The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

Communications on pure and applied mathematics

We use the physics-informed neural network to solve a variety of femtosecond optical soliton solutions of the high-order nonlinear Schrodinger equation, including one-soliton solution, two-soliton solution, rogue wave solution, W-soliton solution and M-soliton solution. The prediction error for one-soliton, W-soliton and M-soliton is smaller. As the prediction distance increases, the prediction error will gradually increase. The unknown physical parameters of the high-order nonlinear Schrodinger equation are studied by using rogue wave solutions as data sets. The neural network is optimized from three aspects including the number of layers of the neural network, the number of neurons, and the sampling points. Compared with previous research, our error is greatly reduced. This is not a replacement for the traditional numerical method, but hopefully to open up new ideas.

Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN

pdf/data-driven-rogue-waves-and-parameter-discovery-in-the-4es2voqyak.pdf

Data-driven rogue waves and parameter discovery in the defocusing nonlinear Schrödinger equation with a potential using the PINN deep learning

With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise a two-stage PINN method which is tailored to the nature of equations by introducing features of physical systems into neural networks. Its remarkable advantage lies in that it can impose physical constraints from a global perspective. In stage one, the original PINN is applied. In stage two, we additionally introduce the measurement of conserved quantities into mean squared error loss to train neural networks. This two-stage PINN method is utilized to simulate abundant localized wave solutions of integrable equations. We mainly study the Sawada-Kotera equation as well as the coupled equations: the classical Boussinesq-Burgers equations and acquire the data-driven soliton molecule, M-shape double-peak soliton, plateau soliton, interaction solution, etc. Numerical results illustrate that abundant dynamic behaviors of these solutions can be well reproduced and the two-stage PINN method can remarkably improve prediction accuracy and enhance the ability of generalization compared to the original PINN method.

A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions

The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the bases of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters are also exhibited in the simulation of the two-order rogue wave solution.

Solving localized wave solutions of the derivative nonlinear Schrodinger equation using an improved PINN method

https://iopscience.iop.org/article/10.1088/1674-1056/abd7e3/pdf

Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints*

The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, and also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the basis of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second-order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters is also exhibited in the simulation of the two-order rogue wave solution.

/pdf/solving-localized-wave-solutions-of-the-derivative-nonlinear-8cflq8urav.pdf

Solving localized wave solutions of the derivative nonlinear Schrödinger equation using an improved PINN method

Protracted amphetamine abstinence syndrome is one of the primary causes of relapse for amphetamine-type drug abusers during withdrawal. However, the importance of the management of protracted amphetamine abstinence syndrome is underestimated. Electro-acupuncture may be a safe and effective alternative therapy for protracted amphetamine abstinence syndrome, but the evidence is limited.The study is a prospective, two-center, randomized, waitlist controlled, open-label pragmatic trial. A total of 300 patients with protracted amphetamine abstinence syndrome will be recruited. All participants will be randomly assigned to an electro-acupuncture group or a waitlist group in a 1:1 ratio. Participants in the electro-acupuncture group will receive the electrical-acupuncture treatment. Waitlist group participants will not receive electro-acupuncture treatment but will be assessed at each visit. Treatments will be administered twice a week for a total of 4 consecutive weeks. The primary outcome in this study is the change in the ACSA between baseline (week 0) and the completion of treatment (week 4), and the secondary outcomes are changes in the Hamilton Depression Scale (HAMD), the visual analog scale (VAS), the Hamilton Anxiety Scale (HAMA), the Pittsburgh Sleep Quality Index (PSQI), the Montreal Cognitive Assessment (MoCA), and the Medical Outcomes Study 36-item Short-Form Health Survey (SF-36).This study will assess the effectiveness of acupuncture in PAAS in real-world settings to provide support for clinical decisions and a basis for subsequent trials comparing acupuncture with other positive regimens.ClinicalTrials.gov ChiCTR2000040619 . Registered on 3 December 2020. 

/pdf/electro-acupuncture-for-protracted-amphetamine-abstinence-2qekfu79.pdf

Electro-acupuncture for protracted amphetamine abstinence syndrome: study protocol for a pragmatic randomized controlled trial

The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to the difficulty of solving this equation, in particular in high dimensions, lots of methods are proposed to effectively obtain different kinds of solutions, such as neural networks, among others. Recently, a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation's dynamical behaviors from spatiotemporal data directly. Compared with traditional neural networks, this method can obtain remarkably accurate solution with extraordinarily less data. Meanwhile, this method also provides a better physical explanation and generalization. In this paper, based on the above method, we present an improved deep learning method to recover the soliton solutions, breather solution and rogue wave solutions to the nonlinear Schrodinger equation. In particular, the dynamical behaviors and error analysis about the one-order and two-order rogue waves of the Schrodinger equation are revealed by the deep neural network for the first time. Moreover, the effects of different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions. Numerical experiments show that the dynamical behaviors of soliton solutions, breather solution and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.

Soliton, Breather and Rogue Wave Solutions for Solving the Nonlinear Schr\"odinger Equation Using a Deep Learning Method with Physical Constraints

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