty}(s)ds$
$\displaystyle=\pi^{2}\left(2-\alpha\right)\left(t-\frac{2}{\pi}\right)+2\pi\alpha\left(t\arcsin\left(\frac{2}{\pi
t}\right)-1+\frac{2}{\pi}\operatorname{arcosh}\left(\frac{\pi}{2}t\right)\right)$
(A.16) $\displaystyle\qquad+4\alpha\left(\sqrt{1-\left(\frac{2}{\pi
t}\right)^{2}}+\ln\left(\frac{2}{\pi}\right)-\ln\left(\sqrt{t^{2}-\left(\frac{2}{\pi}\right)^{2}}+t\right)\right)\\!.$

For
$\xi\in(0,\xi_{1,\textrm{l}}(t))\cup(\xi_{1,\textrm{r}}(t),\xi_{2,\textrm{l}}(t))$,
we will use

(A.17)
$U(t,\xi)=U\left(\frac{2}{\pi},\xi\right)+\frac{1}{2}\int_{\frac{2}{\pi}}^{t}\left(V(s,\xi)-\frac{1}{2}V_{\infty}(s)\right)ds.$

Consequently, for $\xi\in(0,\xi_{1,\textrm{l}}(t))$ we find, equipped with
(A.16) and (A.17),

(A.18)
$U(t,\xi)=\cos(\pi\bar{y}(\xi))-\pi-\frac{1}{4}b(t)+\frac{\pi}{8}\left(2\pi\bar{y}(\xi)-\sin\big{(}2\pi\bar{y}(\xi)\big{)}\right)t.$

For $\xi\in(\xi_{1,\textrm{l}}(t),\xi_{1,\textrm{l}}(\frac{2}{\pi}))$ wave
breaking has already occurred. Hence, instead of integrating (2.3b) from
$\frac{2}{\pi}$ to $t$, we integrate from $\tau(\xi)$ to $t$. Moreover, as $U$
is continuous in time, we compute $U(\tau(\xi),\xi)$ from (A.18). Thus

$\displaystyle U(t,\xi)$
$\displaystyle=\cos(\pi\bar{y}(\xi))-\pi-\frac{1}{8}\left(b(t)+b\left(\tau(\xi)\right)\right)$
$\displaystyle\quad+\frac{\pi}{8}\left(2\pi\bar{y}(\xi)-\sin\left(2\pi\bar{y}(\xi)\right)\right)\left(t-\alpha(t-\tau(\xi))\right)-\frac{\pi^{2}}{8}(2-\alpha)(t-\tau(\xi)).$

Next, for $\xi\in(\xi_{1,\textrm{r}}(t),\xi_{2,\textrm{l}}(t))$ using (A.17)
we obtain

(A.19)
$U(t,\xi)=\cos(\pi\bar{y}(\xi))-\frac{\pi^{2}}{2}t+\frac{\pi}{8}\left(2\pi\bar{y}(\xi)-\sin\left(2\pi\bar{y}(\xi)\right)\right)t.$

Finally, as characteristics labeled by
$\xi\in(\xi_{1,\textrm{l}}(\frac{2}{\pi}),\xi_{1,\textrm{r}}(t))$ have
experienced wave breaking, using (A.19) and integrating (2.3b) from
$\tau(\xi)$ to $t$, we get

$\displaystyle U(t,\xi)$
$\displaystyle=\cos(\pi\bar{y}(\xi))+\frac{1}{8}\big{(}b(t)-b(\tau(\xi))\big{)}-\frac{\pi^{2}}{8}\big{(}(2-\alpha)t+(2+\alpha)\tau(\xi)\big{)}$
$\displaystyle\quad+\frac{\pi}{8}\big{(}2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi))\big{)}\big{(}t-\alpha\big{(}t-\tau(\xi)\big{)}\big{)}.$

Following the same lines of reasoning for the remaining cases, for
$t\geq\frac{2}{\pi}$, yields

