On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation
نویسندگان
چکیده
We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation i Ψt + 2 2 Ψxx + |Ψ |2Ψ = 0, 1, with analytic initial data of the form Ψ (x,0; ) = A(x)e i S(x) is approximately described by a particular solution to the Painlevé-I equation.
منابع مشابه
On universality of critical behaviour in the focusing nonlinear
We argue that the critical behaviour near the point of " gradient catastrophe " of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation ii Ψ t + 2 2 Ψ xx + |Ψ| 2 Ψ = 0, 1, with analytic initial data of the form Ψ(x, 0;) = A(x) e i S(x) is approximately described by a particular solution to the Painlevé-I equation .
متن کاملOn universality of critical behaviour in the focusing
We argue that the critical behaviour near the point of " gradient catastrophe " of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation ii ψ t + 2 2 ψ xx + |ψ| 2 ψ = 0 with analytic initial data of the form ψ(x, 0;) = A(x) e i S(x) is approximately described by a particular solution to the Painlevé-I equation.
متن کاملOn Critical Behaviour in Systems of Hamiltonian Partial Differential Equations
We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by parti...
متن کاملScattering for the focusing Ḣ-critical Hartree equation with radial data
We investigate the focusing Ḣ-critical nonlinear Schrödinger equation (NLS) of Hartree type i∂tu + ∆u = −(| · |−3 ∗ |u|2)u with Ḣ radial data in dimension d = 5. It is proved that if the maximal life-span solution obeys supt ∥|∇| 12 u ∥∥ 2 < √ 6 3 ∥|∇| 12Q ∥∥ 2 , where Q is the positive radial solution to the elliptic equation with nonlocal operator (1.4) which corresponds to a new variational ...
متن کاملAnalytical Soliton Solutions Modeling of Nonlinear Schrödinger Equation with the Dual Power Law Nonlinearity
Introduction In this study, we use a newly proposed method based on the software structure of the maple, called the Khaters method, and will be introducing exponential, hyperbolic, and trigonometric solutions for one of the Schrödinger equations, called the nonlinear Schrödinger equation with the dual power law nonlinearity. Given the widespread use of the Schrödinger equation in physics and e...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Nonlinear Science
دوره 19 شماره
صفحات -
تاریخ انتشار 2009