Asymptotic behavior of solutions of defocusing integrable discrete nonlinear Schrödinger equation

We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schr?dinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of order t^-1/2     

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation

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where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).

     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ?1,K?1,s>1, we construct smooth initial data u ₀ with \(\|u_{0}\|_{{H}^{s}}<\delta\), so that the corresponding time evolution u satisfies \(\|u(T)\|_{{H}^{s}}>K\) at some time T. This growth occurs despite the Hamiltonian’s bound on \(\|u(t)\|_{\dot{H}^{1}}\) and despite the conservation of the quantity \(\|u(t)\|_{L^{2}}\).The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems.     

Classification of solutions of the generalized mixed nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We construct a generalized Darboux transformation for a generalized mixed nonlinear Schrödinger equation and consider a complete reduction classification...     

Existence of solutions to the nonlinear Schrödinger equation on locally finite graphs

Archiv der Mathematik - Let $$G=(V, E)$$ be a locally finite connected graph and $$\Delta $$ be the usual graph Laplacian operator. According to Lin and Yang (Rev. Mat. Complut., 2022), using...     

Integration of a defocusing nonlinear Schrödinger equation with additional terms

Theoretical and Mathematical Physics - The inverse spectral problem method is used to integrate the nonlinear Schrödinger equation with some additional terms in the class of infinite-gap...     

Toward a classification of quasirational solutions of the nonlinear Schrödinger equation

Based on a representation in terms of determinants of the order 2N, we attempt to classify quasirational solutions of the one-dimensional focusing nonlinear Schrödinger equation and also formulate several conjectures about the structure of the solutions. These solutions can be written as a product of a t-dependent exponential times a quotient of two N(N+1)th degree polynomials in x and t depending on 2N?2 parameters. It is remarkable that if all parameters are equal to zero in this representation, then we recover the P_N breathers.     

Multiplicity of positive solutions of a nonlinear Schrödinger equation

We consider the multiple existence of positive solutions of the following nonlinear Schrödinger equation: where if N3 and p(1, ) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well := int a^–1(0) consisting of k components and the first eigenvalues of –+b(x) on _j under Dirichlet boundary condition are positive for all . Under these conditions we show that (PM) has at least 2^k–1 positive solutions for large . More precisely we show that for any given non-empty subset , (P) has a positive solutions u(x) for large . In addition for any sequence _n we can extract a subsequence _ni along which u_ni converges strongly in H¹(R^N). Moreover the limit function u(x)=lim_iu_ni satisfies (i) For jJ the restriction u|_j of u(x) to _j is a least energy solution of –v+b(x)v=v^p in _j and v=0 on _j. (ii) u(x)=0 for .Mathematics Subject Classifications (2000):35Q55, 35J20     

Depencence of solutions of the nonlinear Schrödinger equation on dissipation parameters

We consider an initial-boundary-value problem for the nonlinear Schrödinger equation in the complexvalued functionE=E(x,z): (1) $\partial _z E + i\Delta E + i\alpha \left| E \right|^p E + \beta \left| E \right|^q E = 0, q > p \geqslant 0, \beta > 0,$ (2) $\left. E \right|_{z = 0} = E_0 \in H^2 (\Omega ) \cap H_0^1 (\Omega ), \left. E \right|_{\partial \Omega } = 0, \Omega \subset R^2 , \partial \Omega \in C^2 .$ We investigate the behavior of the solution of problem (1)–(2) as β→0 and its closeness to the solution of the degenerate equation (β=0). Given the consistency conditionq(β)=p+εln(1/β), 00, we establish boundedness of the norm $\left\| E \right\|_{C([0,z_0 ]):H_0^1 (\Omega ))} + \left\| {\partial _z E} \right\|_{C([0,z_0 ]);L^2 (\Omega ))} $ for every finitez ₀>0 as β→0. For α≤0 and a fixedq, we prove uniform (in β) boundness of solutions of problem (1)–(2) on some interval [0,Z] and their convergence as β→0 to the solution of the degenerate problem (β=0) in the normC([0,Z];L ² (Ω)).     

Maximal amplitudes of hyperelliptic solutions of the derivative nonlinear Schrödinger equation

A simple formula is proven for an upper bound for amplitudes of hyperelliptic (finite-gap or N-phase) solutions of the derivative nonlinear Schrödinger equation. The upper bound is sharp, viz, it is attained for some initial conditions. The method used to prove the upper bound is the same method, with necessary modifications, used to prove the corresponding bound for solutions of the focusing NLS equation (Wright OC, III. Sharp upper bound for amplitudes of hyperelliptic solutions of the focusing nonlinear Schrödinger equation. Nonlinearity. 2019;32:1929-1966).     

Pure soliton solutions of the nonlocal Kundu–nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We systematically present an inverse scattering transform for a nonlocal reverse-space higher-order nonlinear Schrödinger equation with nonzero boundary...     

Analytical solutions of nonlinear Schrödinger equation with distributed coefficients

We combine the F-expansion method with the homogeneous balance principle to build a strategy to find analytical solitonic and periodic wave solutions to a generalized nonlinear Schrödinger equation with distributed coefficients, linear gain/loss, and nonlinear gain/absorption. In the case of a dimensionless effective Gross–Pitaevskii equation which describes the evolution of the wave function of a quasi-one-dimensional cigar-shaped Bose–Einstein condensate, the building strategy is applied to generate analytical solutions.     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.