$\displaystyle U(t,\xi)$
$\displaystyle=\begin{cases}1-\pi-\frac{1}{4}b(t),&\xi\leq 0,\\\
\cos(\pi\bar{y}(\xi))-\pi-\frac{1}{4}b(t)\\\
\quad+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))t,&0<\xi\leq\xi_{1,\textrm{l}}(t),\\\
\cos(\pi\bar{y}(\xi))-\pi-\frac{1}{8}(b(t)+b(\tau(\xi)))\\\
\quad+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))(t-\alpha(t-\tau(\xi)))\\\
\quad-\frac{\pi^{2}}{8}(2-\alpha)(t-\tau(\xi)),&\xi_{1,\textrm{l}}(t)<\xi\leq\xi_{1,\textrm{l}}(\tfrac{2}{\pi}),\\\
\cos(\pi\bar{y}(\xi))-\frac{1}{8}(b(t)-b(\tau(\xi)))\\\
\quad+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))(t-\alpha(t-\tau(\xi)))\\\
\quad-\frac{\pi^{2}}{8}((2-\alpha)t+(2+\alpha)\tau(\xi)),&\xi_{1,\textrm{l}}(\tfrac{2}{\pi})<\xi\leq\xi_{1,\textrm{r}}(t),\\\
\cos(\pi\bar{y}(\xi))-\frac{\pi^{2}}{2}t+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))t,&\xi_{1,\textrm{r}}(t)<\xi\leq\xi_{2,\textrm{l}}(t),\\\
\cos(\pi\bar{y}(\xi))+\frac{1}{8}(b(t)-b(\tau(\xi)))\\\
\quad+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))(t-\alpha(t-\tau(\xi)))\\\
\quad-\frac{\pi^{2}}{8}((6-5\alpha)t+(-2+5\alpha)\tau(\xi)),&\xi_{2,\textrm{l}}(t)<\xi\leq\xi_{2,\textrm{l}}(\tfrac{2}{\pi}),\\\
\cos(\pi\bar{y}(\xi))+\pi+\frac{1}{8}(b(t)+b(\tau(\xi)))\\\
\quad+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))(t-\alpha(t-\tau(\xi)))\\\
\quad-\frac{\pi^{2}}{8}((6-5\alpha)t+(2+5\alpha)\tau(\xi)),&\xi_{2,\textrm{l}}(\tfrac{2}{\pi})<\xi\leq\xi_{2,\textrm{r}}(t),\\\
\cos(\pi\bar{y}(\xi))+\pi+\frac{1}{4}b(t)\\\
\quad+\frac{\pi}{8}(2\pi\bar{y}(\xi)-\sin(2\pi\bar{y}(\xi)))t-\pi^{2}t,&\xi_{2,\textrm{r}}(t)<\xi\leq
4+2\pi^{2},\\\ 1+\pi+\frac{1}{4}b(t),&4+2\pi^{2}<\xi,\end{cases}$

where $\xi_{1,\textrm{l}}(\frac{2}{\pi})=\frac{1}{4}(2+\pi^{2})$ and
$\xi_{2,\textrm{l}}(\frac{2}{\pi})=\frac{5}{4}(2+\pi^{2})$. Define $B(t)$ by
the following

$\displaystyle B(t)$ $\displaystyle:=\int_{\frac{2}{\pi}}^{t}b(s,\alpha)ds$
$\displaystyle=\frac{\pi^{2}}{8}(2-\alpha)\left(t-\frac{2}{\pi}\right)^{2}+\frac{1}{2}\alpha(1-\pi
t)+\alpha
t\ln\left(\frac{2}{\pi}\right)+2\alpha\sqrt{t^{2}-\left(\frac{2}{\pi}\right)^{2}}$
$\displaystyle\quad+\frac{\alpha}{4}\left(\pi t^{2}\arcsin\left(\frac{2}{\pi
t}\right)-2\sqrt{t^{2}-\left(\frac{2}{\pi}\right)^{2}}+4t\operatorname{arcosh}\left(\frac{\pi}{2}t\right)\right)$
$\displaystyle\quad-\alpha\bigg{(}\frac{2}{\pi}\arctan\left(\frac{\pi}{2}\sqrt{t^{2}-\left(\frac{2}{\pi}\right)^{2}}\right)+t\ln\left(\sqrt{t^{2}-\left(\frac{2}{\pi}\right)^{2}}+t\right).$

