Ljudmila A. Bordag, Alina Z. Chmakova
Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving a nonlinear Black-Scholes equation.
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We study the derivative nonlinear Schrödinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.
We study the orbital stablity and instability of solitary wave solutions for nonlinear Schrödinger equations of derivative type.
Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
本文提出了一种通过机器学习在蒙特卡洛模拟中学习随机变量的方法，用于参数偏微分方程和金融衍生品定价，以克服维度灾难。
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schrödinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result up to the scaling for small initial data with some spherical symmetry structure.
We consider the nonlinear derivative Schrodinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties consist of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. This phenomena arrests energy transfers between low and high modes, which are quantified by a growth in the Sobolev norm.
An explicit lifespan estimate is presented for the derivative Schrödinger equations with periodic boundary condition.
A polynomial-in-time growth bound is established for global Sobolev $H^s(\mathbb T)$ solutions to the derivative nonlinear Schrödinger equation on the circle with $s>1$. These bounds are derived as a consequence of a nonlinear smoothing effect for an appropriate gauge-transformed version of the periodic Cauchy problem, according to which a solution with its linear part removed possesses higher spatial regularity than the initial datum associated with that solution.
We consider the initial value problem for cubic derivative nonlinear Schrödinger equation in one space dimension. Under a suitable weakly dissipative condition on the nonlinearity, we show that the small data solution has a logarithmic time decay in $L^2$.
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schrödinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result close to scaling for small initial data.
We prove that the derivative nonlinear Schrödinger equation is globally well-posed in $H^{\frac 12} (\mathbb{R})$ when the mass of initial data is strictly less than $4π$.
For semilinear wave equations on Lorentzian manifolds with quadratic derivative non-linear terms, we study the inverse problem of determining the background Lorentzian metric. Under some conditions on the nonlinear term, we show that from the source-to-solution map, one can determine the Lorentzian metric up to diffeomorphisms.
We study the orbital instability of solitary waves for a generalized derivative nonlinear Schrödinger equation. We give sufficient conditions for instability of a two-parameter family of solitary waves in a degenerate case.
Global well-posedness for the derivative nonlinear Schrödinger equation on the circle
研究圆周上导数非线性薛定谔方程，证明其在Sobolev空间$H^1$中全局适定。
In [19] and [26], the authors proved the stability of multi-solitons for derivative nonlinear Schr{ö}dinger equations. Roughly speaking, sum of finite stable solitons is stable. We predict that if there is one unstable solition then multi-soliton is unstable. This prediction is proved in [7] for classical nonlinear Schr{ö}dinger equations. In this paper, we proved this prediction for derivative nonlinear Schr{ö}dinger equations by using the method of C{ô}te-Le Coz [7] with the help of Gauge transformation.
We consider the Schrödinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves.
We present a numerical study of a derivative nonlinear Schrödinger equation with a general power nonlinearity, $|ψ|^{2σ}ψ_x$. In the $L^2$-supercritical regime, $σ>1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation with supercritical power law nonlinearity.
Consider the initial value problem for systems of cubic derivative nonlinear Schrödinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under which the small data solution gains an additional logarithmic decay as $t \to +\infty$ compared with the corresponding free evolution.
In this paper, we consider a weakly coupled system of a wave and damped Klein-Gordon equation with nonlinearities of derivative type. We prove a blow-up result for the Cauchy problem associated with this system for nonnegative and compactly supported data by means of an iteration argument.
One of the most interesting problems discerned when applying the Black--Scholes model to financial derivatives, is reconciling the deviation between expected and observed values. In our recent work, we derived a new model based on the Black--Scholes model and formulated a new mathematical approach to an inverse problem in financial markets. In this paper, we apply microlocal analysis to prove a uniqueness of the solution to our inverse problem. While microlocal analysis is used for various models in physics and engineering, this is the first attempt to apply it to a model in financial markets.…
In this note, we study the ill-posedness problem for the derivative nonlinear Schrödinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in $L^2(\mathbb{R})$. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data.
