Pavel I. Naumkin's research while affiliated with Universidad Nacional Autónoma de México and other places

We study the Cauchy problem for the higher-order KdV–BBM type equation where \(\varvec{\Lambda }\) \(=\mathcal {F}^{-1}\Lambda \mathcal {F}\) and \(\Theta \) \(=\mathcal {F}^{-1}\Theta \mathcal {F}\) are the pseudodifferential operators, defined by their symbols \(\Lambda \left( \xi \right) \) and \( \Theta \left( \xi \right) \), respectively. The aim of the present paper is to develop a general approach through the Factorization Techniques of evolution operators which can be applied for finding the large time asymptotics of small solutions to a wide class of nonlinear dispersive KdV- type equations including the KdV or the improved version of the KdV with higher order dispersion terms.

The Cauchy problem of the form $$ \begin{cases} i \partial_{t}(u-\partial_{x}^{2}u)+\partial_{x}^{2}u -a \partial_{x}^{4}u=u^{3}, & t>0, x\in\mathbb{R}, u(0,x) =u_{0}(x),& x\in\mathbb{R}, \end{cases} $$ is considered for a Sobolev-type nonlinear equation with cubic nonlinearity, where $a>1/5$, $a\neq1$. It is shown that the asymptotic behaviour of the solution is characterized by an additional logarithmic decay in comparison with the corresponding linear case. To find the asymptotics of solutions of the Cauchy problem for a nonlinear Sobolev-type equation, factorization technique is developed. To obtain estimates for derivatives of the defect operators, $\mathbf{L}^{2}$-estimates of pseudodifferential operators are used. Bibliography: 20 titles.

We study the large-time asymptotics of solutions to the fractional modified Korteweg–de Vries equation $$\begin{aligned} \left\{ \begin{array}{c} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=\partial _{x}\left( u^{3}\right) ,~ t>0,\ x\in {\mathbb {R}}\textbf{,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,\ x\in {\mathbb {R}}\textbf{,} \end{array} \right. \end{aligned}$$ (0.1) where \(\alpha \in \left( 1,2\right) ,\) \(\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}\) is the fractional derivative. The case of \(\alpha =3\) corresponds to the classical modified KdV equation. In the case of \(\alpha =2\), it is the modified Benjamin–Ono equation. Our aim is to extend the results in [10, 16] for \(\alpha \in \left( 0,1\right) \) to \(\alpha \in \left( 1,2\right) \). We develop the method based on the factorization techniques, which was started in [11], and apply the known results on the \({\textbf{L}}^{2}\) - boundedness of pseudodifferential operators to get the large-time asymptotics of solutions.

Рассмотрена задача Коши для нелинейного уравнения типа Соболева с кубической нелинейностью $$ \begin{cases} i \partial_{t}(u-\partial_{x}^{2}u)+\partial_{x}^{2}u -a \partial_{x}^{4}u=u^{3}, & t>0, x\in\mathbb{R}, u(0,x) =u_{0}(x),& x\in\mathbb{R}, \end{cases} $$ где $a>1/5$, $a\neq1$. Доказано, что асимптотика решения обладает дополнительным логарифмическим убыванием по сравнению с соответствующим линейным случаем. Для нахождения асимптотики решений задачи Коши для нелинейного уравнения типа Соболева развивается техника факторизации. Также для получения оценок производных операторов дефекта применяются $\mathbf{L}^{2}$-оценки псевдодифференциальных операторов. Библиография: 20 названий.

We study the Cauchy problem for the fractional nonlinear Schrödinger equation $$\left\{{\matrix{{i{\partial _t}u + {1 \over \alpha}{{\left| {{\partial _x}} \right|}^\alpha}u = \lambda |u{|^2}u,\,\,t>0,} \hfill\;\;\;\;\;\;\;\;\;\;\;\; {x \in \mathbb{R},} \hfill \cr {u(0,x) = {u_0}(x),} \hfill\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{x \in \mathbb{R},} \hfill \cr}} \right.$$ where λ ∈ ℝ, the fractional derivative \(|{\partial _x}{|^\alpha} = {{\cal F}^{-1}}|\xi {|^\alpha}{\cal F}\), the order \(\alpha \in ({3 \over 2},2)\). Our aim is to prove the modified scattering for solutions of the fractional nonlinear Schrödinger equation. We develop the factorization techniques which we proposed in our previous works.

We study the large time asymptotics of solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation $$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}w+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}w=\partial _{x}\left( w^{3}\right) ,\text { }t>0,\, x\in {\mathbb {R}}\textbf{,}\\ w\left( 0,x\right) =w_{0}\left( x\right) ,\,x\in {\mathbb {R}} \textbf{,} \end{array} \right. \end{aligned}$$ ∂ t w + 1 α ∂ x α - 1 ∂ x w = ∂ x w 3 , t > 0 , x ∈ R , w 0 , x = w 0 x , x ∈ R , where $$\alpha \in \left[ 4,5\right) ,$$ α ∈ 4 , 5 , and $$\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}$$ ∂ x α = F - 1 ξ α F is the fractional derivative. The case of $$\alpha =3$$ α = 3 corresponds to the classical modified KdV equation. In the case of $$\alpha =2$$ α = 2 it is the modified Benjamin–Ono equation. Our aim is to find the large time asymptotic formulas for the solutions of the Cauchy problem for the fractional modified KdV equation. We develop the method based on the factorization techniques which was started in our previous papers. Also we apply the known results on the $${\textbf{L}}^{2}$$ L 2 —boundedness of pseudodifferential operators.

We continue to study the large time asymptotics of solutions for the fractional modified Korteweg–de Vries equation $$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=\partial _{x}\left( u^{3}\right) ,&{}\quad t>0,\ x\in {\mathbb {R}}, \\ u\left( 0,x\right) =u_{0}\left( x\right) ,&{}\quad x\in {\mathbb {R}}, \end{array} \right. \end{aligned}$$where \(\alpha \in \left( 2,3\right) ,\) \(\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}\) is the fractional derivative. This is a sequel to the previous works in which the cases \(\alpha \in \left( 0,1\right) \cup \left( 1,2\right) \) were studied. It is known that the case of \(\alpha =3\) corresponds to the classical modified KdV equation. In the case of \(\alpha =2\) it is called the modified Benjamin–Ono equation. In the case \(\alpha =1,\) it is the nonlinear wave equation and the exceptional case. Our aim is to find the large time asymptotic formulas of solutions. Main difference between the previous works and our result is in the order of fractional derivative \(\alpha .\) The order \(\alpha =2\) is a critical point which divides the smoothing property and the derivative loss of solutions.