Then, by integrating (2.3a), we obtain after some tedious calculations the
following expression for $y(t,\xi)$, where we for convenience set
$\bar{y}=\bar{y}(\xi)$ and $\tau=\tau(\xi)$,

$\displaystyle y(t,\xi)$
$\displaystyle=\begin{cases}\xi+1+(1-\pi)t-\frac{1}{4}B(t),&\xi\leq 0,\\\
\bar{y}+1+\cos(\pi\bar{y})t-\pi t-\frac{1}{4}B(t)\\\ \hskip
1.42271pt+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))t^{2},&0<\xi<\xi_{1,\textrm{l}}(t),\\\
\bar{y}+1+\cos(\pi\bar{y})t-\pi t\\\ \hskip
1.42271pt-\frac{1}{8}\left(B(t)+B(\tau)+b(\tau)(t-\tau)\right)\\\ \hskip
1.42271pt+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))(t^{2}-\alpha(t-\tau)^{2})\\\
\hskip
1.42271pt-\frac{\pi^{2}}{16}(2-\alpha)(t-\tau)^{2},&\xi_{1,\textrm{l}}(t)\leq\xi\leq\xi_{1,\textrm{l}}(\tfrac{2}{\pi}),\\\
\bar{y}+\cos(\pi\bar{y})t-\frac{1}{8}(B(t)-B(\tau)-b(\tau)(t-\tau))\\\ \hskip
1.42271pt+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))(t^{2}-\alpha(t-\tau)^{2})\\\
\hskip
1.42271pt-\frac{\pi^{2}}{16}((2-\alpha)t^{2}+2(2+\alpha)t\tau+(-2-\alpha)\tau^{2}),&\xi_{1,\textrm{l}}(\tfrac{2}{\pi})<\xi\leq\xi_{1,\textrm{r}}(t),\\\
\bar{y}+\cos(\pi\bar{y})t-\frac{\pi^{2}}{4}t^{2}+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))t^{2},&\xi_{1,\textrm{r}}(t)<\xi<\xi_{2,\textrm{l}}(t),\\\
\bar{y}+\cos(\pi\bar{y})t+\frac{1}{8}(B(t)-B(\tau)-b(\tau)(t-\tau))\\\ \hskip
1.42271pt+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))(t^{2}-\alpha(t-\tau)^{2})\\\
\hskip
1.42271pt-\frac{\pi^{2}}{16}((6-5\alpha)t^{2}+(-4+10\alpha)t\tau+(2-5\alpha)\tau^{2}),&\xi_{2,\textrm{l}}(t)\leq\xi\leq\xi_{2,\textrm{l}}(\tfrac{2}{\pi}),\\\
\bar{y}-1+\cos(\pi\bar{y})t+\pi t\\\ \hskip
1.42271pt+\frac{1}{8}(B(t)+B(\tau)+b(\tau)(t-\tau))\\\ \hskip
1.42271pt+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))(t^{2}-\alpha(t-\tau)^{2})\\\
\hskip
1.42271pt-\frac{\pi^{2}}{16}((6-5\alpha)t^{2}+(4+10\alpha)t\tau+(-2-5\alpha)\tau^{2}),&\xi_{2,\textrm{l}}(\tfrac{2}{\pi})<\xi\leq\xi_{2,\textrm{r}}(t),\\\
\bar{y}-1+\cos(\pi\bar{y})t+\pi t+\frac{1}{4}B(t)-\frac{1}{2}\pi^{2}t^{2}\\\
\hskip
1.42271pt+\frac{\pi}{16}(2\pi\bar{y}-\sin(2\pi\bar{y}))t^{2},&\xi_{2,\textrm{r}}(t)<\xi\leq
4+2\pi^{2},\\\
\xi-2\pi^{2}-1+(1+\pi)t+\frac{1}{4}B(t),&4+2\pi^{2}<\xi.\end{cases}$