Large time asymptotics of solutions for the subcritical fractional modified Korteweg–de Vries equation

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation $u_t+(-\partial^2_x)^{\alpha/2}u+uu_x=0$ with $\alpha\in(0,1]$, supplemented with an initial datum approaching the constant states $u_\pm$ ($u_-<u_+$) as $x\to\pm\infty$, respectively. It was shown by Karch, Miao, and Xu [SIAM J. Math. Anal., 39 (2008), pp. 1536–1549] that, for $\alpha\in(1,2)$, the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for $\alpha\leq1$. If $\alpha=1$, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case $\alpha\in(0,1)$, we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.

Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations.- Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations #x2013 Some Constructive Approaches.- Self-Similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations.- Asymptotics in Fluid Mechanics.

We study the large time asymptotics of solutions to the Cauchy problem for the modified Korteweg-de Vries equation ∂tu–13∂x3u=∂x(u3), t>0,x∈R,u(0,x)=u0(x), x∈R. We develop the factorization technique to obtain the large time asymptotics of solutions in the neighborhood of the self-similar solution in the case of nonzero total mass initial data. Our result is an improvement of the previous work [18].

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov–Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.

Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

Large Time Asymptotics of Solutions to the Generalized Korteweg–de Vries Equation

We study large time asymptotics of solutions to the Korteweg–de Vries–Burgers equation $$ u_{t} + uu_{x} - u_{{xx}} + u_{{xxx}} = 0,x \in {\text{R}},t > 0. $$ We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u 0 ∈ H s (R) ∩ L 1 (R) , where \( s > - \frac{1} {2}, \) then there exists a unique solution u (t, x) ∈ C ∞ ((0,∞) ;H ∞ (R)) to the Cauchy problem for the Korteweg–de Vries–Burgers equation, which has asymptotics $$ u{\left( t \right)} = t^{{ - \frac{1} {2}}} f_{M} {\left( {{\left( \cdot \right)}t^{{ - \frac{1} {2}}} } \right)} + o{\left( {t^{{ - \frac{1} {2}}} } \right)} $$ as t → ∞, where f M is the self–similar solution for the Burgers equation. Moreover if xu 0 (x) ∈ L 1 (R) , then the asymptotics are true $$ u{\left( t \right)} = t^{{ - \frac{1} {2}}} f_{M} {\left( {{\left( \cdot \right)}t^{{ - \frac{1} {2}}} } \right)} + O{\left( {t^{{ - \frac{1} {2} - \gamma }} } \right)}, $$ where \( \gamma \in {\left( {0,\frac{1} {2}} \right)}. \)

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: u t + ( | u | ρ − 1 u ) x + H u x x = 0 u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal {H} u_{xx} = 0 , where H \mathcal {H} is the Hilbert transform, x , t ∈ R x, t \in {\mathbf {R}} , when the initial data are small enough. If the power ρ \rho of the nonlinearity is greater than 3 3 , then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case ρ = 3 \rho =3 critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same L ∞ L^{\infty } time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.

In this survey, we present modern approaches to the construction and justification of large time asymptotics for solutions of main soliton equations with step-like initial condition whose boundary conditions as O→±∞are finite-gap, quasi-periodic solutions. The principal term of the asymptotic is also a finite-gap, quasi-periodic solution whose phase vectors are modulated with respect to the slow space-like variable. The Whitham equations describing this modulation are studied in detail. For the KdV equations, we construct and justify the principal term of the asymptotic for arbitrary finite-gap boundary conditions. By examining the sine-Gordon equation, we study the case of boundary conditions with complex-valued, self-conjugated quasi-periods. We prove the existence and uniqueness theorems in the case of the complexvalued group velocities that appear in the Whitham equations in the case considered. We present a complete picture (uniform in x) of the Whitham deformation for the case of 1-gap boundary conditions in the sine-Gordon equation.

We study large time asymptotics of solutions to the BBM–Burgers equation $$\partial_{t} (u - u_{xx}) + \beta u_{x} - \mu u_{xx} + uu_{x} = 0$$ . We are interested in the large time asymptotics for the case, when the initial data have an arbitrary size. Let the initial data \(u_{0} \in {\bf H}^{1} ({\bf R}) \cap {\bf W}^{1}_{1} ({\bf R})\), and \(\theta = \int _{\bf R} u_{0} (x) dx \neq 0\). Then we prove that there exists a unique solution \(u (t, x) \in {\bf C} ([0,\infty); {\bf H}^{1} ({\bf R}) \cap {\bf W}^{1}_{1} ({\bf R}))\) to the Cauchy problem for the BBM–Burgers equation. We also find the large time asymptotics for the solutions

We introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton–Jacobi equations, which modifies and simplifies the previous ones (Barles et al. in Arch Ration Mech Anal 204(2):515–558, 2012; Barles and Souganidis in SIAM J Math Anal 31(4):925–939, 2000), under a refined “strict convexity” assumption on the Hamiltonians. Not only such “strict convexity” conditions generalize the corresponding requirements on the Hamiltonians in Barles and Souganidis (SIAM J Math Anal 31(4):925–939, 2000), but also one of the most refined our conditions covers the situation studied in Namah and Roquejoffre (Commun Partial Differ Equ 24(5–6):883–893, 1999).

Large time asymptotics for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the reduced Ostrovsky equation $$u_{tx} = u + \left(u^{3}\right)_{xx}$$ with real valued initial data \({u \left(0\right) = u_{0}}\). We introduce the factorization for the free evolution group to prove the global existence of solutions. Also, we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case.

We consider the Cauchy problem for the nonlinear Schrodinger equations of fractional order $$\left\{\begin{array}{l}i\partial _{t}u-2\left( -\partial _{x}^{2} \right)^{\frac{1}{4}} \, u=F\left( u\right) \\ u\left( 0,x\right) =u_{0} \left( x\right),\end{array}\right.$$ where \({F\left( u\right) }\) is the cubic nonlinearity $$F\left( u\right) =\lambda \left| u\right| ^{2}u$$ with \({\lambda \in \mathbf{R}}\). We find the large time asymptotics of solutions to the Cauchy problem. We use the factorization technique similar to that developed for the Schrodinger equation.

We consider a model semilinear reaction-diffusion system with cubic nonlinear reaction terms and small spatially decaying initial data on R1. The model system is motivated by the thermal-diffusive system in combustion, and it reduces to a scalar reaction-diffusion equation with Zeldovich nonlinearity when the Lewis number is one and proper initial data are prescribed. For scalar equations of similar type it is well known that while a nonlinearity of degree greater than three (supercritical case) has no effect for large times a cubic nonlinearity qualitatively changes the long time behaviour. The latter case has been treated in the literature by a rescaling method under the additional assumption of smallness of the nonlinearity. Although for our system the cubic nonlinearity is also critical we establish large time behaviour when the nonlinearity is not necessarily small which essentially differs from the supercritical case. This is possible because of the interaction between nonlinear terms which have different signs. We show that although the system admits no self-similar solutions their do exist self-similar super (sub) solutions with new scaling exponents. This allows us to use a renormalization group analysis combined with a maximum principle.