Global Existence for Coupled 3-D Nonlinear Wave and Klein-Gordon Equations with Large Derivatives of Initial Data
该研究针对带二次非线性项的三维波动与Klein-Gordon耦合方程的柯西问题，在大导数初始数据、零条件等假设下证明了解的全局存在性，推进了相关偏微分方程理论研究。
The nonlinear Schr{ö}dinger equation with derivative cubic nonlinearity admits a family of solitons, which are orbitally stable in the energy space. In this work, we prove the orbital stability of multi-solitons configurations in the energy space, under suitable assumptions on the speeds and frequencies of the composing solitons. The main ingredients of the proof are modulation theory, energy coercivity and monotonicity properties.
The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.
Consider the initial value problem for cubic derivative nonlinear Schrödinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors where the gauge-invariant nonlinearity was treated.
Global well-posedness for the derivative nonlinear Schrödinger equation in $L^2(\mathbb{R})$
本文证明了一维导数非线性薛定谔方程在直线上尺度临界空间 $L^2(\mathbb{R})$ 中的整体适定性。
As a continuation of the previous work \cite{Wu}, we consider the global well-posedness for the derivative nonlinear Schrödinger equation. We prove that it is globally well-posed in energy space, provided that the initial data $u_0\in H^1(\mathbb{R})$ with $\|u_0\|_{L^2}< 2\sqrtπ$.
A class of self-similar solutions to the derivative nonlinear Schrödinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions.
We study a derivative nonlinear Schrödinger equation, allowing non-integer powers in the nonlinearity, $|u|^{2σ} u_x$. Making careful use of the energy method, we are able to establish short-time existence of solutions with initial data in the energy space, $H^1$. For more regular initial data, we establish not just existence of solutions, but also well-posedness of the initial value problem. These results hold for real-valued $σ\geq 1,$ while prior existence results in the literature require integer-valued $σ$ or $σ$ sufficiently large ($σ\geq 5/2$), or use higher-regularity function spaces.
In this paper we construct a Gibbs measure for the derivative Schrödinger equation on the circle. The construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation.
Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities
本文针对两类含二次时间导数非线性项的一维非线性波动模型，构造分类五参数广义自相似爆破解族，证明相关分支渐近稳定性，统一刻画其爆破机制。
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a W^{1,1}_0 solution which is distributional or entropic, according to the growth assumptions on a lower order term in divergence form.
We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm.
This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.
There are few approaches to the solution of a system of nonlinear differential equations in partial derivatives, for example $\cite{NK87} - \cite{EK98}$. In our paper we propose an approach that was used to solve the Navier-Stokes equations in three dimensional space. This solution is described in details in article "Existence, uniqueness and smoothness of solution for 3D Navier-Stokes equations with any smooth initial velocity" $\cite{TT12}$. The authors expect that it can be successfully applied to other systems of nonlinear differential equations in partial derivatives.
It is shown that the cubic derivative nonlinear Schrödinger equation is locally well-posed in Besov spaces $B^{s}_{2,\infty}(\mathbb X)$, $s\ge\tfrac12$, where we treat the non-periodic setting $\mathbb X=\mathbb R$ and the periodic setting $\mathbb X=\mathbb T$ simultaneously. The proof is based on the strategy of Herr for initial data in $H^{s}(\mathbb T)$, $s\ge\tfrac12$.
This paper provides a small data global existence result for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions. This is an extension of the previous results by Li [Discrete Contin. Dyn. Syst., 32 (2012), 4265--4285] and Ikeda--Katayama--Sunagawa [Ann. H. Poincaré 16 (2015), 535--567].
We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects.
This paper is painfully withdrawn.
We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, their velocity. We illustrate these results with numerical simulations.
