Normalized solutions of nonlinear Schrödinger equations

We consider the problem $$\left\{\begin{array}{ll}-\Delta u - g(u) = \lambda u,\\ u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{array}\right.$$ in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L ²-unit sphere, and we show the existence of infinitely many solutions.     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

Classical solutions of nonlinear Schrödinger equations

We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iu_t–u+q(|u|²)u=0 in iu_t - u + (|u|²)u = in (t, x)xⁿ for 6n11.     

Small solutions to nonlinear Schrödinger equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev spaceH ^s=H^s(R ⁿ) of fractional orders>n/2 under the following assumptions. (1) Concerning the Cauchy data ?∈H ^s: ‖?;L ²‖ is relatively small with respect to ‖?;H ^σ‖ for any fixed σ withn/2<σ≤s. (2) Concerning the nonlinearityf: f(u) behaves as a conformal poweru ¹⁺⁴/n near zero and has an arbitrary growth rate at infinity.     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      8. Global strong solutions for nonlinear Schrödinger equations      《Nonlinear Analysis: Theory, Methods & Applications》1987,11(10):1143-1154       9. Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics      Feng-Hua Qi  Bo Tian  Xing Lü  Rui Guo  Yu-Shan Xue 《Communications in Nonlinear Science & Numerical Simulation》2012,17(6):2372-2381 In this paper, by virtue of the Darboux transformation (DT) and symbolic computation, the quintic generalization of the coupled cubic nonlinear Schrödinger equations from twin-core nonlinear optical fibers and waveguides are studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained and the corresponding DT is constructed. Moreover, one-, two- and three-soliton solutions are presented in the forms of modulus. Features of solitons are graphically discussed: (1) head-on and overtaking elastic collisions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) energy-exchanging collisions of the three solitons.      10. The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation      《Communications in Nonlinear Science & Numerical Simulation》2014,19(6):1706-1722 In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schrödinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.      11. Multi-rogue waves and rational solutions of the coupled nonlinear Schrödinger equations      Bao-Guo Zhai  Wei-Guo Zhang  Xiao-Li Wang  Hai-Qiang Zhang 《Nonlinear Analysis: Real World Applications》2013,14(1):14-27       12. Blow-up solutions of inhomogeneous nonlinear Schrödinger equations      Peter Y. H. Pang  Hongyan Tang  Youde Wang 《Calculus of Variations and Partial Differential Equations》2006,26(2):137-169 In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schr?dinger equation on . We present existence and non-existence results and investigate qualitative properties of the solutions when they exist. Mathematics Subject Classification (2000) 35Q55, 35G25 Dedicated respectfully to Professor Weiyue Ding on the occasion of his sixtieth birthday.      13. The elliptic-in-t solutions of the nonlinear Schrödinger equation      A. O. Smirnov 《Theoretical and Mathematical Physics》1996,107(2):568-578 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.      14. The derivative nonlinear Schrödinger equation on the half line      M.B. Erdoğan  T.B. Gürel  N. Tzirakis 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(7):1947-1973 We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\mathbb{R}$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation.      15. Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations      Silvia Cingolani  Louis Jeanjean  Kazunaga Tanaka 《Journal of Fixed Point Theory and Applications》2017,19(1):37-66 We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations $$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$ (0.1) where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by $$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$ (0.2) Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that $$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$ and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).       16. Multibump solutions for discrete periodic nonlinear Schrödinger equations      Shiwang Ma  Zhi-Qiang Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(5):1413-1442 In this paper, we study the existence of multibump solutions for discrete nonlinear Schrödinger equations with periodic potentials. We first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group. Then, we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear Schrödinger equations, and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case. As an application, we get, without any symmetry assumptions, infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity.      17. Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations      Yang  Jianfu  Yang  Jinge 《中国科学 数学(英文版)》2022,65(7):1383-1412 Science China Mathematics - In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation $$left{ {matrix{ { -...      18. Special derivative nonlinear Schrödinger (DNLS) systems exhibiting 2-soliton solutions      《Chaos, solitons, and fractals》2000,11(13):2055-2066 Various versions of the derivative nonlinear Schrödinger (DNLS) equation occur frequently in applied science. Modified DNLS systems with fifth order nonlinearity are studied here via the Hirota bilinear transformation. 2-Soliton solutions are constructed as a preliminary signal for the special nature of the partial differential equations. Cases treated include a (2+1) (2 spatial and 1 temporal)-dimensional system, an equation incorporating third order dispersion, and a coupled (multi-component) system. Relevance and potential applications to hydrodynamics and fiber optics are discussed.      19. Superposition principle and composite solutions to coupled nonlinear Schrödinger equations      Laila Al Sakkaf  Usama Al Khawaja 《Mathematical Methods in the Applied Sciences》2020,43(17):10168-10189 We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.      20. Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations      Dario Bambusi 《Mathematische Zeitschrift》1999,230(2):345-387 We prove a Nekhoroshev type result [26,27] for the nonlinear Schr?dinger equation with vanishing or periodic boundary conditions on ; here is a parameter and is a function analytic in a neighborhood of the origin and such that , . More precisely, we consider the Cauchy problem for (0.1) with initial data which extend to analytic entire functions of finite order, and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum. The proof is obtained by a method which applies to Hamiltonian perturbations of linear PDE's with the following properties: (i) the linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. Eq. (0.1) satisfies (i) and (ii) when restricted to a level surface of , which is an integral of motion. The main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here. Received June 23, 1997; in final form June 1, 1998

Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics

In this paper, by virtue of the Darboux transformation (DT) and symbolic computation, the quintic generalization of the coupled cubic nonlinear Schrödinger equations from twin-core nonlinear optical fibers and waveguides are studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained and the corresponding DT is constructed. Moreover, one-, two- and three-soliton solutions are presented in the forms of modulus. Features of solitons are graphically discussed: (1) head-on and overtaking elastic collisions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) energy-exchanging collisions of the three solitons.     

The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schrödinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.     

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schr?dinger equation on . We present existence and non-existence results and investigate qualitative properties of the solutions when they exist. Mathematics Subject Classification (2000) 35Q55, 35G25 Dedicated respectfully to Professor Weiyue Ding on the occasion of his sixtieth birthday.     

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.     

The derivative nonlinear Schrödinger equation on the half line

We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\mathbb{R}$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation.     

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$

(0.1)

where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by

$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$

(0.2)

Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).

     

Multibump solutions for discrete periodic nonlinear Schrödinger equations

In this paper, we study the existence of multibump solutions for discrete nonlinear Schrödinger equations with periodic potentials. We first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group. Then, we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear Schrödinger equations, and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case. As an application, we get, without any symmetry assumptions, infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity.     

Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

Science China Mathematics - In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation $$left{ {matrix{ { -...     

Special derivative nonlinear Schrödinger (DNLS) systems exhibiting 2-soliton solutions

Various versions of the derivative nonlinear Schrödinger (DNLS) equation occur frequently in applied science. Modified DNLS systems with fifth order nonlinearity are studied here via the Hirota bilinear transformation. 2-Soliton solutions are constructed as a preliminary signal for the special nature of the partial differential equations. Cases treated include a (2+1) (2 spatial and 1 temporal)-dimensional system, an equation incorporating third order dispersion, and a coupled (multi-component) system. Relevance and potential applications to hydrodynamics and fiber optics are discussed.     

Superposition principle and composite solutions to coupled nonlinear Schrödinger equations

We show that the superposition principle applies to coupled nonlinear Schrödinger equations with cubic nonlinearity where exact solutions may be obtained as a linear combination of other exact solutions. This is possible due to the cancelation of cross terms in the nonlinear coupling. First, we show that a composite solution, which is a linear combination of the two components of a seed solution, is another solution to the same coupled nonlinear Schrödinger equation. Then, we show that a linear combination of two composite solutions is also a solution to the same equation. With emphasis on the case of Manakov system of two-coupled nonlinear Schrödinger equations, the superposition is shown to be equivalent to a rotation operator in a two-dimensional function space with components of the seed solution being its coordinates. Repeated application of the rotation operator, starting with a specific seed solution, generates a series of composite solutions, which may be represented by a generalized solution that defines a family of composite solutions. Applying the rotation operator to almost all known exact seed solutions of the Manakov system, we obtain for each seed solution the corresponding family of composite solutions. Composite solutions turn out, in general, to possess interesting features that do not exist in the seed solution. Using symmetry reductions, we show that the method applies also to systems of N-coupled nonlinear Schrödinger equations. Specific examples for the three-coupled nonlinear Schrödinger equation are given.     

Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations

We prove a Nekhoroshev type result [26,27] for the nonlinear Schr?dinger equation with vanishing or periodic boundary conditions on ; here is a parameter and is a function analytic in a neighborhood of the origin and such that , . More precisely, we consider the Cauchy problem for (0.1) with initial data which extend to analytic entire functions of finite order, and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum. The proof is obtained by a method which applies to Hamiltonian perturbations of linear PDE's with the following properties: (i) the linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. Eq. (0.1) satisfies (i) and (ii) when restricted to a level surface of , which is an integral of motion. The main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here. Received June 23, 1997; in final form June 1, 1998