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.     

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.     

Investigating the exact integrability of the multiwave nonlinear Schrödinger equation

It is shown that the multiwave nonlinear Schrödinger equation describing the evolution of several quasimonochromatic waves having the same group velocities is not exactly integrable (in the sense that no infinite sequence of local conservation laws and symmetries exists). The exact integrability for systems of the form w _t ⁱ =α_iw _xx ⁱ +a _klm ⁱ w^kw^lw^m is investigated, where α_i are different from zero.     

Autoresonance excitation of a soliton of the nonlinear Schrödinger equation

The action of an external parametric perturbation with slowly changing frequency on a soliton of the nonlinear Schrödinger equation is studied. Equations for the time evolution of the parameters of the perturbed soliton are derived. Conditions for the soliton phase locking are found, which relate the rate of change of the perturbation frequency, its amplitude, the wave number, and the phase to the initial data of the soliton. The cases when the initial amplitude of the soliton is small and when the amplitude of the soliton is of the order of unity are considered.     

Allowable graphs of the nonlinear Schrödinger equation and their applications

We construct the definition of allowable graphs of the nonlinear Schrödinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving 3-dimensional blocks of the normal form of the nonlinear Schrödinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schrödinger equation.     

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.     

Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity

We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.     

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

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).     

Maximum decay rate for the nonlinear Schrödinger equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation in with and We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that u(0) lies in the weighted Sobolev space in the energy space, namely or in according to the different cases.     

Next hierarchy of mixed derivative nonlinear Schrödinger equation

In this paper, we derive the next hierarchy of the mixed derivative nonlinear Schrödinger (MDNLS) equation. Considering the Wadati–Konno–Ichikawa eigen value problem, the Lax Pair for the above equation is explicitly constructed. Obtained results are in agreement with the results derived through other methods in the recent past. We also briefly discuss the construction of Bäcklund transformation.     

On a class of nonlinear inhomogeneous Schrödinger equation

In this paper, we study the inhomogeneous Schrödinger equation $$i\varphi_{t}=-\triangle\varphi -|x|^{b}|\varphi|^{p-1}\varphi,\quad x\in \mathbb{R}^{N}.$$ By using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.     

Hirota bilinearization of coupled higher-order nonlinear Schrödinger equation

In this paper, we present the Hirota bilinearization of the coupled Sasa–Satsuma equation. The procedure employed here generates a more general solution than the one reported earlier. We also discuss the soliton solutions of the equation and show that the solutions found earlier are only special cases of the solution discussed here.     

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