Blow up for nonlinear wave-type equations with perturbed derivatives
本文研究可转化为导数受零阶项扰动的半线性波动型方程，改进了径向初始数据的爆破结果，并确定了控制导数扰动、初始数据大小与非线性指数相互作用的零阶项条件。
Modified Scattering and Asymptotic Completeness for the Derivative Nonlinear Schrödinger equation
本文首次在拟线性情形下证明导数非线性薛定谔方程的修正散射与渐近完备性，方法上结合波包检验、bootstrap 论证与 Klainerman Sobolev 向量场方法。
In this note, we consider the derivative nonlinear Schrödinger equation on the circle. In particular, by adapting Wu's recent argument to the periodic setting, we prove its global well-posedness in $H^1(\mathbb T)$, provided that the mass is less than $4π$. Moreover, this mass threshold is independent of spatial periods.
In the present paper, we consider the Cauchy problem of nonlinear Schrödinger equations with a derivative nonlinearity which depends only on $\bar{u}$. The well-posedness of the equation at the scaling subcritical regularity was proved by A. Grünrock (2000). We prove the well-posedness of the equation and the scattering for the solution at the scaling critical regularity by using $U^{2}$ space and $V^{2}$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).
In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.
We look for solutions to derivative nonlinear Schrodinger equations built upon solitons. We prove the existence of multi-solitons i.e. solutions behaving at large time as the sum of finite solitons. We also show that one can attach a kink at the begin of the sum of solitons i.e multi kink-solitons. Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates.
We prove global existence of small solutions to the initial value problem for a class of cubic derivative nonlinear Schrödinger systems with the masses satisfying suitable non-resonance relations. The large-time asymptotics of the solutions are also shown. This work is intended to provide a counterpart of the previous paper (arXiv:1507.07617) in which the mass resonance case was treated.
We show that the derivative nonlinear Schrödinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our result exploits the complete integrability of DNLS and removes certain spectral conditions on the initial data, thanks to Xin Zhou's analysis on spectral singularities in the context of inverse scattering.
Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron's method, unique existence for the Cauchy-Dirichlet and Cauchy-Neumann problems are proved.
In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{xx}+iλ\abs{u}^k u_x=0$ with non-zero $λ\in \Real$ and any real number $k\gs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
By an extension of of some estimates due to Crandall and Pierre and Di Benedetto we derive consequences for fully nonlinear parabolic equations of the form $\dt v + F(t,x,D^2v)=0$, where $F$ can be both singular and degenerate elliptic and also non-homogeneous. Such equations appear in the theory of option pricing with market impact.
On a generalized derivative nonlinear Schrödinger equation
本文研究广义导数非线性薛定谔方程，在渐近态小性条件下证明波算子存在，并证明小初值解在规范变换意义下散射。
We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided.
In the present paper, we determine the estimations on Atangana-Baleanu-Caputo fractional derivative at extreme points. With the assistance of the estimations obtained, we derive the comparison results. Peano's type existence results established for nonlinear fractional differential equations involving Atangana-Baleanu-Caputo fractional derivative. The acquired comparison results are then utilized to deal with the existence of local, extremal and global solution.
We derive asymptotic formulas for the solution of the derivative nonlinear Schrödinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.
A new integrable system of two symmetrically coupled derivative nonlinear Schroedinger equations is detected by means of the singularity analysis. A nonlinear transformation is proposed which uncouples the equations of the new system.
A refinement of uniform resolvent estimate is given and several smoothing estimates for Schrodinger equations in the critical case are induced from it. The relation between this resolvent estimate and radiation condition is discussed. As an application of critical smoothing estimates, we show a global existence results for derivative nonlinear Schrodinger equations.
The Derivative Nonlinear Schrödinger equation is an $L^2$-critical nonlinear dispersive equation model for Alfvén waves in space plasmas. Recent numerical studies on an $L^2$-supercritical extension of this equation provide evidence of finite time singularities. Near the singular point, the solution is described by a universal profile that solves a nonlinear elliptic eigenvalue problem depending only on the strength of the nonlinearity. In the present work, we describe the deformation of the profile and its parameters near criticality, combining asymptotic analysis and numerical simulations.
In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.
We consider the semi-classical limit of nonlinear Schrodinger equations in the presence of both a polynomial nonlinearity and thederivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolicstructure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time dependent analyticregularity.
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