Data-driven parametric soliton-rogon state transitions for nonlinear wave equations using deep learning with Fourier neural operator

In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.

Deep learning neural networks for the third-order nonlinear Schrodinger equation: bright solitons, breathers, and rogue waves

The dimensionless third-order nonlinear Schrodinger equation(alias the Hirota equation) is investigated via deep leaning neural networks. In this paper, we use the physics-informed neural networks(PINNs) deep learning method to explore the data-driven solutions(e.g. bright soliton,breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and perturbated(a 2% noise) training data are considered. Moreover, we use the PINNs deep learning to study the data-driven discovery of parameters appearing in the Hirota equation with the aid of bright solitons.

Vector semi-rational rogon-solitons and asymptotic analysis for any multi-component Hirota equations with mixed backgrounds

The Hirota equation can be used to describe the wave propagation of an ultrashort optical field.In this paper,the multi-component Hirota(alias n-Hirota,i.e.n-component third-order nonlinear Schrodinger)equations with mixed non-zero and zero boundary conditions are explored.We employ the multiple roots of the characteristic polynomial related to the Lax pair and modified Darboux transform to find vector semi-rational rogon-soliton solutions(i.e.nonlinear combinations of rogon and soliton solutions).The semi-rational rogon-soliton features can be modulated by the polynomial degree.For the larger solution parameters,the first m(m<n)components with non-zero backgrounds can be decomposed into rational rogons and grey-like solitons,and the last n-m components with zero backgrounds can approach bright-like solitons.Moreover,we analyze the accelerations and curvatures of the quasi-characteristic curves,as well as the variations of accelerations with the distances to judge the interaction intensities between rogons and grey-like solitons.We also find the semi-rational rogon-soliton solutions with ultrahigh amplitudes.In particular,we can also deduce vector semi-rational solitons of the ncomponent complex mKdV equation.These results will be useful to further study the related nonlinear wave phenomena of multi-component physical models with mixed background,and even design the related physical experiments.

A NEW HIERARCHY OF LAX AND LIOUVILLE INTEGRABLE EVOLUTION EQUATIONS ASSOCIATED WITH AN ISOSPECTRAL PROBLEM IN THE LOOP ALGEBRA■

在这篇论文，有五个潜力的一个 isospectral 问题在环代数学 A_2 被调查以便有五任意的功能的进化方程的一个新层次被获得。然后由修理五任意的功能是某些功能并且使用踪迹身份，进化方程的层次的概括 Hamiltonian 结构被给。方程的这个层次是 Liouville integrable，这被显示出。最后， isospectral 问题的一些特殊情况也被给。